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\title{On Uncertain Seismic Wave Propagation}


%\subtitle
%{Include Only If Paper Has a Subtitle}

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%{F.~Author\inst{1} \and S.~Another\inst{2}}
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\author[Jeremi{\'c} and Sett] % (optional, use only with lots of authors)
{Boris~Jeremi{\'c} and Kallol Sett}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.


\institute[Computational Geomechanics Group \hspace*{0.3truecm}
\pgfuseimage{university-logo} \hspace*{-0.3truecm}] % (optional, but mostly needed)
 { Department of Civil and Environmental Engineering\\
  University of California, Davis }
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.




\date
{Stein Sture \\ Symposium on Geomechanics \\
EMI Conference, \\ 
May 2008}
% %\date[] % (optional, should be abbreviation of conference name)
% %{
% %\small Graduate Student: Guanzhou Jie 
% %\\ ~\\
% %Funding: NSF--CMS--0324661, NSF--TeraGrid, NSF--EEC--9701568 \\
% %%\texttt{http://sokocalo.engr.ucdavis.edu/$\tilde{~}$jeremic/}
% }
% % - Either use conference name or its abbreviation.
% % - Not really informative to the audience, more for people (including
% %   yourself) who are reading the slides online

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% is a graphic format that can be processed by latex or pdflatex,
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%\logo{\pgfuseimage{university-logo}}

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% \logo{\pgfuseimage{university-logo}}




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%  --   
%  --   \documentclass{beamer}
%  --   
%  --   % This file is a solution template for:
%  --   
%  --   % - Talk at a conference/colloquium.
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%  --   %\graphicspath{{/home/kallol/publication/2007/Presentation/USNCCM9/}}
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%  --   % Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
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%  --   % In principle, this file can be redistributed and/or modified under
%  --   % the terms of the GNU Public License, version 2.
%  --   %
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%  --   % for your own needs. For this reason, if you use this file as a
%  --   % template and not specifically distribute it as part of a another
%  --   % package/program, I grant the extra permission to freely copy and
%  --   % modify this file as you see fit and even to delete this copyright
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%  --   %\title[Short Paper Title] % (optional, use only with long paper titles)
%  --   %{Title As It Is In the Proceedings}
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%  --   \title{Uncertain Elasto--Plasticity}
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%  --   %\ \\
%  --   %\ \\
%  --   \Large{Boris~Jeremi{\'c}} \\
%  --   \large{Department of Civil and Environmental Engineering}\\
%  --   \large{University of California, Davis}
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%  --   %with guidance from \\
%  --   %\vspace*{1.0truecm} \\
%  --   %\ \\
%  --   %\hspace*{-1.9truecm} guidance from 
%  --   %Professors Boris Jeremi\'{c}}, M. Levent Kavvas, Ross Boulanger, and Niels Gr{\o}nbech Jensen 
%  --   %\hspace*{-1.9truecm}
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%  --   %  Department of Civil and Environmental Engineering \\
%  --   %  University of California, Davis
%  --   %  \and
%  --   %  \inst{2}%
%  --   %  Department of Theoretical Philosophy\\
%  --   %  University of Elsewhere
%  --   }
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%  --   %\begin{footnotesize}
%  --   %This work is supported by NSF (through Award \#) \\
%  --   %\end{footnotesize}
%  --   
%  --   %\vspace*{-1.4truecm}
%  --   \date
%  --   {
%  --   USC Seminar Series, December 2007 
%  --   %\vspace*{-1.1truecm}
%  --   \begin{footnotesize}
%  --   %\begin{tiny}
%  --   %\hspace*{-1.9truecm} 
%  --   %This work is supported by grant from NSF (CMMI-0600766) 
%  --   %\& EEC-9701568)
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\begin{document}

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\begin{frame}
  \titlepage
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\begin{frame}
  \frametitle{Outline}
  \tableofcontents
  % You might wish to add the option [pausesections]
\end{frame}

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% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
% solution: 

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

% - A conference audience is likely to know very little of what you
%   are going to talk about. So *simplify*!
% - In a 20min talk, getting the main ideas across is hard
%   enough. Leave out details, even if it means being less precise than
%   you think necessary.
% - If you omit details that are vital to the proof/implementation,
%   just say so once. Everybody will be happy with that.





\section{Motivation}



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\subsection{Personal and Otherwise}
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\begin{frame} 
  
  
  \frametitle{Personal Motivation}

 

\begin{itemize}


\item Probabilistic fish counting
\item Stein Sture's lectures, MGM project
\item David Muir Wood's book
%\item Emir Macari's dissertation 
\item John Williams' DEM sims., displacement vortexes 
\item Kaspar Willam's spec. topics course, comments at EM98
%\item Carlos Felippa's lectures on Hopf bifurcation
\item Kenneth Runesson's explanations on bifurcation
\item Lev Kavvas' probabilistic hydrology 




\end{itemize}

 

\end{frame}



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%- 
%- 
%- 
%- 
%- \begin{frame}
%- \frametitle{Failure Mechanisms of Geomaterials} 
%- 
%- \begin{flushleft}
%- \vspace*{-0.5cm}
%- \includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
%- \\
%- \small{
%- Soil: Inside Failure of "Uni,form" MGM Specimen \\
%- (After Swanson et al. 1998)
%- }
%- \end{flushleft}
%- \vspace*{-6.0cm}
%- %
%- \end{frame}
%- 
%- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%- 
%- 
%- \begin{frame}
%- \frametitle{Geomaterials are inherently Uncertain}
%- \begin{flushleft}
%- %\begin{center}
%- %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%- \vspace*{-0.1cm}
%- \includegraphics[height=4.0cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/FrictionAngleProfile.jpg}
%- %\vspace*{-5.0cm}
%- %\mbox{\tiny{Lambe, T. W. and Whitman, R. V.,1969. Soil Mechanics. New York, John Wiley \& Sons}}
%- %\vspace*{-6.5cm}
%- \end{flushleft}
%- \vspace*{-4.0cm}
%- %
%- \begin{flushleft}
%- \begin{small}
%- \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Spatial Variation of Friction Angle\\
%- %\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Undrained Shear Strength \\
%- \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (After Mayne et al. (2000))
%- \end{small}
%- \end{flushleft}
%- \vspace*{0.2cm}
%- \begin{flushright}
%- \includegraphics[width=7.1cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/TableTypicalCOV.jpg}
%- \end{flushright}
%- 
%- \begin{flushleft}
%- \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ Typical COVs of Different Soil Properties \\
%- \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (After Lacasse and Nadim 1996)
%- \end{flushleft}
%- 
%- \end{frame}
%- 

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%- 
%- 
%- \begin{frame}
%- \frametitle{Brownian Motion}
%- 
%- \begin{itemize}
%- 
%- %\vspace*{1.5cm}
%- \item Governing equation (Langevin equation):
%- 
%- 
%- %\vspace*{-0.5cm}
%- 
%- \begin{equation}
%- \nonumber
%- m \frac{dv}{dt} = F(x) - \beta v + \eta(t)
%- \end{equation} 
%- 
%- 
%- %\vspace*{-0.5cm}
%- \item Probability density function (PDF) of particle displacement obeys a 
%- simple diffusion equation (Einstein (1905)):
%- 
%- %\vspace*{-0.5cm}
%- 
%- \begin{equation}
%- \nonumber
%- \frac{\partial f (x,t)}{\partial t} = D \frac{\partial^2 f(x,t)}{\partial x^2}
%- \end{equation}
%- 
%- %\vspace*{0.5cm}
%- \item Addition of external forces (gravity, elastic or magnetic attraction) $\rightarrow$ Fokker-Planck-Kolmogorov (FPK)
%- equation governs the PDF (Kolmogorov 1941)
%- 
%- %\vspace*{0.5cm}
%- \item Alternately, Monte Carlo method can be used for solution of Langevin equation  $\rightarrow$ computationally very expensive
%- 
%- \end{itemize} 
%- 
%- 
%- \end{frame}
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%- 
%- \begin{frame}
%- \frametitle{Classical Second-Order Analysis}
%- 
%- 
%- \begin{itemize}
%- %\vspace*{1.5cm}
%- 
%- \item Classical approach to second-order analysis of random processes 
%- 
%- \begin{itemize}
%- 
%- %\vspace*{-0.5cm}
%- \item Relationship between the autocorrelation function and spectral 
%- density function (Wiener 1930) $\rightarrow$ Paved the way to the solution of general 
%- stochastic differential equation (SDE) 
%- 
%- \end{itemize}
%- 
%- %\vspace*{-0.5cm}
%- \item SDEs with random forcings $\rightarrow$ Highly developed mathematical theory for It{\^o} type equation:
%- 
%- \vspace*{-0.2cm}
%- \begin{equation}
%- \nonumber
%- dx = a(x,t) dt + b(x,t) dW
%- \end{equation}
%- 
%- \begin{itemize}
%- 
%- \item Solution is a Markov process
%- 
%- \item PDF of solution process satisfies a FPK PDE
%- 
%- \vspace*{-0.3cm}
%- \begin{equation}
%- \nonumber
%- \frac{\partial p\left(x,t\right)}{\partial t} = -\frac{\partial}{\partial x} \left[a(x,t)p\left(x,t \right)\right]
%- + \frac{1}{2}\frac{\partial^2}{\partial x^2}\left[b^2(x,t)p\left(x,t \right) \right]
%- \end{equation}
%- 
%- \end{itemize}
%- \end{itemize}
%- 
%- \end{frame}
%- 
%- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%- 
%- \begin{frame}
%- \frametitle{SDEs with Random Coefficients}
%- 
%-   \begin{itemize}
%-   
%-   \item Solution methods
%-          
%-  \begin{itemize}
%- 
%-         \item Functional integration approach (Hopf 1952)
%- 
%-         \item Averaged equation approach (Bharrucha-Reid 1968) 
%- 
%-         \item Numerical approaches (perturbation method, stochastic finite element etc.) 
%-  
%-         \item Monte Carlo method.
%- 
%-         \end{itemize}
%- 
%-   
%- %  \vspace*{1.0cm}
%-   \item FPK equation for the characteristic functional of the solution for problem of 
%-         wave propagation in random elastic media (Lee 1974)
%- 
%- %  \vspace*{1.0cm}
%- %  \item Eulerian-Lagrangian form of FPK equation for probabilistic solution of flow through porous media (Kavvas 2003)
%- 
%-   \end{itemize}
%- 
%- 
%- \end{frame}
%- 
%-     
%- 
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\begin{frame} 

\frametitle{On Uncertainties}

 

\begin{itemize}



\item Epistemic uncertainty - due to lack of knowledge

   \begin{itemize}

     \item Can be reduced by \\ collecting more data

     \item Mathematical tools not well \\
           developed, trade-off with \\
           aleatory uncertainty

   \end{itemize}

\vspace*{-2.9cm}
\begin{figure}[!hbpt]
\begin{flushright}
\includegraphics[height=4cm,angle=-90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/uncertain03.pdf}
\end{flushright}
\end{figure}



\vspace*{0.15cm}
\item Aleatory uncertainty - inherent variation of physical system 

    \begin{itemize}

      \item Can not be reduced

      \item Has highly developed mathematical tools 

    \end{itemize}



\item Ergodicity -- exchange ensemble average
for time average? 

    \begin{itemize}

      \item Possibly  yes 

      \item Issues up for discussion (soil, concrete, rock, biomaterials...)

    \end{itemize}





\end{itemize}

 

% 
% \vspace*{-0.5cm}
% \begin{figure}[!hbpt]
% %\nonumber
% \begin{center}
% %\begin{center}
% %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
% \includegraphics[height=4cm,angle=-90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/uncertain03.pdf}
% %
% %\mbox{\tiny{Lambe, T. W. and Whitman, R. V.,1969. Soil Mechanics. New York, John Wiley \& Sons}}
% %
% \end{center}
% %\begin{flushright}
% %Soil Variability in Relatively \\ Homogeneous Soil Deposit \\ (Clay Deposit of the Valley \\ of Mexico at a Typical \\ Spot in Mexico City)
% %\includegraphics[width=14cm]{TypicalSoilCOV.jpg}
% %
% %\nonumber
% %\caption{Drucker-Prager Associative Linear Hardening Response with Random $G$}
% %{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
% %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
% %\end{flushright}
% %\end{center}
% \end{figure}
% 

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\end{frame}






% 
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%  \begin{frame} 
%  
%  \frametitle{Ergodicity}
%  
%   
%  
%  \begin{itemize}
%  
%  
%  
%  \item Exchange ensemble averages for time averages
%  
%  \vspace*{0.2cm}
%  \item Is soil elasto-plasticity ergodic? 
%  
%      \begin{itemize}
%  
%        \item  Can  soil  elastic--plastic  statistical  properties be obtained by
%        temporal averaging?
%  
%  
%        \item Will soil  elastic--plastic  statistical  properties "renew" at each
%        occurrence?
%  
%        \item  Are  soil   elastic--plastic  statistical  properties statistically
%        independent?
%  
%      \end{itemize}
%  
%  
%  \item  Claim  in  literature  that structural nonlinear behavior is non--ergodic
%  while earthquake characteristics are (?!)
%  
%  \item However, earthquake characteristics is representing mechanics (fault slip)
%  on a different scale...
%  
%  \end{itemize}
%  
%   
%  \end{frame}
%  
%  





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\begin{frame} 

\frametitle{Historical Overview}


 

\begin{itemize}

%
\item Brownian motion, Langevin equation $\rightarrow$  PDF governed by simple diffusion Eq. (Einstein 1905)


%
%  
%  \begin{equation}
%  \nonumber
%  \frac{\partial f (x,t)}{\partial t} = D \frac{\partial^2 f(x,t)}{\partial x^2}
%  \end{equation}
%  

\vspace*{0.2cm}
\item With external forces $\rightarrow$ Fokker-Planck-Kolmogorov (FPK) for the PDF (Kolmogorov 1941)

\vspace*{0.2cm}
\item  Approach  for  random  forcing  $\rightarrow$  relationship  between  the
autocorrelation function and spectral density function (Wiener 1930)

\vspace*{0.2cm}
\item Approach for random coefficient $\rightarrow$ 
Functional integration approach (Hopf 1952), Averaged equation approach (Bharrucha-Reid 1968), 
Numerical approaches, Monte Carlo method


%\item Alternately, Monte Carlo method can be used for solution of Langevin equation  $\rightarrow$ computationally very expensive

\end{itemize}

 


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\end{frame}






%tempout - 
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%tempout -  \begin{frame} 
%tempout -  
%tempout -  \frametitle{Brownian Motions}
%tempout -  
%tempout -  
%tempout -   
%tempout -  
%tempout -  \begin{itemize}
%tempout -  
%tempout -  %
%tempout -  \item Governing (Langevin) equation:
%tempout -  
%tempout -  
%tempout -  
%tempout -  \begin{equation}
%tempout -  \nonumber
%tempout -  m \frac{dv}{dt} = F(x) - \beta v + \eta(t)
%tempout -  \end{equation} 
%tempout -  
%tempout -  
%tempout -  
%tempout -  %
%tempout -  \item Probability density function (PDF) of particle displacement obeys a 
%tempout -  simple diffusion equation (Einstein (1905)):
%tempout -  
%tempout -  %
%tempout -  
%tempout -  \begin{equation}
%tempout -  \nonumber
%tempout -  \frac{\partial f (x,t)}{\partial t} = D \frac{\partial^2 f(x,t)}{\partial x^2}
%tempout -  \end{equation}
%tempout -  
%tempout -  
%tempout -  \item Addition of external forces (gravity, elastic or magnetic attraction) $\rightarrow$ Fokker-Planck-Kolmogorov (FPK)
%tempout -  equation governs the PDF (Kolmogorov 1941)
%tempout -  
%tempout -  
%tempout -  \item Alternately, Monte Carlo method can be used for solution of Langevin equation  $\rightarrow$ computationally very expensive
%tempout -  
%tempout -  \end{itemize}
%tempout -  
%tempout -   
%tempout -  
%tempout -  
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%tempout -  
%tempout -  \end{frame}
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%tempout -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%tempout -  \begin{frame} 
%tempout -  
%tempout -  \frametitle{Stochastic Systems: Random Forcing}
%tempout -  
%tempout -  %
%tempout -   
%tempout -  
%tempout -  \begin{itemize}
%tempout -  %
%tempout -  
%tempout -  \item  Classical approach: relationship between the autocorrelation function and
%tempout -  spectral  density  function  (Wiener  1930)  $\rightarrow$  Paved the way to the
%tempout -  solution of general stochastic differential equation (SDE)
%tempout -  
%tempout -  
%tempout -  \item SDEs with random forcing $\rightarrow$ Highly developed mathematical theory for It{\^o} type equation:
%tempout -  %
%tempout -  %
%tempout -  \begin{equation}
%tempout -  \nonumber
%tempout -  dx = a(x,t) dt + b(x,t) dW
%tempout -  \end{equation}
%tempout -  
%tempout -    \begin{itemize}
%tempout -    
%tempout -      \item Solution is a Markov process
%tempout -      
%tempout -      \item PDF of solution process satisfies a FPK PDE
%tempout -      %
%tempout -      %
%tempout -      \begin{equation}
%tempout -      \nonumber
%tempout -      \frac{\partial p\left(x,t\right)}{\partial t} = -\frac{\partial}{\partial x} \left[a(x,t)p\left(x,t \right)\right]
%tempout -      + \frac{1}{2}\frac{\partial^2}{\partial x^2}\left[b^2(x,t)p\left(x,t \right) \right]
%tempout -      \end{equation}
%tempout -      
%tempout -    \end{itemize}
%tempout -  
%tempout -  \end{itemize}
%tempout -  
%tempout -   
%tempout -  
%tempout -  
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%tempout -  
%tempout -  \end{frame}
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%tempout -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%tempout -  \begin{frame} 
%tempout -  
%tempout -  \frametitle{Stochastic Systems: Random Coefficient}
%tempout -  
%tempout -  %
%tempout -   
%tempout -  
%tempout -  
%tempout -  \begin{itemize}
%tempout -  
%tempout -  %\vspace*{0.5cm}
%tempout -  \item Approximate Solution methods
%tempout -                
%tempout -        \begin{itemize}
%tempout -  
%tempout -        \item Functional integration approach (Hopf 1952)
%tempout -  
%tempout -        \item Averaged equation approach (Bharrucha-Reid 1968) 
%tempout -  
%tempout -        \item Numerical approaches  
%tempout -  
%tempout -        \item Monte Carlo method
%tempout -  
%tempout -        \end{itemize}
%tempout -  
%tempout -  
%tempout -  
%tempout -  \vspace*{0.5cm}
%tempout -  \item FPK equation for the characteristic functional of the solution for problem of 
%tempout -        wave propagation in random media (Lee 1974)
%tempout -  
%tempout -  
%tempout -  \vspace*{0.5cm}
%tempout -  \item Eulerian-Lagrangian form of FPK equation for probabilistic solution of flow through porous media (Kavvas 2003)
%tempout -  
%tempout -  \end{itemize}
%tempout -  
%tempout -  
%tempout -   
%tempout -  
%tempout -             
%tempout -  
%tempout -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%tempout -  
%tempout -  \end{frame}
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\begin{frame} 

\frametitle{Material Behavior Inherently Uncertainties}


%\begin{itemize}

%\vspace*{0.5cm}
%\item  
%Material  behavior  is  inherently uncertain  (concrete,  metals,  soil,  rock, 
%bone, foam, powder etc.)

        \begin{itemize}

\vspace*{0.5cm}
        \item Spatial \\ 
              variability

\vspace*{0.5cm}
        \item Point-wise \\ 
              uncertainty, \\
              testing \\
              error, \\
              transformation \\
              error

        \end{itemize}

% \vspace*{0.5cm}
% \item  Failure mechanisms related to spatial variability (strain localization and
%        bifurcation of response)
% 
% \vspace*{0.5cm}
% \item Inverse problems
% 
%       \begin{itemize}
%       
%       \item New material design, ({\it point-wise})
% 
%       \item Solid and/or structure design (or retrofits),  ({\it spatial})
% 
%       \end{itemize}
                                  
%\end{itemize}

\vspace*{-5cm}
\begin{figure}[!hbpt]
%\nonumber
%\begin{center}
\begin{flushright}
%\includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
%\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/FrictionAngleProfile.jpg}
\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/FrictionAngleProfile.jpg}
\\
\mbox{(Mayne et al. (2000) }
\end{flushright}
%\end{center}
%\end{center}
\end{figure}


% - %\vspace*{2cm}
% - %
% - {%
% - \begin{beamercolorbox}{section in head/foot}
% - \usebeamerfont{framesubtitle}\tiny{
% - Stein Sture, Nicholas C. Costes, Susan N. Batiste, Mark R. Langton, Khalid A. Al-Shibli, Boris
% - Jeremi{\'c}, Roy A. Swanson and Melissa Frank. Mechanics of granular materials at low effective
% - stresses. \textit{ASCE Journal of Aerospace Engineering}, vol. 11, No. 3, pages 67-72, 1998.}
% - %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
% - \end{beamercolorbox}%
% - }

 


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%\slide{\Large{Motivation}}

%
%\begin{figure}[!hbpt]
%\begin{flushleft}
%\includegraphics[height=17cm]{TypicalSoilProfile.jpg}
%
%\mbox{\tiny{Lambe, T. W. and Whitman, R. V.,1969. Soil Mechanics. New York, John Wiley \& Sons}}
%
%\end{flushleft}
%\begin{flushright}
%Soil Variability in Relatively \\ Homogeneous Soil Deposit \\ (Clay Deposit of the Valley \\ of Mexico at a Typical \\ Spot in Mexico City)
%\end{flushright}
%\end{figure}


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% -- 
% -- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -- \begin{frame} \frametitle{{Soil: Inside Failure (MGM)}}
% -- 
% -- 
% -- \begin{figure}[!hbpt]
% -- %\nonumber
% -- \begin{center}
% -- \includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
% -- \end{center}
% -- %\end{center}
% -- \end{figure}
% -- 
% -- {%
% -- \begin{beamercolorbox}{section in head/foot}
% -- \usebeamerfont{framesubtitle}\tiny{
% -- Stein Sture, Nicholas C. Costes, Susan N. Batiste, Mark R. Langton, Khalid A. Al-Shibli, Boris
% -- Jeremi{\'c}, Roy A. Swanson and Melissa Frank. Mechanics of granular materials at low effective
% -- stresses. \textit{ASCE Journal of Aerospace Engineering}, vol. 11, No. 3, pages 67-72, 1998.}
% -- %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
% -- \end{beamercolorbox}%
% -- }
% -- 
% -- 
% -- 
% -- 
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% -- 
% -- \end{frame}
% -- 
% -- 


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% --   \begin{frame} 
% --   
% --   \frametitle{{Soil: Spatial and Point Variation}}
% --   
% --   
% --   \begin{figure}[!hbpt]
% --   %\nonumber
% --   \begin{center}
% --   %\includegraphics[height=4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/FrictionAngleProfile.jpg}
% --   \includegraphics[height=5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/FrictionAngleProfile.jpg}
% --   %\includegraphics[height=4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ShearStrengthProfile.jpg}
% --   %\includegraphics[height=4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ShearStrengthProfile.jpg}
% --   \end{center}
% --   %\end{center}
% --   \end{figure}
% --   
% --   (Mayne et al. (2000)
% --   
% --   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% --   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% --   
% --   \end{frame}
% --   
% --   
% --   
% --   


%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame} \frametitle{\Large{Motivation}}
%  
%  
%  \center{Typical Coefficients of Variation of Different Soil Properties} 
%  
%  \center{(After Lacasse and Nadim 1996)}
%  \begin{figure}[!hbpt]
%  %\nonumber
%  %\begin{flushleft}
%  \begin{center}
%  %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%  %\includegraphics[height=12cm]{TypicalSoilProfile.jpg}
%  %\caption{Soil Variability in Relatively Homogeneous Soil Deposit}
%  %
%  %\end{flushleft}
%  %\begin{flushright}
%  \includegraphics[width=25cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/TableTypicalCOV.jpg}
%  %\tiny{Lacasse and Nadim (1996), "Uncertainties in Characterizing Soil Properties", Proceedings of Uncertainty '96, July 31-August 3, 
%  %1996, Madison, Wisconsin } 
%  %
%  %\nonumber
%  %\caption{Drucker-Prager Associative Linear Hardening Response with Random $G$}
%  %{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%  %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%  %\end{flushright}
%  \end{center}
%  \end{figure}
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%  \end{frame}
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%  





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%  \begin{frame} 
%  
%  \frametitle{Soil Uncertainties and Quantification}
%  
%  
%   
%  
%  \begin{itemize}
%  
%  %
%  %\vspace*{0.5cm}
%  \item Natural variability of soil deposit  (Fenton 1999) 
%  
%  \begin{itemize}
%  
%  \item Function of soil formation process 
%   
%  \end{itemize}
%  
%  
%  %
%  \vspace*{0.5cm}
%  \item Testing error (Stokoe et al. 2004)
%  
%  \begin{itemize}
%  
%  \item Imperfection of instruments 
%   
%  \item Error in methods to register quantities
%  
%  \end{itemize}
%  
%
%  \vspace*{0.5cm}
%  \item Transformation error (Phoon and Kulhawy 1999)
%  
%  \begin{itemize}
%  
%  \item Correlation by empirical data fitting (e.g. CPT data $\rightarrow$ friction angle etc.)
%  
%  \end{itemize}
%  
%  \end{itemize}
%  
%   
%  
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  
%  \end{frame}
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%  \begin{frame} 
%  
%  \frametitle{Probabilistic material (Soil Site) Characterization}
%  
%   
%  
%  
%  \begin{itemize}
%  
%  \item Ideal: complete probabilistic site characterization
%  %  a very large amount of data is needed...... Need alternate strategies!!!}
%  
%  
%  
%  
%  \item Large (physically large but not statistically) amount of data 
%  
%    \begin{itemize}
%    
%    
%      \item Site specific mean and coefficient of variation (COV)
%      
%      \item  Covariance  structure from similar sites (e.g. Fenton 1999)
%    
%    \end{itemize}
%  
%  
%  \item Moderate amount of data $\rightarrow$ Bayesian updating (e.g. Phoon and Kulhawy 1999, Baecher and Christian 2003)
%  
%  \item Minimal data: general guidelines for typical sites and test methods (Phoon and Kulhawy (1999))
%  
%    \begin{itemize}
%    
%      \item COVs and covariance structures of inherent variability
%      
%      \item COVs of testing errors and transformation uncertainties.
%    
%    \end{itemize}
%  
%  %\item  Marosi  and Hiltunen (2004) and Stokoe et al. (2004) extended the general
%  %       guidelines for SASW method and $G/G_{max}$ curve
%  %
%  %%
%  %\end{itemize}
%  
%  
%  
%  \end{itemize}
%  
%   
%  
%  
%  
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%  
%  \end{frame}
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%  \begin{frame}
%  
%  
%  \frametitle{Recent State-of-the-Art}
%  
%  \begin{itemize}
%  
%  %\vspace*{-0.5cm}
%  \item Governing equation
%  
%  %       \vspace*{-0.5cm}
%         \begin{itemize}
%  
%         \item Dynamic problems  $\rightarrow$ $ M \ddot u + C \ddot u + K u = \phi $
%  
%         \item Static problems  $\rightarrow$   $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \   K u = \phi $
%  
%         \end{itemize}
%  
%  %\vspace{-0.4cm}
%  \item Existing solution methods
%  
%  %       \vspace*{-0.5cm}
%         \begin{itemize}
%  
%         \item \textbf{Random r.h.s} (external force random)
%  
%               \begin{itemize}
%  
%        \item FPK equation approach 
%  
%        \item Use of fragility curves with deterministic FEM (DFEM)
%  
%        \end{itemize}
%  
%  %       \vspace*{0.2cm}
%         \item \textbf{Random l.h.s} (material properties random)
%  
%               \begin{itemize}
%  
%        \item Monte Carlo approach with DFEM $\rightarrow$ CPU expensive
%  
%  %      \item Stochastic finite element method (e.g. Perturbation method  
%  %             $\rightarrow$ a linearized expansion! Error increases as a function
%  %      of COV; Spectral method 
%  %             $\rightarrow$ developed for elastic materials so far)
%  
%        \item Perturbation method 
%               $\rightarrow$ a linearized expansion! Error increases as a function
%        of COV 
%        
%        \item Spectral method 
%               $\rightarrow$ developed for elastic materials so far
%  
%  %             \begin{itemize}
%  %
%  %             \item Perturbation method  $\rightarrow$ fails if COVs of soil $>$ 20\%
%  %
%  %      \item Spectral method $\rightarrow$ only for elastic material
%  %
%  %             \end{itemize}
%  
%        \end{itemize}
%               
%         \end{itemize}      
%  
%  \item  New developments for elasto--plastic applications
%  
%  
%  \end{itemize}
%  
%  \end{frame}
%  
%  
%  % -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  % -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  % -  \begin{frame}
%  % -  \frametitle{Objectives of Proposed Work}
%  % -  
%  % -  \begin{itemize}
%  % -  
%  % -  \item Constitutive Problem: Obtain second-order (mean and variance) exact (analytical) PDF 
%  % -  of stress response $\rightarrow$ \textbf{Fokker--Planck--Kolmogorov (FPK) Equation}
%  % -  
%  % -  \item Boundary Value Problem: Model uncertain spatial variability of elastic--plastic soil
%  % -  $\rightarrow$ \textbf{Spectral Stochastic Elastic--Plastic Finite Element Method (SSEPFEM)}
%  % -  
%  % -  
%  % -  \vspace*{0.5truecm}
%  % -          
%  % -  \item Overcome the drawbacks of \textit{Monte Carlo Technique} and \textit{Perturbation Method}
%  % -  
%  % -  \item Obtain complete probabilistic description (PDF): Materials often fail at low 
%  % -  probability (tails of PDF)
%  % -  
%  % -  %         \begin{itemize}
%  % -  
%  % -  %          \item Materials often fail at low probability (tails of PDF)
%  % -  
%  % -  %         \end{itemize} 
%  % -  
%  % -  \item Carry out sensitivity analysis: Advanced constitutive models are sometimes highly sensitive to 
%  % -  fluctuations in material parameters
%  % -  
%  % -  %          \begin{itemize}
%  % -  
%  % -  %        \item Advanced models are highly sensitive to fluctuations in soil parameters
%  % -  
%  % -  \end{itemize}
%  % -  
%  % -  %\end{itemize}
%  % -  
%  % -  \end{frame}
%  % -  %
%  % -  
%  % -  
%  % -  
%  % -  
%  % -  
%  
%  
%  
%  
%  
%  \section{Probabilistic Elasto--Plasticity}
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  
%  
%  
%  \subsection{PEP Formulations}
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame} 
%  
%  \frametitle{Uncertainty Propagation through Constitutive Eq.}
%  
%  % 
%  
%  \begin{itemize}
%  
%   
%  
%  
%  \item Incremental el--pl constitutive equation  
%       $\displaystyle \frac{d\sigma_{ij}}{dt} = D_{ijkl} \displaystyle \frac{d\epsilon_{kl}}{dt}$
%  
%   
%    
%  %\begin{normalsize}
%  
%  %    
%  %    \begin{equation}
%  %    \nonumber
%  %    \frac{d\sigma_{ij}}{dt} = D_{ijkl} \frac{d\epsilon_{kl}}{dt}
%  %    \end{equation}
%      
%      
%      \begin{eqnarray}
%      \nonumber
%       D_{ijkl} = \left\{\begin{array}{ll}
%  %
%       D^{el}_{ijkl}
%  %
%  %
%          \;\;\; & \mbox{\large{~for elastic}} \\
%  %
%             \\
%  %
%             D^{el}_{ijkl}
%             -
%             \frac{\displaystyle  D^{el}_{ijmn} m_{mn} n_{pq} D^{el}_{pqkl}}
%             {\displaystyle n_{rs} D^{el}_{rstu} m_{tu} - \xi_* r_*}
%  
%             \;\;\; & \mbox{\large{~for elastic--plastic}}
%  %
%             \end{array} \right.
%             \end{eqnarray}
%  
%  %\end{normalsize}
%  
%   
%  
%  
%  
%  %\vspace{0.5cm}
%  % \item Non--linear coupling in the El--Pl modulus
%  
%  
%  %-  \item Focus on 1-D $\rightarrow$ a nonlinear ODE with random coefficient and random forcing
%  %-  
%  %-  
%  %- 
%  %-  \begin{eqnarray}
%  %-  \nonumber
%  %-  \frac{d\sigma(x,t)}{dt} &=& \beta(\sigma(x,t),D^{el}(x),q(x),r(x);x,t) \frac{d\epsilon(x,t)}{dt} \\
%  %-  \nonumber
%  %-  &=& \eta(\sigma,D^{el},q,r,\epsilon; x,t)  \mbox{\ \ \ \ with an I.C. $\sigma(0)=\sigma_0$}
%  %-  \end{eqnarray}
%  %- 
%   
%  
%  \end{itemize}
%  
%  % 
%  
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%  
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%  
%  \begin{itemize}
%  
%  \item 
%  Linear algebraic or differential equations  $\rightarrow$ Analytical solution:
%  
%           \begin{itemize}
%          \item Variable Transf. Method (Montgomery and Runger 2003)
%          \item Cumulant Expansion Method (Gardiner 2004)
%          \end{itemize}
%  
%  \item 
%  Nonlinear differential equations (elasto-plastic/viscoelastic-viscoplastic):
%       
%        \begin{itemize}
%  
%        \item  Monte  Carlo Simulation (Schueller 1997, De Lima et al 2001, Mellah
%        et al. 2000, Griffiths et al. 2005...) \\ $\rightarrow$ accurate, very costly
%  
%        \item  Perturbation  Method  (Anders and Hori 2000, Kleiber and Hien 1992,
%        Matthies  et  al. 1997) \\ $\rightarrow$ first and second order Taylor series
%        expansion  about mean - limited to problems with small C.O.V. and inherits
%        "closure problem"
%  
%        \end{itemize}
%  
%  % 
%  %  \item 
%  %  Monte Carlo method: accurate, very costly
%  %  
%  %  \item 
%  %  Perturbation method: 
%  
%  \end{itemize}
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\section{Stochastic Elasto--Plasticity}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Constitutive and Finite Element Levels}
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\begin{frame} 

\frametitle{Problem Setup}
 


 
 
\begin{itemize}

\vspace*{0.3cm}
\item Incr. 3D el--pl:
%
%
%    \begin{equation}
$
    \nonumber
      d\sigma_{ij}/d t = \left \{ 
    D^{el}_{ijkl}
           -
           \frac{\displaystyle  D^{el}_{ijmn} m_{mn} n_{pq} D^{el}_{pqkl}}
           {\displaystyle n_{rs} D^{el}_{rstu} m_{tu} - \xi_* r_*}
           \right \}  
          {d\epsilon_{kl}}/{d t}
$    
%   \end{equation}
   

\vspace*{0.3cm}
\item phase density $\rho$ of $\sigma(x,t)$ varies in time according to a continuity 
Liouville equation (Kubo 1963) 

\vspace*{0.3cm}
\item 
Continuity  equation  written  in  ensemble  average  form  (eg. cumulant
expansion method (Kavvas and Karakas 1996))


\vspace*{0.3cm}
\item 
van  Kampen's  Lemma (van Kampen 1976) $\rightarrow$  $  <\rho(\sigma,t)>=P(\sigma,t) $, 
ensemble average of phase density is the probability density

%  
%  
%   \item  Focus  on  1D  $\rightarrow$  a  nonlinear  ODE  with random coefficient
%   (material) and random forcing ($\epsilon$)
%  %
%  % 
%  %
%   \begin{eqnarray}
%   \nonumber
%   \frac{d\sigma(x,t)}{dt} &=& \beta(\sigma(x,t),D^{el}(x),q(x),r(x);x,t) \frac{d\epsilon(x,t)}{dt} \\
%   \nonumber
%   &=& \eta(\sigma,D^{el},q,r,\epsilon; x,t)
%   \end{eqnarray}
%  %
%  with initial condition $\sigma(0)=\sigma_0$

 \end{itemize}

\end{frame}






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%  \begin{frame} 
%  
%  
%  
%  \frametitle{Evolution of the Density $\rho(\sigma,t)$ }
%  
%   
%  
%  \begin{itemize}
%  
%  \vspace*{1cm}
%    \item  From each initial point in \\
%           $\sigma$-space a trajectory \\
%           starts out describing \\
%           the corresponding solution \\
%           of the stochastic process
%    
%  \vspace*{0.3cm}
%    \item  Movement of a cloud of initial\\
%           points  described  by density \\
%           $\rho(\sigma,0)$ in $\sigma$-space, \\
%           is governed by the \\
%           constitutive equation,
%  
%  \end{itemize}
%  
%  
%  %\begin{figure}[!hbpt]
%  %\begin{center}
%  \vspace*{-5cm}
%  \hspace*{6cm}
%  \includegraphics[height=4.5cm,angle=-90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/Cloud_of_Points.pdf}
%  %\end{center}
%  %\end{figure}
%                
%  
%  
%  \end{frame}
%  
%  
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame} 
%  
%  
%  \frametitle{Stochastic Continuity (Liouville) Equation}
%  
%   
%  
%  \begin{itemize}
%  
%  
%  \item phase density $\rho$ of $\sigma(x,t)$ varies in time according to a continuity 
%  Liouville equation (Kubo 1963): 
%  %
%  \begin{eqnarray}
%  \frac{\partial \rho (\sigma(x,t),t)}{\partial t}
%  = 
%  \nonumber
%  \\
%  -\frac{\partial  \eta (\sigma(x,t), D^{el}(x), q(x), r(x), \epsilon(x,t))  }{\partial \sigma}
%  \;\; 
%  \rho[\sigma(x,t),t]
%  \nonumber
%  \end{eqnarray}
%  
%  \vspace{0.5cm}
%  \item with initial conditions $\rho(\sigma,0) = \delta(\sigma-\sigma_0)$
%  
%  
%  \end{itemize}
%   
%  \end{frame}
%  
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%  \begin{frame} 
%  
%  
%  \frametitle{{Ensemble Average form of Liouville Equation}}
%  
%  
%  
%   
%  
%  
%  
%  \noindent
%  Continuity  equation  written  in  ensemble  average  form  (eg. cumulant
%  expansion method (Kavvas and Karakas 1996)):
%  
%   
%  %\vspace*{-0.5cm}
%  
%  \begin{footnotesize}
%  
%  \begin{eqnarray}
%  \nonumber
%  &&\displaystyle \frac{\partial \left < \rho(\sigma(x_t,t), t) \right >}{\partial t}=
%  - \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
%  q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
%  \nonumber
%  &+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
%  \epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
%  \nonumber
%  & & \left. \left. \left. \eta(\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
%  \epsilon(x_{t-\tau},t-\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} \left < \rho (\sigma(x_t,t),t) \right > \right] \\
%  \nonumber
%  &+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
%  \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
%  \nonumber
%  & & \left. \left. \left. \eta (\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
%  \epsilon(x_{t-\tau},t-\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} \left < \rho (\sigma (x_t,t),t) \right > \right] \\
%  \nonumber
%  \end{eqnarray}
%  
%  \end{footnotesize}
%  
%  
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  
%  
%  \end{frame}
%  





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\begin{frame} 

\frametitle{Eulerian--Lagrangian FPK Equation}

%

%\begin{itemize}

\begin{footnotesize}

% %\noindent
% van  Kampen's  Lemma (van Kampen 1976) $\rightarrow$  $  <\rho(\sigma,t)>=P(\sigma,t) $, 
% ensemble average of phase density 
% %(in stress space here) 
% is the probability density;
% 

\begin{eqnarray}
\nonumber
&&\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=
- \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
\nonumber
&+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
\epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta(\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
\epsilon(x_{t-\tau},t-\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} P(\sigma(x_t,t),t) \right] \\
\nonumber
&+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
\eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta (\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
\epsilon(x_{t-\tau},t-\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right] \\
\nonumber
\end{eqnarray}

\end{footnotesize}


\vspace*{0.5cm}
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{B. Jeremi\'{c}, K. Sett, and M. L. Kavvas, "Probabilistic
Elasto--Plasticity: Formulation in 1--D", \textit{Acta Geotechnica}, Vol. 2, No. 3, 2007, In press (published
online in the \textit{Online First} section)}
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}


\end{frame}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 

\frametitle{Euler--Lagrange FPK Equation}

 

\begin{itemize}

\item Advection-diffusion equation 
%
\begin{equation}                
\nonumber
\frac{\partial P(\sigma,t)}{\partial t} = -\frac{\partial}{\partial \sigma}\left[N_{(1)}P(\sigma,t)-\frac{\partial}{\partial \sigma}
\left\{N_{(2)} P(\sigma,t)\right\} \right]
\end{equation}

%

\item Complete probabilistic description of response 


\item Solution PDF is second-order exact to covariance of time (exact mean and variance)


\item It is deterministic equation in probability density space

\item It is linear PDE in probability density space 
$\rightarrow$ Simplifies the numerical solution process

\item Template FPK diffusion--advection equation is applicable to any material model $\rightarrow$ 
only the coefficients $N_{(1)}$ and $N_{(2)}$ are different for different material models

%\vspace*{0.2truecm}

\end{itemize}



  
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{K. Sett, B. Jeremi{\'c} and M.L.  Kavvas, "The Role of Nonlinear 
Hardening/Softening in Probabilistic Elasto--Plasticity", \textit{International Journal for Numerical 
and Analytical Methods in Geomechanics}, Vol. 31, No. 7, pp. 953-975, 2007}
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}



\end{frame}



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% %\begin{frame} 
% %
% %
% %
% %\frametitle{Template Solution of FPK Equation}
% %
% % 
% %
% %\begin{itemize}
% %
% %
% %
% %
% %\item FPK diffusion--advection equation is applicable to any material model $\rightarrow$ 
% %only the coefficients $N_{(1)}$ and $N_{(2)}$ are different for different material models
% %%
% %%
% %%
% %%\begin{normalsize}
% %\begin{equation}                
% %\nonumber
% %\frac{\partial P(\sigma,t)}{\partial t} = -\frac{\partial}{\partial \sigma}\left[N_{(1)}P(\sigma,t)-\frac{\partial}{\partial \sigma}
% %\left\{N_{(2)} P(\sigma,t)\right\} \right]
% %%\nonumber
% % = -\frac{\partial \zeta}{\partial \sigma}
% %\end{equation}
% %%\end{normalsize}
% %
% %%
% %
% %\item Initial condition 
% %
% %\begin{itemize}
% %
% %\item Deterministic $\rightarrow$ Dirac delta function $\rightarrow$ $ P(\sigma,0)=\delta(\sigma) $
% %
% %\item Random $\rightarrow$ Any given distribution
% %
% %\end{itemize}
% %
% %\item Boundary condition: Reflecting BC $\rightarrow$ conserves probability mass
% %$\zeta(\sigma,t)|_{At \ Boundaries}=0$
% %
% %\item Finite Differences used for solution (among many others)
% %
% %
% %\end{itemize}
% %
% % 
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% %
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% %\end{frame}
% %
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%  %==\begin{frame} \frametitle{Solution of FPK Equation: Numerical Scheme}
%  %==
%  %== 
%  %==
%  %==\begin{itemize}
%  %==
%  %==  
%  %==  \item \textit{Method of Lines} $\rightarrow$ semi-discretization of stress domain by \textit{Finite Difference Technique}
%  %==
%  %==    \begin{itemize}
%  %==
%  %==    \item Has inherent drawbacks $\rightarrow$ nor very efficient
%  %==
%  %==
%  %==\begin{center}
%  %==  \begin{figure}[!hbpt]
%  %==  \nonumber
%  %==%--    \begin{center}
%  %==%--      \begin{flushleft}
%  %==%--        \begin{center}
%  %==          \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/AnticipatedInfluence.jpg}
%  %==          \includegraphics[width=16.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/NumericalScheme01.jpg}
%  %==          
%  %==%--      \end{flushleft}
%  %==%--      \begin{flushright}
%  %==\includegraphics[height=13cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG-m.pdf}
%  %==
%  %==\nonumber
%  %==\caption{Low OCR Cam Clay Response with Random $G$}
%  %=={\includegraphics[width=10.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/QualSchematics4.jpg}}
%  %==\includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/AnticipatedInfluence.jpg}
%  %==%--      \end{flushleft}
%  %==
%  %==%--      \end{flushright}
%  %==%--
%  %==%--\begin{flushright}
%  %==
%  %==\begin{itemize}
%  %==
%  %==\* Adaptive technique  \\
%  %==
%  %==\ \\
%  %==
%  %==\* Krylov subspace \ \ \ \ \ \\
%  %==
%  %==\ \\
%  %==
%  %==\* Meshfree technique
%  %==
%  %==\end{itemize}
%  %==
%  %==\end{flushright}
%  %==
%  %==\end{center}
%  %==\end{figure}
%  %==
%  %==\item Possible improvements through:
%  %==
%  %==\begin{itemize}
%  %==
%  %==\item Adaptive techniques (FEM, meshfree etc.)
%  %==
%  %==\item Krylov subspace technique (reduced-order modeling)
%  %==
%  %==\end{itemize}
%  %==
%  %==\end{itemize}
%  %==
%  %==\end{itemize}
%  %==
%  %== 
%  %==
%  %==%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  %==
%  %==
%  %==\end{frame}
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%  \begin{frame} 
%  
%  \frametitle{Application of FPK equation to Material Models}
%  
%   
%  
%  \begin{itemize}
%  
%  \vspace*{0.5cm}
%  \item  FPK  equation  is applicable to any incremental elastic--plastic material
%  model
%  
%  %\item  Only the coefficients $N_{(1)}$ and $N_{(2)}$ differ
%  
%  
%  \vspace*{0.5cm}
%        \item Solution in terms of PDF, not a single value of stress
%  
%  \vspace*{0.5cm}
%        \item Influence of initial condition on the PDF of stress
%  
%  \vspace*{0.5cm}
%     \item Mean stress yielding or 
%   
%  \vspace*{0.5cm}
%     \item Probabilistic yielding 
%        
%  
%  %       \item Differences in mean, mode and deterministic solution of stress
%  
%  
%  \end{itemize}
%  
%   
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  
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%  \subsection{Probabilistic Elastic--Plastic Response}
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  \begin{frame} 
%  
%  \frametitle{Elastic Response with Random $G$}
%  
%   
%  
%  \begin{itemize}
%  
%  \item General form of elastic constitutive rate equation
%  
%  \begin{eqnarray}
%  \nonumber
%  \frac{d \sigma_{12}}{dt} &=& 2G \frac{d \epsilon_{12}}{dt} \\
%  \nonumber
%  &=& \eta(G, \epsilon_{12};t)
%  \end{eqnarray}
%  
%  \item Advection and diffusion coefficients of FPK equation
%  
%  \begin{equation}
%  \nonumber
%  N_{(1)}=2\frac{d \epsilon_{12}}{dt} <G>
%  \end{equation}
%  
%  \begin{equation}
%  \nonumber
%  N_{(2)}=4t\left(\displaystyle \frac{d \epsilon_{12}}{dt} \right)^2 Var[G]
%  \end{equation}
%  
%  \end{itemize}
%   
%  
%  
%  
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  \begin{frame} 
%  
%  \frametitle{Elastic Response with Random $G$}
%  
%  
%  \begin{figure}[!hbpt]
%  %\nonumber
%  %\begin{flushleft}
%  \begin{center}
%  %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%  %\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFElastic_RandomG-m.pdf}
%  \includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2005/8USNCCM/Probabilistic_Elasto_Plasticity/PDFElastic_RandomG-m.pdf}
%  %
%  %\end{flushleft}
%  %\begin{flushright}
%  %\includegraphics[height=9cm]{ReasonOfNotMatchingMonteCarlo-m.pdf}
%  %
%  %\nonumber
%  %\caption{Drucker-Prager Associative Linear Hardening Response with Random $G$}
%  %{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%  %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%  %\end{flushright}
%  \end{center}
%  \end{figure}
%  $<G>$ = 2.5 MPa;
%  Std. Deviation$[G]$ = 0.5 MPa
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  \begin{frame} 
%  
%  \frametitle{Verification -- Variable Transformation Method}
%  
%  
%  \begin{figure}[!hbpt]
%  \begin{center}
%  %\includegraphics[height=6cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/MonteCarlo_Elastic-m.pdf}
%  \includegraphics[height=6cm]{/home/jeremic/tex/works/Conferences/2005/8USNCCM/Probabilistic_Elasto_Plasticity/MonteCarlo_Elastic-m.pdf}
%  %  %
%  %  \hspace*{-3cm}
%  %  \vspace*{4cm}
%  %  %
%  %  \includegraphics[height=4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ReasonOfNotMatchingMonteCarlo-m.pdf}
%  \end{center}
%  \end{figure}
%  %  
%  %  \vspace*{-4cm}
%  %  
%  %  \begin{flushleft}
%  %  Effect of Approximation of I.C on the \\ PDF of Stress at 0.0426 \% Strain
%  %  \end{flushleft}
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame} 
%  
%  \frametitle{Drucker-Prager Linear Hardening with Random $G$}
%  
%   
%  
%  %
%  %  
%  %  
%  %  General form of Drucker-Prager elastic-plastic linear hardening constitutive rate equation
%  %  
%  %  
%  
%  \begin{eqnarray}
%  \nonumber
%  \frac{d \sigma_{12}}{dt} = 
%  G^{ep} \frac{d \epsilon_{12}}{dt} = 
%  \eta(\sigma_{12}, G, K, \alpha, \alpha', \epsilon_{12};t)
%  \end{eqnarray}
%  
%  
%  
%  \noindent
%  Advection and diffusion coefficients of FPK equation
%  
%  
%  
%  \begin{equation}
%  \nonumber
%  N_{(1)}=\displaystyle \frac{d \epsilon_{12}}{dt} \left< 2G - \displaystyle \frac{G^2}{G+9K\alpha^2+\displaystyle \frac{1}{\sqrt{3}}I_1
%  \alpha'} \right>
%  \end{equation}
%  
%  
%  
%  \begin{equation}
%  \nonumber
%  N_{(2)}=t\left(\displaystyle \frac{d \epsilon_{12}}{dt} \right)^2 Var\left[2G - \displaystyle \frac{G^2}{G+9K\alpha^2+\displaystyle 
%  \frac{1}{\sqrt{3}}I_1 \alpha'} \right]
%  \end{equation}
%  
%  %  \end{itemize}
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%  % -- \begin{frame} 
%  % -- 
%  % -- \frametitle{{Drucker-Prager Linear Hardening \\ with Random $G$}}
%  % -- 
%  % -- \vspace*{-1.5cm}
%  % -- 
%  % -- \begin{figure}[!hbpt]
%  % -- \begin{center}
%  % -- \includegraphics[height=3.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFDruckerPrager_RandomG-m.pdf}
%  % -- \hfill
%  % -- \includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourDruckerPrager_RandomG-m.pdf}
%  % -- \end{center}
%  % -- \end{figure}
%  % -- 
%  % -- 
%  % -- 
%  % -- 
%  % -- \begin{itemize}
%  % -- 
%  % -- \item Mean stress yielding
%  % -- 
%  % -- \item Approximation of I.C.
%  % -- 
%  % -- 
%  % -- \item Smooth transition between el. \& el.-pl.
%  % -- 
%  % -- 
%  % -- \item Symmetry in probability distribution
%  % -- 
%  % -- \end{itemize}
%  % -- 
%  % -- %
%  % -- % \begin{flushright}
%  % -- % \includegraphics[height=13cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourDruckerPrager_RandomG-m.pdf}
%  % -- % %
%  % -- % \nonumber
%  % -- % %\caption{Drucker-Prager Associative Linear Hardening Response with Random $G$}
%  % -- % %{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%  % -- % %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%  % -- % %\end{flushleft}
%  % -- % \end{flushright}
%  % -- % \end{figure}
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%  \begin{frame} 
%  
%  \frametitle{Verification of D--P E--P Response - Monte Carlo}
%  
%  
%  \begin{figure}[!hbpt]
%  %\nonumber
%  %\begin{flushleft}
%  \begin{center}
%  %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%  %\includegraphics[height=6cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/MonteCarlo_DruckerPrager-m.pdf}
%  \includegraphics[height=6cm]{/home/jeremic/tex/works/Conferences/2005/8USNCCM/Probabilistic_Elasto_Plasticity/MonteCarlo_DruckerPrager-m.pdf}
%  %
%  %\end{flushleft}
%  %\begin{flushright}
%  %\includegraphics[height=11.5cm]{ContourDruckerPrager_RandomG-m.pdf}
%  %
%  %\nonumber
%  %\caption{Drucker-Prager Associative Linear Hardening Response with Random $G$}
%  %{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%  %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%  %\end{flushleft}
%  %\end{flushright}
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%  \begin{frame} 
%  
%  \frametitle{Modified Cam Clay Constitutive Model}
%  
%   
%  
%  \begin{small}
%  \begin{equation}
%  \nonumber
%  \frac{d \sigma_{12}}{dt} = G^{ep} \frac{d \epsilon_{12}}{dt} = \eta(\sigma_{12}, G, M, e_0, p_0, \lambda, \kappa, \epsilon_{12};t)
%  \end{equation}
%  %
%  %
%  %
%  %
%  %
%  \begin{equation}
%  \nonumber
%  \eta = \left[2G - \displaystyle \frac{\left(36 \displaystyle \frac{G^2}{M^4} \right) \sigma_{12}^2}
%  {\displaystyle \frac{(1+e_0)p(2p-p_0)^2}{\kappa} + \left(18 \displaystyle \frac{G}{M^4}\right) \sigma_{12}^2
%  + \displaystyle \frac{1+e_0}{\lambda-\kappa} p p_0 (2p-p_0)} \right] 
%  \end{equation}
%  \end{small}
%  
%  
%  
%  \noindent
%  Advection and diffusion coefficients of FPK equation
%  
%  
%  
%  \begin{equation}
%  \nonumber
%  N_{(1)}^{(i)}=\left<\eta^{(i)}(t)\right> + \int_0^t d\tau cov\left[\displaystyle \frac{\partial \eta^{(i)}(t)}{\partial t};
%  \eta^{(i)} (t-\tau)\right]
%  \end{equation}
%  
%  
%  
%  \begin{equation}
%  \nonumber
%  N_{(2)}^{(i)} = \int_0^t d\tau cov\left[\eta^{(i)}(t); \eta^{(i)} (t-\tau)\right]
%  \end{equation}
%  
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%  % -- \begin{frame} 
%  % -- 
%  % -- \frametitle{Low OCR Cam Clay with \\ Random $G$}
%  % -- 
%  % -- \vspace*{-1.5cm}
%  % -- 
%  % -- \begin{figure}[!hbpt]
%  % -- %\nonumber
%  % -- \begin{center}
%  % -- \includegraphics[height=3.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFLowOCR_RandomG-m.pdf}
%  % -- \hfill
%  % -- \includegraphics[height=5.4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG-m.pdf}
%  % -- \end{center}
%  % -- \end{figure}
%  % -- 
%  % -- \begin{itemize}
%  % -- 
%  % -- \item Approximation of I.C.
%  % -- 
%  % -- 
%  % -- %\item Wide transition between el. \& el.-pl.
%  % -- 
%  % -- 
%  % -- \item Non-symmetry in probability distribution!
%  % -- 
%  % -- 
%  % -- \item Response at critical state fairly certain but different than deterministic
%  % -- 
%  % -- \end{itemize}
%  % -- 
%  % -- 
%  % -- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  % -- \end{frame}
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%  \begin{frame} 
%  
%  \frametitle{{Low OCR Cam Clay  with  Random $G$, $M$ and $p_0$}}
%  
%  
%  
%  
%  
%  \begin{itemize}
%  
%  %\item Narrow transition between el. \& el.-pl.
%  
%  
%  \vspace*{0.3cm}
%  \item Non-symmetry in \\
%        probability \\
%        distribution
%  
%  
%  \vspace*{0.3cm}
%  \item Difference \\
%        between \\
%        mean, mode and \\
%        deterministic 
%  
%  
%  
%  \vspace*{0.3cm}
%  \item Divergence at \\
%        critical state \\
%        because $M$ is \\
%        uncertain
%  
%  \end{itemize}
%  
%  
%  \vspace*{-6.3cm}
%  %\hspace*{0.5cm}
%  \begin{figure}[!hbpt]
%  \begin{flushright}
%  %\includegraphics[height=3.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFLowOCR_RandomG_RandomM_Randomp0-m.pdf}
%  %\hfill
%  %\includegraphics[height=6.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0-m.pdf}
%  \includegraphics[height=6.5cm]{/home/jeremic/tex/works/Conferences/2005/8USNCCM/Probabilistic_Elasto_Plasticity/ContourLowOCR_RandomG_RandomM_Randomp0-m.pdf}
%  \end{flushright}
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%  % --  
%  % --  \begin{frame} 
%  % --  
%  % --  \frametitle{{Low OCR Cam Clay  with \\ Random $G$, $M$ and $p_0$}}
%  % --  
%  % --  \vspace*{-1.5cm}
%  % --  
%  % --  \begin{figure}[!hbpt]
%  % --  \begin{center}
%  % --  \includegraphics[height=3.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFLowOCR_RandomG_RandomM_Randomp0-m.pdf}
%  % --  \hfill
%  % --  \includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0-m.pdf}
%  % --  \end{center}
%  % --  \end{figure}
%  % --  
%  % --  
%  % --  
%  % --  \vspace*{-0.5cm}
%  % --  
%  % --  \begin{itemize}
%  % --  
%  % --  %\item Narrow transition between el. \& el.-pl.
%  % --  
%  % --  
%  % --  \item Non-symmetry in probability distribution
%  % --  
%  % --  
%  % --  \item Difference between mean, mode and deterministic 
%  % --  
%  % --  
%  % --  
%  % --  \item Divergence at critical state because $M$ is uncertain
%  % --  
%  % --  \end{itemize}
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%  \begin{frame} 
%  
%  \frametitle{Comparison of Low OCR Cam Clay at $\epsilon$ = 1.62 \%}
%  
%  
%  
%  %\vspace*{-4.50cm}
%  
%  \begin{figure}[!hbpt]
%  \begin{center}
%  %\includegraphics[height=14cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/CamClayPDFComparison-m.pdf}
%  %\includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/LowOCR_ComparisonPDF-m_ps.pdf}
%  \includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/LowOCR_ComparisonPDF-m_ps.pdf}
%  \end{center}
%  \end{figure}                                                          
%  
%  \vspace*{-0.5cm}
%  
%  \begin{itemize}
%  
%  \item None coincides with deterministic
%  
%  \item Some  very uncertain, some very certain
%  
%  \item Either on safe or unsafe side
%  
%  \end{itemize}
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%  \begin{frame} 
%  
%  \frametitle{{High OCR Cam Clay with Random $G$ and $M$}}
%  
%  
%  \begin{itemize}
%  
%  %\item Approximation of I.C.
%  
%  %
%  %\item Very uncertain transition between el. \& el.-pl.
%  
%  
%  \vspace*{0.5cm}
%  \item Large non-symmetry \\
%       in probability \\
%       distribution
%  
%  
%  \vspace*{0.5cm}
%  \item Significant \\
%        differences in \\
%        mean, mode, \\
%        and deterministic 
%  
%  
%  \vspace*{0.5cm}
%  \item Divergence at \\
%        critical state, \\
%        $M$ is uncertain
%  
%  \end{itemize}
%  
%  
%  \vspace*{-6.5cm}
%  %
%  \begin{figure}[!hbpt]
%  \begin{flushright}
%  %\includegraphics[height=3.4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFHighOCR_RandomG_RandomM-m.pdf}
%  %\hfill
%  \includegraphics[height=6.5cm]{/home/jeremic/tex/works/Conferences/2005/8USNCCM/Probabilistic_Elasto_Plasticity/ContourHighOCR_RandomG_RandomM-m.pdf}
%  \end{flushright}
%  \end{figure}                                                          
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\begin{frame} 


\frametitle{Probabilistic Yielding}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{itemize}

%
\vspace*{0.2cm}
\item Weighted elastic and elastic--plastic Solution
${\partial P(\sigma,t)}/{\partial t} 
= 
-{\partial \left(N^w_{(1)}P(\sigma,t)
-{\partial \left(N^w_{(2)} P(\sigma,t)\right) }/{\partial \sigma} \right)}/
{\partial \sigma}$






\vspace*{0.1cm}
\item Weighted advection and diffusion coefficients are then 
$N_{(1,2)}^{w} (\sigma) 
= 
(1 - P[\Sigma_y \leq \sigma]) N_{(1)}^{el} + P[\Sigma_y \leq \sigma] N_{(1)}^{el-pl} $
\begin{footnotesize}
(Black--Scholes options \\
 pricing model '73, \\
Nobel Economics Prize '97)
\end{footnotesize}


\vspace*{0.1cm}
%\vspace*{0.5cm}
\item   
Cumulative Probability \\
Density  function  (CDF) \\
of the yield function 

\end{itemize}

\vspace*{-3.5cm}
\begin{flushright}
%\begin{figure}[!h]
%\hspace*{7cm}
\includegraphics[width=5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/vonMises_YieldCDF_Combined-edited.pdf} 
%\end{figure}
\end{flushright}
\vspace*{-0.5cm}


%\vspace*{0.5cm}
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{B.  Jeremi{\'c} and K. Sett. On Probabilistic
Yielding  of  Materials.  in print in  \textit{Communications in Numerical Methods in
Engineering}, 2007.}
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}



\end{frame}



%--  
  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}



\frametitle{Transformation of a Bi--Linear (von Mises) Response}


\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDF-edited.pdf}
\end{center}
\end{figure}                                                          

\vspace*{-0.3cm}
linear elastic -- linear hardening plastic von Mises


\end{frame}



%--  
  





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Boundary Value Problem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section[SEPFEM]{Stochastic Elastic--Plastic Finite Element Method}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{SEPFEM Formulations}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% 
% 
% \frametitle{Governing Equations \& Discretization Scheme}
% 
% \begin{itemize}
% 
% \item Governing equations of mechanics:
% 
% 
% \begin{equation}
% \nonumber
% A\sigma = \phi(t);~~~ Bu = \epsilon; ~~~\sigma = D \epsilon
% \end{equation}
% 
% \item  Discretization (spatial and stochastic) schemes
% 
% \begin{itemize}
% 
% \item    Input    random   field   material   properties   ($D$)   $\rightarrow$
% Karhunen--Lo{\`e}ve (KL) expansion, optimal expansion, error minimizing property
% 
% \item  Unknown  solution  random field ($u$) $\rightarrow$ Polynomial Chaos (PC)
% expansion
% 
% \item  Deterministic  spatial  differential operators ($A$ \& $B$) $\rightarrow$
% Regular shape function method with Galerkin scheme
% 
% 
% \end{itemize}
% 
% 
% 
% \end{itemize}
% 
% \end{frame}
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - % - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - 
% - \begin{frame}
% - 
% - 
% - \frametitle{Truncated Karhunen--Lo{\`e}ve (KL) expansion}
% -               
% -        \begin{itemize}                                                                                                                         
% -                                                                                                                                                       
% -        \item Representation of input random fields in eigen-modes of covariance kernel                                                                
% -                                                                                                                                                       
% - %       \vspace*{-0.1cm}                                                                                                                               
% - %              \begin{figure}[!hbpt]                                                                                                                   
% -               \begin{flushleft}                                                                                                                       
% -               \includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2008/GeoCongress/Probabilistic/Paper/TypicalDataPlotBH1-Edited.jpg}
% - %       ShearStrengthProfile.jpg}                                                                                
% -               \end{flushleft}                                                                                                                         
% -               \vspace*{-3.15cm}                                                                                                                        
% -        \begin{flushright}                                                                                                                             
% - %       \begin{equation}                                                                                                                              
% - %       \nonumber                                                                                                                                     
% - %       \begin{normalsize}                                                                                                                            
% -        $ q_T(x,\theta) = \bar q_T(x) + \sum_{n=1}^M \sqrt{\lambda_n} \xi_n(\theta) f_n(x) $ \\                                                        
% -        \ \\                                                                                                                                           
% -        $ \int_D C(x_1, x_2) f (x_2) dx_2 = \lambda f (x_1) \ \ \ \ \ \ \ \ \ \  \ \ $ \\                                                              
% -        \ \\                                                                                                                                           
% -        $ \xi_i(\theta) = \displaystyle \frac{1}{\sqrt \lambda_i} \int_D \left [q_T(x,\theta) - \bar q_T (x) \right] f_i (x) dx $                                   
% - %       \end{equation}                                                                                                                                
% - %              \end{normalsize}                                                                                                                       
% -        \end{flushright}                                                                                                                               
% - %       \end{figure}                                                                                                                                   
% -                                                                                                                                                       
% - %       \vspace{-6.0cm}                                                                                                                               
% - %       \begin{flushright}                                                                                                                            
% - %       \begin{equation}                                                                                                                              
% - %       \nonumber                                                                                                                                     
% - %        w(x,\theta) = \bar w(x) + \sum_{n=0}^M \sqrt{\lambda_n} \zeta_n(\theta) f_n(x)                                                               
% - %       \end{equation}                                                                                                                                
% - %        \end{flushright}                                                                                                                             
% -                                                                                                                                                       
% - %        \vspace*{0.8cm}                                                                                                                               
% -         \item Error minimizing property                                                                                                               
% -                                                                                                                                                       
% -         \item Optimal expansion $\rightarrow$ minimization of number of stochastic dimensions                                                         
% -                                                                                                                                                       
% -                                                                                                                                                       
% -          \end{itemize}                                                                                                                                                                                                                                                                   
% -                                                                                                                                                                                                                                                                                          
% - 
% - 
% - \end{frame}
% - 
% - 
% - % - % - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - % - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - % - \begin{frame}
% - % - \frametitle{KL Expansion (of Covariance Kernel)}
% - % - 
% - % - \begin{flushleft}
% - % - %\begin{center}
% - % - %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
% - % - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/ActualExponentialCovarianveSurface.jpg}
% - % - \hspace*{0.3cm}
% - % - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_1Term_CovarianveSurface.jpg}
% - % - %\vspace*{1.0cm}
% - % - %\mbox{Exact covariance surface}
% - % - %\vspace*{-4.0cm}
% - % - \end{flushleft}
% - % - \vspace*{-0.4cm}
% - % - \small{\ \ \ \ \ \ \ \ \ \ \ \ \ Exact \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 term approx.} \\
% - % - \small{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (8.49\% error)}
% - % - %\begin{flushright}
% - % - %\end{center}
% - % - %\end{figure}
% - % - 
% - % - %\vspace*{-0.5cm}
% - % - %\small{Exact covariance surface \ \ \ \ \ \ \ \ \ \ \ \ \ \ One-term approximation}
% - % - %
% - % - %\vspace*{0.5cm}
% - % - %
% - % - %\begin{figure}[!hbpt]
% - % - %\begin{center}
% - % - \begin{flushleft}
% - % - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_2Terms_CovarianveSurface.jpg}
% - % - \hspace*{-0.2cm}
% - % - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_3Terms_CovarianveSurface.jpg}
% - % - \end{flushleft}
% - % - %\vspace*{-0.5cm}
% - % - \small{\ \ \ \ \ 2 terms approx. \ \ \ \ \ \ \ \ \  3 terms approx.} \\
% - % - \small{\ \ \ \ \ \ (1.15\% error) \ \ \ \ \ \ \ \ \ \ \ \ \ (1.13\% error)} 
% - % - %
% - % - \vspace{-6.5cm}
% - % - \begin{flushright}
% - % - 
% - % - \includegraphics[height=2.0cm]{/home/jeremic/tex/works/Reports/2006/SEPFEM/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScale-Edited.jpg} \hspace*{-0.7cm}
% - % - %ShearStrengthProfile.jpg}
% - % - 
% - % - covariance function \\
% - % - (exponential):  \ \ \ \ \  \\
% - % - %\ \\
% - % - $ C(x_1, x_2) = \sigma^2 e^{-| x_1 - x_2 | /b} $  \hspace*{-0.5cm} \\
% - % - \ \\
% - % - \ \\
% - % - KL approximation: \ \ \ \ \ \ \ \  \\
% - % - \ \\
% - % - $ C(x_1, x_2) \ \ \ \ \ \ \ \ \ \ \ \ \  $ \\
% - % -     $ = \sum_{k =1}^M \lambda_k f_k(x_1) f_k(x_2) $
% - % - 
% - % - \end{flushright}
% - % - %\end{figure}
% - % - 
% - % - %\vspace*{2.5cm}
% - % - %\small{Two-terms approximation \ \ \ \ \ \ \ \ \ \ \ \  Three-terms approximation}
% - % - 
% - % - \vspace*{4.0cm}
% - % - \begin{center}
% - % - \large{KL Expansion of Covariance Kernel}
% - % - \end{center}
% - % - 
% - % - \end{frame}
% - % - 
% - % - 
% - % - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - 
% - \begin{frame}
% - 
% - \frametitle{Polynomial Chaos (PC) Expansion}
% - 
% - %\begin{itemize}
% - 
% - %\vspace*{-0.35cm}
% - %\item Solution (displacement) random field $\rightarrow$ Can not use K-L expansion directly
% - 
% -        \begin{itemize}
% - 
% -        \item Covariance kernel is not known a priori
% - 
% -        \vspace*{-0.5cm}
% -        \begin{normalsize}
% -        \begin{equation}
% -        \nonumber
% -        u(x,\theta)=\sum_{j=1}^L e_j \chi_j(\theta)b_j(x)
% -        \end{equation}
% -        \end{normalsize}
% - 
% -        \vspace*{-0.3cm}
% -        \item Can be expressed as functional of known random variables and unknown deterministic function
% - 
% -        \vspace*{-0.5cm}
% -        \begin{normalsize}
% -        \begin{equation}
% -        \nonumber
% -        u(x,\theta)=\zeta[\xi_i(\theta),x]
% -        \end{equation}
% -        \end{normalsize}
% - 
% -        \vspace*{-0.5cm}
% -        \item Need a basis of known random variables $\rightarrow$ PC expansion
% - 
% -        \vspace*{-0.2cm}
% -        \begin{normalsize}
% -        \begin{equation}
% -        \nonumber
% -        \chi_j(\theta)=\sum_{i=0}^P\gamma_i^{(j)}\psi_i\left[\left\{\xi_r\right\}\right]
% -        \end{equation}
% - 
% -        \vspace*{-0.5cm}
% -        \begin{equation}
% -        \nonumber
% -        u(x,\theta)=\sum_{j=1}^L \sum_{i=0}^P \gamma_i^{(j)} \psi_i[\{\xi_r\}]e_j b_j(x) = \sum_{i=0}^P \psi_i[\{\xi_r\}] d_i(x)
% -        \end{equation}
% -        \end{normalsize}
% - 
% - %       \vspace*{-0.6cm}
% - %       \item Deterministic coefficients can be found by minimizing norm of error of finite 
% - %            representation (e.g. using Galerkin scheme) 
% -        
% -         
% - 
% - 
% - %       \end{itemize}
% - \end{itemize}
% - 
% - 
% - %\end{itemize}
% - 
% - \end{frame}
% - 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Spectral Stochastic Elastic--Plastic FEM}

\begin{itemize}

\item  Minimizing    norm  of  error  of  finite  representation  using Galerkin
technique (Ghanem and Spanos 2003):

      \vspace*{-0.6truecm}
      \begin{flushright}
      \begin{equation}
      \nonumber
      \sum_{n = 1}^N K_{mn} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K'_{mnk}  = \left< F_m \psi_i[\{\xi_r\}] \right >
      \end{equation}
      \end{flushright}

%      \begin{itemize}
%
%      \vspace*{0.5cm}
%      \item Final eqn.:
%      
%      \vspace*{-0.4cm}
%      \begin{flushright}
%      \begin{normalsize}
%      \begin{equation}
%      \nonumber
%      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sum_{n = 1}^N K_{mn} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K'_{mnk}  = \left< F_m \psi_i[\{\zeta_r\}] \right >
%      \end{equation}
%      \end{normalsize}
%      \end{flushright}

      \vspace*{-0.5cm}
      \begin{equation}
      \nonumber
      K_{mn} = \int_D B_n \textcolor{mycolor}{D}  B_m dV    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ K'_{mnk} = \int_D B_n {\sqrt \lambda_k h_k} B_m dV
      \end{equation}
      
      \vspace*{-1.0cm}
      \begin{equation}
      \nonumber
      C_{ijk} = \left < \xi_k(\theta) \psi_i[\{\xi_r\}] \psi_j[\{\xi_r\}] \right >  \ \ \ \ \ \ \ \ \ \ \ \  F_m = \int_D \phi N_m dV  \ \ \ \ \ \ \ \ \ \ \ \
      \end{equation}

%\item Fokker--Planck--Kolmogorov approach based probabilistic constitutive integration
%      at Gauss integration points


\end{itemize}

%  \noindent Salient Features:
       
%      \begin{itemize}
%
%      \item Efficient representation of input random fields into finite number of random
%            variables using KL-expansion
%      
%      \item Representation of (unknown) solution random variables using polynomial chaos of
%            (known) input random variables
%
%      \item Fokker--Planck--Kolmogorov approach based probabilistic constitutive integration
%      at Gauss integration points
%
%      \end{itemize}
%      
%%   \end{itemize}
%
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Inside SEPFEM}

\begin{itemize}

\item Explicit stochastic elastic--plastic finite element computations

\vspace*{0.2cm}
\item FPK probabilistic constitutive integration at Gauss integration points

\vspace*{0.2cm}
\item   Increase   in  (stochastic)  dimensions  (KL  and  PC)  of  the  problem
(parallelism)

\vspace*{0.2cm}
\item Development of the probabilistic elastic--plastic stiffness tensor


\end{itemize}

%  \noindent Salient Features:
       
%      \begin{itemize}
%
%      \item Efficient representation of input random fields into finite number of random
%            variables using KL-expansion
%      
%      \item Representation of (unknown) solution random variables using polynomial chaos of
%            (known) input random variables
%
%      \item Fokker--Planck--Kolmogorov approach based probabilistic constitutive integration
%      at Gauss integration points
%
%      \end{itemize}
%      
%%   \end{itemize}
%
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{SEPFEM Verification Example}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% \begin{frame}
% 
% 
% \frametitle{1--D Static Pushover Test Example}
% 
% 
% 
% 
% \begin{itemize}
% 
% \vspace*{-0.3cm}
%   \item  Linear elastic model: \\
%          $<G> = 2.5$~kPa, \\
%          $Var[G]=0.15~{\rm kPa}^2$,\\ 
%          correlation length for $G=0.3$~m.
%   
% \vspace*{0.3cm}
%   \item  Elastic--plastic material model,\\
%          von Mises, linear hardening,\\
%          $<G> = 2.5$~kPa, \\
%          $Var[G]=0.15~{\rm kPa}^2$,\\ 
%          correlation length for $G=0.3$~m, \\
%          $C_u = 5$~kPa, \\
%          $C^{'}_u = 2$~kPa.
% 
% \end{itemize}
% 
% 
% %\begin{figure}[!hbpt]
% %\begin{center}
% \vspace*{-6cm}
% \hspace*{7.5cm}
% \includegraphics[height=2.5cm,angle=-90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/ShearBeamModel.pdf}
% %\end{center}
% %\end{figure}
% 
% % % 
% % \begin{center}
% % %\hspace*{-1.7cm}
% % \includegraphics[height=5.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_SchematicStaticProblem.jpg}
% % %\hspace*{0.7cm}
% % %\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Elastic-Edited.jpg} \\
% % %\vspace*{0.1truecm}
% % %%\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Base Displacement}
% % %\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_ElasticPlastic-Edited.pdf}
% % %\hspace*{0.0cm}
% % %\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/PushOverTestPDF.jpg}
% % \end{center}
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% 
% \frametitle{Linear Elastic FEM Verification}
% 
% 
% \vspace*{-0.5cm}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.8\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Elastic-Edited.pdf}
% \label{figure:StaticElasticSimulation}
% \end{center}
% \end{figure}
% \vspace*{-2.5cm}
% 
% Mean and standard deviations of displacement at the top node,\\
% linear elastic material model, \\
% KL-dimension=2, order of PC=2. 
% %Monte Carlo simulation
% %is also shown.
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% 
% \frametitle{SEPFEM verification}
% 
% 
% 
% %\vspace*{-0.2cm}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.75\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_ElasticPlastic-Edited.pdf}
% \label{figure:StaticElasticPlasticSimulation}
% \end{center}
% \end{figure}
% \vspace*{-0.2cm}
% 
% Mean and standard deviations of displacement at the top node,\\
% von Mises elastic-plastic linear hardening material model,\\ 
% KL-dimension=2, order of PC=2.
% % Monte Carlo
% %simulation is also shown}
% 
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % -- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % -- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % -- \begin{frame}
% % -- 
% % -- 
% % -- \frametitle{Stochastic Response at the Top}
% % -- 
% % -- 
% % -- 
% % -- 
% % -- 
% % -- \begin{figure}[!htbp]
% % -- \begin{center}
% % -- \includegraphics[width=0.65\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_PDFComparison-Edited.pdf}
% % -- \label{figure:StaticProblem_PDFComparison}
% % -- \end{center}
% % -- \end{figure}
% % -- 
% % -- 
% % -- Comparison of PDF of top node displacement.
% % -- 
% % -- 
% % -- 
% % -- \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- \begin{frame}
% %-- \frametitle{1--D Static Pushover Test Example}
% %-- 
% %-- 
% %-- 
% %-- \begin{figure}[!htbp]
% %-- \begin{center}
% %-- \includegraphics[width=0.35\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_CorLength_andKL_LargeCorLength_Mean-Edited.pdf}
% %-- \includegraphics[width=0.35\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_CorLength_andKL_LargeCorLength_SD-Edited.pdf}
% %-- 
% %-- \label{figure:CorLength_andKL_LargeCorLength_Mean}
% %-- \end{center}
% %-- \end{figure}
% %-- 
% %-- 
% %-- 
% %-- Correlation length and KL dimension: Mean and stadard deviation of displacement along depth of the 1--D soil column with
% %-- linear elastic material model, having very small variance (COV = 1\%) of shear modulus and very large 
% %-- ratio of correlation length of shear modulus to domain length (= 100)}
% %-- 
% %-- Correlation length and KL dimension: Standard deviation of displacement along depth of the 1--D 
% %-- soil column with linear elastic material model, having very small variance (COV = 1\%) of shear modulus 
% %-- and very large ratio of correlation length of shear modulus to domain length (= 100)}
% %-- 
% %-- \end{frame}
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- \begin{frame}
% %-- \frametitle{1--D Static Pushover Test Example}
% %-- 
% %-- 
% %-- 
% %-- \begin{figure}[!htbp]
% %-- \begin{center}
% %-- \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_CorLength_andKL_SmallCorLength_Mean-Edited.pdf}
% %-- \caption{Correlation length and KL dimension: Mean displacement along depth of the 1--D soil column with
% %-- linear elastic material model, having very small variance (COV = 1\%) of shear modulus and very small 
% %-- ratio of correlation length of shear modulus to domain length (= 0.0001)}
% %-- \label{figure:CorLength_andKL_SmallCorLength_Mean}
% %-- \end{center}
% %-- \end{figure}
% %-- 
% %-- 
% %-- 
% %-- \end{frame}
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- \begin{frame}
% %-- \frametitle{1--D Static Pushover Test Example}
% %-- 
% %-- 
% %-- 
% %-- \begin{figure}[!htbp]
% %-- \begin{center}
% %-- \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_CorLength_andKL_SmallCorLength_SD-Edited.pdf}
% %-- \caption{Correlation length and KL dimension: Standard deviation of displacement along depth of the 1--D 
% %-- soil column with linear elastic material model, having very small variance (COV = 1\%) of shear modulus 
% %-- and very small ratio of correlation length of shear modulus to domain length (= 0.0001)}
% %-- \label{figure:CorLength_andKL_SmallCorLength_SD}
% %-- \end{center}
% %-- \end{figure}
% %-- 
% %-- 
% %-- 
% %-- 
% %-- \end{frame}
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- \begin{frame}
% %-- \frametitle{1--D Static Pushover Test Example}
% %-- 
% %-- 
% %-- 
% %-- \begin{figure}[!htbp]
% %-- \begin{center}
% %-- \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Variance_andPC_LargeVariance_Mean-Edited.pdf}
% %-- \caption{Variance and order of PC: Mean displacement along depth of the 1--D soil column with
% %-- linear elastic material model, having large variance (COV = 20\%) of shear modulus and  
% %-- ratio of correlation length of shear modulus to domain length = 0.1}
% %-- \label{figure:Variance_andPC_LargeVariance_Mean}
% %-- \end{center}
% %-- \end{figure}
% %-- 
% %-- 
% %-- \end{frame}
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- \begin{frame}
% %-- \frametitle{1--D Static Pushover Test Example}
% %-- 
% %-- 
% %-- 
% %-- \begin{figure}[!htbp]
% %-- \begin{center}
% %-- \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Variance_andPC_LargVariance_SD-Edited.pdf}
% %-- \caption{Variance and order of PC: Standard deviation of displacement along depth of the 1--D 
% %-- soil column with linear elastic material model, having large variance (COV = 20\%) of shear modulus 
% %-- and ratio of correlation length of shear modulus to domain length = 0.1}
% %-- \label{figure:Variance_andPC_LargeVariance_SD}
% %-- \end{center}
% %-- \end{figure}
% %-- 
% %-- 
% %-- \end{frame}
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- \begin{frame}
% %-- \frametitle{1--D Static Pushover Test Example}
% %-- 
% %-- 
% %-- 
% %-- \begin{figure}[!htbp]
% %-- \begin{center}
% %-- \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Variance_andPC_SmallVariance_Mean-Edited.pdf}
% %-- \caption{Variance and order of PC: Mean displacement along depth of the 1--D soil column with
% %-- linear elastic material model, having very small variance (COV = 1\%) of shear modulus and  
% %-- ratio of correlation length of shear modulus to domain length = 0.1}
% %-- \label{figure:Variance_andPC_SmallVariance_Mean}
% %-- \end{center}
% %-- \end{figure}
% %-- 
% %-- 
% %-- 
% %-- \end{frame}
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %-- \begin{frame}
% %-- \frametitle{1--D Static Pushover Test Example}
% %-- 
% %-- 
% %-- 
% %-- \begin{figure}[!htbp]
% %-- \begin{center}
% %-- \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Variance_andPC_SmallVariance_SD-Edited.pdf}
% %-- \caption{Variance and order of PC: Standard deviation of displacement along depth of the 1--D 
% %-- soil column with linear elastic material model, having very small variance (COV = 1\%) of shear modulus 
% %-- and ratio of correlation length of shear modulus to domain length = 0.1}
% %-- \label{figure:Variance_andPC_SmallVariance_SD}
% %-- \end{center}
% %-- \end{figure}
% %-- 
% %-- % \begin{table}
% %-- % \begin{center}
% %-- % \caption{Comparison of results (at top node) of FPKE-based spectral stochastic finite element with
% %-- % direct spectral stochastic finite element, for 1--D soil column example, with linear elastic material}
% %-- % \label{Table:Comparison}
% %-- % \begin{tabular}{|c|ccc|ccc|}
% %-- %  \hline
% %-- % Load Step & \multicolumn{3}{c|}{Mean of solution} & \multicolumn{3}{c|}{Standard deviation of solution} \\
% %-- %  \cline{2-7}
% %-- % & Direct (mm) & FPKE (mm) & Error (\%) & Direct (mm) & FPKE (mm) & Error (\%) \\
% %-- %  \hline \hline
% %-- % 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
% %-- %  \hline
% %-- % 1 & 2.015 & 2.015 & 0 & 0.2005 & 0.2005 & 0 \\
% %-- %  \hline
% %-- % 2 & 4.03 & 4.00 & 0.7 & 0.40 & 0.39 & 0.7 \\
% %-- %  \hline
% %-- % 3 & 6.04 & 5.95 & 1.4 & 0.60 & 0.59 & 1.4 \\
% %-- %  \hline
% %-- % 4 & 8.06 & 7.89 & 2.1 & 0.80 & 0.78 & 2.1 \\
% %-- %  \hline
% %-- % 5 & 10.00 & 9.84 & 1.7 & 1.00 & 0.98 & 1.7 \\
% %-- %  \hline
% %-- % 6 & 12.09 & 11.88 & 1.7 & 1.20 & 1.21 & 0.6 \\
% %-- %  \hline
% %-- % \end{tabular}        
% %-- % \end{center}
% %-- % \end{table}
% %--  
% %-- 
% %-- %\begin{figure}[htbp]
% %-- %\begin{center}
% %-- %\includegraphics[width=\textwidth]{EffectOfCorLength_6_3_01}
% %-- %\caption{}
% %-- %\label{figure:EffectOfKL_on_CorLength_Mean}
% %-- %\end{center}
% %-- %\end{figure}
% %-- %
% %-- %
% %-- %\begin{figure}[htbp]
% %-- %\begin{center}
% %-- %\includegraphics[width=\textwidth]{EffectOfCorLength_6_3_02}
% %-- %\caption{}
% %-- %\label{figure:EffectOfKL_on_CorLength_Mean}
% %-- %\end{center}
% %-- %\end{figure}
% %-- %
% %-- %\begin{figure}[htbp]
% %-- %\begin{center}
% %-- %\includegraphics[width=\textwidth]{EffectOfCorLength_6_3_03}
% %-- %\caption{}
% %-- %\label{figure:EffectOfKL_on_CorLength_Mean}
% %-- %\end{center}
% %-- %\end{figure}
% %-- %
% %-- %\begin{figure}[htbp]
% %-- %\begin{center}
% %-- %\includegraphics[width=\textwidth]{EffectOfCorLength_6_3_04}
% %-- %\caption{}
% %-- %\label{figure:EffectOfKL_on_CorLength_Mean}
% %-- %\end{center}
% %-- %\end{figure}
% %-- %
% %-- %
% %-- %\begin{figure}[htbp]
% %-- %\begin{center}
% %-- %\includegraphics[width=\textwidth]{EffectOfCorLength_4_4_01}
% %-- %\caption{}
% %-- %\label{figure:EffectOfKL_on_CorLength_Mean}
% %-- %\end{center}
% %-- %\end{figure}
% %-- %
% %-- %\begin{figure}[htbp]
% %-- %\begin{center}
% %-- %\includegraphics[width=\textwidth]{EffectOfCorLength_4_4_02}
% %-- %\caption{}
% %-- %\label{figure:EffectOfKL_on_CorLength_Mean}
% %-- %\end{center}
% %-- %\end{figure}
% %-- %
% %-- %
% %-- %\begin{figure}[htbp]
% %-- %\begin{center}
% %-- %\includegraphics[width=\textwidth]{EffectOfCorLength_4_4_03}
% %-- %\caption{}
% %-- %\label{figure:EffectOfKL_on_CorLength_Mean}
% %-- %\end{center}
% %-- %\end{figure}
% %-- %
% %-- %
% %-- %\begin{figure}[htbp]
% %-- %\begin{center}
% %-- %\includegraphics[width=\textwidth]{EffectOfKL_on_CorLength_Mean}
% %-- %\caption{}
% %-- %\label{figure:EffectOfKL_on_CorLength_Mean}
% %-- %\end{center}
% %-- %\end{figure}
% %-- %
% %-- %
% %-- %\begin{figure}[htbp]
% %-- %\begin{center}
% %-- %\includegraphics[width=\textwidth]{EffectOfKL_on_CorLength_SD}
% %-- %\caption{}
% %-- %\label{figure:EffectOfKL_on_CorLength_SD}
% %-- %\end{center}
% %-- %\end{figure}
% %-- %
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\section{An Application}


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\subsection{Seismic Wave Propagation Through Uncertain Soils}


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% \vspace*{0.3cm}
% \item Probabilistic inverse problems
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% \item Geotechnical site characterization design
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% - + 
% - + \frametitle{Random Field Modeling of Uncertain Soil Properties}
% - + 
% - + \begin{itemize}
% - + 
% - + \item Finite scale model 
% - + 
% - + \begin{itemize}
% - + 
% - + \item Short memory, finite correlation length
% - + 
% - + \item Common autocovariance model $\rightarrow$ exponential, spherical, triangular, linear--exponential
% - + 
% - + \end{itemize}
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% - + \item Fractal model 
% - + 
% - + \begin{itemize}
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% - + \item long memory, infinite correlation length $\rightarrow$ more realistic for modeling horizontal 
% - + spatial uncertainty
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% - + \item 1/f-type noise process with power spectral density, $P(\omega)~=~P_0~\omega^{-\gamma}$, with
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\frametitle{Seismic Wave Propagation through Stochastic Soil}

%\begin{flushleft}
%\includegraphics[height=5.0cm]{PEER2007_3.jpg}
%\end{flushleft}
%\vspace*{-0.5truecm}

\begin{itemize}

\item Soil as 12.5 m deep 1--D soil column (von Mises Material) 

\begin{itemize}

\item Properties (including testing uncertainty) obtained through random field modeling of CPT $q_T$
%
$\left<q_T\right> = 4.99 ~MPa;~~Var[q_T] = 25.67 ~MPa^2; $\\
Cor. ~Length $[q_T] = 0.61 ~m; $ Testing~Error $= 2.78 ~MPa^2$

\end{itemize}

\vspace*{0.2cm}
\item $q_T$ was transformed to obtain $G$: ~~$G/(1-\nu)~=~2.9q_T$ 

\begin{itemize}
\item Assumed transformation uncertainty = 5\%
%
$\left<G\right> = 11.57MPa; Var[G] = 142.32 MPa^2$ \\
Cor.~Length $[G] = 0.61 m$

\end{itemize}


%\begin{center}
%\hspace*{-1.7cm}
%\includegraphics[height=3.5cm]{Chapter9_Schematic.jpg}
%\hspace*{0.0cm}
%\includegraphics[height=3.5cm]{Chapter9_BaseDisplacement.jpg} \\
%\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Base Displacement}
%\end{center}

\vspace*{0.2cm}
\item Input motions: modified 1938 Imperial Valley

% \vspace*{-0.2cm}
% \begin{center}
% \includegraphics[height=2.0cm]{Chapter9_BaseDisplacement.jpg} 
% \end{center}

\end{itemize}

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\frametitle{Random Field Parameters from Site Data}

\begin{itemize}

%\item maximizing the log--likelihood of observing the spatial data under assumed joined distribution (for finite 
%scale model) or maximizing the log--likelihood of observing the periodogram estimates (for fractal model)

\item Maximum likelihood estimates

\vspace*{0.3truecm}

%\begin{figure}
\begin{flushleft}
\hspace*{-1.7cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/SamplingPlan-Edited.jpg}
\hspace*{0.0cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalDataPlotBH1-Edited.jpg} \\
\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Typical CPT $q_T$}
\end{flushleft}
%\end{figure}

\vspace*{-4.9truecm}

%\begin{figure}
\begin{flushright}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScale-Edited.jpg} \\
\vspace*{0.01truecm}
\small{Finite Scale}
\end{flushright}
%\end{figure}

\vspace*{0.02truecm}

%\begin{figure}
\begin{flushright}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_Fractal-Edited.jpg} \\
\small{Fractal}
\end{flushright}
%\end{figure}

\end{itemize}

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\frametitle{``Uniform'' CPT  Site Data}


\vspace*{-0.7cm}
%\begin{figure}
\begin{center}
\includegraphics[height=6.7cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/EastWestProfile-Edited.pdf}
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%\end{figure}


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%--  \subsection{Seismic Wave Propagation Through Uncertain Soils}



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\frametitle{Surface Displacement Time History}

\vspace*{-12.0truecm}

%\hspace*{-0.7cm} \uncover<1->{\includegraphics[width=12.0cm]{Chapter9_ElasticPlasticResponse_Mean-New.pdf}} 
%%
%\hspace*{-6.0cm} \uncover<2->{\includegraphics[width=12.0cm]{Chapter9_ElasticPlasticResponse_SD-New.pdf}} \hspace*{-2.0cm} \\
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\frametitle{Mean $\pm$ Standard Deviation}

\vspace*{-19.0truecm}

\hspace*{0.15truecm} \includegraphics[width=19.0cm]{Chapter9_ElasticPlasticResponse-New.pdf}

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%\frametitle{Evolution of Statistics of Top Node Displacement}
%
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%
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%
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%\tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Mean~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Standard Deviation}
%
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%
%
%\hspace*{-0.05cm} \includegraphics[width=9.0cm]{Chapter9_ElasticPlasticResponse-New.pdf}
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%
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%\tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~Mean$\pm$ Standard Deviation~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~COV}
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\begin{frame}
\frametitle{PDF of Surface Displacement Time History}

\begin{itemize}

\item PDF at the finite \\
element nodes can be \\
obtained using, e.g., \\
Edgeworth expansion \\
(Ghanem and Spanos \\ 
2003)

\vspace*{0.1truecm}

\item Numerous applications,  \\
especially where extreme \\
statistics are critical

\end{itemize}

\vspace*{-11.0cm}
\hspace*{5.5truecm} \includegraphics[height=12.0cm]{Chapter9_ElasticPlasticResponse_PDF-New-Edited.pdf} \hspace*{-2.0cm}


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\begin{frame}
\frametitle{Most Probable Solution}

\vspace*{-10.25cm}

\includegraphics[height=16.0cm]{SamplePDF_DynamicProblem_Mode.pdf}  \hspace*{-5.0truecm} At $t~=~1.1$ sec \hspace*{-2.0truecm}

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%\frametitle{Sensitivity Analysis}
%
%\vspace*{-10.25cm}
%
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\frametitle{Tails of PDF}

\vspace*{-10.25cm}

\includegraphics[height=16.0cm]{SamplePDF_DynamicProblem_TailsOfPDF.pdf}  \hspace*{-5.0truecm} At $t~=~1.1$ sec \hspace*{-2.0truecm}

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\begin{frame}
\frametitle{Probability of Exceedance}

\vspace*{-10.25cm}

\includegraphics[height=16.0cm]{SampleCDF_DynamicProblem.pdf}  \hspace*{-5.0truecm} At $t~=~1.1$ sec \hspace*{-2.0truecm}

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\begin{frame}
\frametitle{Derivative Applications}

\vspace*{-0.25truecm}

\begin{itemize}

\item Performance based engineering

\begin{itemize}

\item Reliability index ($\beta$)

\item Probability of failure ($p_f$)

\end{itemize}

\vspace*{-12.0truecm}

\hspace*{-1.0truecm} \includegraphics[height=15.0cm]{SchematicReliability.pdf} \hspace*{-1.5truecm}

\vspace*{0.5truecm}

\item Financial risk analysis

\item Sensitivity analysis

%\vspace*{-8.0truecm}

%\hspace*{4.0truecm} \includegraphics[angle=90,height=10.0cm]{SchematicFinancialRisk.pdf} \hspace*{-1.5truecm}

\end{itemize}

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% 
% 
% 
% \begin{figure}
% \begin{center}
% \hspace*{-0.75cm}
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter9Plots/Chapter9_ElasticPlasticResponse-New.pdf}
% %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% \end{center}
% \end{figure}
% 
% Mean$\pm$ Standard Deviation
% 
% 
% 
% %\begin{flushleft}
% %\includegraphics[height=5.0cm]{PEER2007_3.jpg}
% %\end{flushleft}
% 
% % \hspace*{-1.0cm} \noindent Statistics of Top Node Displacement:
% % 
% % \vspace*{-0.5truecm}
% % 
% % \begin{figure}
% % \begin{flushleft}
% % \hspace*{-1.0cm}
% % \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_Mean-New.jpg}
% % \hspace*{-0.1cm}
% % \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_SD-New.jpg}
% % \end{flushleft}
% % \end{figure}
% % \vspace*{-0.5truecm}
% % \hspace*{-1.0cm} \tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~~~Mean~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Standard Deviation}
% % 
% % \vspace*{-0.3truecm}
% % 
% % \begin{figure}
% % \begin{flushleft}
% % \hspace*{-0.75cm}
% % \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse-New.jpg}
% % \hspace*{0.4cm}
% % \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_COV-New.jpg}
% % \end{flushleft}
% % \end{figure}
% % \vspace*{-0.3truecm}
% % \hspace*{-0.5cm} \tiny{~~~~~~~Mean$\pm$ Standard Deviation~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~COV}
% % 
% % 
% % \vspace*{-6.0cm}
% % \begin{flushright}
% % \includegraphics[height=4.5cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_PDF-New-Edited.jpg} \hspace*{-1.0cm}
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\section{Summary}

\begin{frame}


\frametitle{Summary}

\begin{itemize}

\item  Work  on  probabilistic  elasto--plasticity  as continuation of work with
Stein Sture at CU Bolder

\vspace*{0.5cm}
\item  Getting  older,  supposed  to  have  more  answers,  in  fact I have more
questions



\end{itemize}


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\section{Summary}

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\frametitle{}


\begin{center}
\vspace*{1cm}
 {\Huge Thank you Stein!}
\end{center}

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