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% % (used in a file _Chapter_SoftwareHardware_Domain_Specific_Language_English.tex
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%  \usetheme{Antibes}  % ima sadrzaj gore i kao graf ...
%  \usetheme{Berkeley} % ima sadrzaj desno
%  \usetheme{Berlin} % ima sadrzaj gore i tackice
%  \usetheme{Goettingen} % ima sadrzxaj za desne strane
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%      Site Specific Dynamics of Structures: 
%From Seismic Source to 
%the Safety of Occupants and Content


\title[]
        {Probabilistic Seismic Risk Analysis for\\
         Inelastic Soil-Structure Systems  }


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

%\author[Author, Another] % (optional, use only with lots of authors)
%{F.~Author\inst{1} \and S.~Another\inst{2}}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.
\pgfdeclareimage[height=0.2cm]{university-logo}{/home/jeremic/BG/amblemi/ucdavis_logo_blue_sm}
\pgfdeclareimage[height=0.7cm]{lbnl-logo}{/home/jeremic/BG/amblemi/lbnl-logo}


%\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
%{Boris~Jeremi{\'c}}
{Boris Jeremi{\'c} 
\\ \vspace*{1mm}
Hexiang Wang, Han Yang}


%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
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%{ Professor, University of California, Davis\\
{ University of California, Davis}
% %  and\\
% %  Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley }
%   Lawrence Berkeley National Laboratory, Berkeley, CA}
% % - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.


\date[] % (optional, should be abbreviation of conference name)
{\small UCLA Lifelines Conference  \\
        Jan/Feb 2022}           

\subject{}
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% may not be suitable. Here are some rules that apply for this
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% - 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*!
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%   enough. Leave out details, even if it means being less precise than
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%   just say so once. Everybody will be happy with that.



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\section{Introduction}




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%\subsection{Motivation}
\subsection{\ }



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\begin{frame}
 \frametitle{Motivation}

\begin{itemize}



 

%\vspace*{0.3cm}
  \item[]  Improve  modeling  and  simulation  of infrastructure objects
 

% \vspace*{2mm}
%  \item[]  Expert numerical modeling and simulation tool
% 
% \vspace*{1mm}
%   \item[]      Use of numerical models to
%   analyze statics and dynamics of soil/rock-structure systems
% 

\vspace*{4mm}
  \item[] Modeling, epistemic uncertainty

\vspace*{4mm}
  \item[] Parametric, aleatory  uncertainty



\vspace*{4mm}
  \item[] Goal is to  Predict and Inform 
% rather than (force) fit

%\vspace*{4mm}
%  \item[] Engineer needs to know!



%
%  
%  
%  \vspace*{1mm}
%    \item[]  Follow  the  flow, input and dissipation,  of seismic  energy, 

%  \vspace*{2mm}
%    \item[]  
%  %System  for 
%   {\bf Real}istic modeling and simulation of
%    {\bf E}arthquakes and/or    
%    {\bf S}oils and/or    
%    {\bf S}tructures    and    their    
%    {\bf I}nteraction:\\
%     Real-ESSI 
%   \hspace*{5mm} 
%    \url{http://real-essi.info/} 
%  % % %  \hspace*{25mm} 
%  %     \url{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
%  % %    \href{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}{{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
%  % %  % \url{http://ms-essi.info/}
%  % %  
% 




\end{itemize}
\end{frame}
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%       
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{Parametric, Aleatory Uncertainty}
\frametitle{Aleatory Uncertainties, Material, Motions}



\vspace*{2mm}
%\vspace*{-5mm}
\begin{figure}[!hbpt]
\begin{center}
%
\hspace*{-7mm}
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01-Ed.pdf}
\hspace*{-3mm}
% \hfill
\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_Histogram_Normal_01-Ed.pdf}
%
\end{center}
\end{figure}


\vspace*{-5mm}
%\vspace*{-1.8cm}
%\hspace*{-3.3cm}
\begin{flushright}
{\tiny
(cf. Phoon and Kulhawy (1999B))\\
~}
\end{flushright}
%





   

\vspace*{-9mm}
\begin{figure}[!hbpt]
\begin{center}
%
%\hspace*{-7mm}
\includegraphics[width=5.00truecm]{/home/jeremic/tex/works/Thesis/HexiangWang/time_series_motionsn_06ug2019_SMIRT/Acc_realization_200.pdf}
%\hspace*{-3mm}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion70.pdf}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion100.pdf}
%% \includegraphics[width=0.31\textwidth]{Figures/Acc_time_series_realiztion350.pdf}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion367.pdf}
\includegraphics[width=4cm]{/home/jeremic/tex/works/Papers/2019/1D_risk/version2/Figures/SA_GMPE_verification_std_08_no_smooth.pdf}
%
\end{center}
\end{figure}


\vspace*{-7mm}
\begin{flushright}
{\tiny
(cf. Wang et al. (2019))\\
~}
\end{flushright}








\end{frame}















      
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\section{Probabilistic Seismic Risk}

      
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%\subsection{Stochastic Modeling}
\subsection{Uncertainty Propagation}



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\begin{frame}
\frametitle{Forward Uncertainty Propagation}

\begin{itemize}

\vspace*{3mm}
  \item[-] Given uncertain material

\vspace*{3mm}
  \item[-] Given uncertain loads

\vspace*{3mm}
  \item[-] Determine uncertain response, $u_i, \dot{u}_i, \ddot{u}_i,
  \epsilon_{ij}, \sigma_{ij}$, PDFs/CDFs

\vspace*{3mm}
  \item[-] Direct, intrusive, analytic development
  
\vspace*{3mm}
  \item[-] Circumvent Monte Carlo inefficiencies, inaccuracies
 
\end{itemize}


\end{frame}




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% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame} 
% 
% \frametitle{Forward Uncertain Inelasticity}
% 
% % 
% 
% \begin{itemize}
% 
% 
% 
% \item[-] Incremental el--pl constitutive equation
% %  
%     \begin{eqnarray}
%     \nonumber
%   \Delta \sigma_{ij} 
%   = 
% %  E^{EP}_{ijkl} 
%      E^{EP}_{ijkl} \; \Delta \epsilon_{kl}  
%      =  
%      \left[
%      E^{el}_{ijkl}
%        -
%        \frac{\displaystyle  E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
%            {\displaystyle n_{rs} E^{el}_{rstu} m_{tu} - \xi_* h_*}
%            \right]
%     \Delta \epsilon_{kl}
%     \end{eqnarray}
% 
%  
%   
% 
% \vspace*{2mm}
% \item[-] Dynamic Finite Elements
% %
% \begin{equation}
% { M} \ddot{ u_i} +
% { C} \dot{ u_i} +
% { K}^{ep} { u_i} =
% { F(t)}
%     \nonumber
% \end{equation}
% 
% 
% \vspace*{2mm}
% \item[-] Material and loads are uncertain
% 
% 
% 
% \end{itemize}
% 
% 
% 
% \end{frame}
% 







  

% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame} 
%  
%  \frametitle{Probabilistic Elastic-Plastic 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}
%  \includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOE-LLNL-workshop-27-28-Feb-2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDF-edited.pdf}
%  \end{center}
%  \end{figure}
%  
%  \end{frame}
%  








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\begin{frame} 

\frametitle{{Cam Clay  with  Random $G$, $M$ and $p_0$}}


\begin{figure}[!hbpt]
\begin{center}
\hspace*{-15mm}
\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0-m.pdf}
%\hspace*{-2mm}
\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomM-m.pdf}
\hspace*{-15mm}
\end{center}
\end{figure}


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\end{frame}
% --  %%%%%%









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


\vspace*{2mm}
Dynamic Finite Elements 
$
{ M} \ddot{ u_i} +
{ C} \dot{ u_i} +
{ K}^{ep} { u_i} =
{ F(t)}$


\begin{itemize}
\vspace*{2mm}
\item[-] Input random field/process{\normalsize{(non-Gaussian, heterogeneous/ non-stationary)}}:
          Multi-dimensional Hermite Polynomial Chaos (PC) with {known coefficients}
%\vspace{0.05in}
\vspace*{2mm}
\item[-] Output response process: Multi-dimensional Hermite PC with {unknown coefficients}
% \vspace{0.05in}
\vspace*{2mm}
 \item[-] Galerkin projection: minimize the error to compute unknown coefficients of response process
% %\vspace{0.05in}
% \vspace*{2mm}
%  \item[-] Time integration using Newmark's method
% % : Update coefficients following
% % an elastic-plastic constitutive law at each time step
 
\end{itemize}

%\scriptsize 
%Note: PC = Polynomial Chaos

\end{frame}





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

%\frametitle{State of the Art Probabilistic Seismic Risk Analysis}
\frametitle{Probabilistic Seismic Risk Analysis}

\begin{itemize}

% \item Target: safe design, acceptably small failure probability $\lambda (EDP)$

\item[-] 
%Powerful tool 
%Allows for 
Objective, quantitative decision making based on exceedance rate
$\lambda (EDP>z)$

\item[-] PSRA: convolution of PSHA and fragility 

% \[\lambda(EDP>z) = \int_{IM} \underbrace{|\frac{d\lambda(IM)}{dIM}|}_\text{PSHA} \underbrace{G(EDP|IM)}_{\text{fragility}} dIM \]

\vspace{-0.1cm}

\[\lambda(EDP>z) = \int \underbrace{|\frac{d\lambda(IM>x)}{dx}|}_\text{\textbf{PSHA}} \underbrace{G(EDP>z|IM=x)}_{\text{\textbf{fragility analysis}}} dx\]

\small{$\lambda(\cdot)$ : rate of exceedance\\
\vspace{0.07cm}
$EDP$: engineering demand parameter\\
\vspace{0.07cm}
$PSHA$: probabilistic seismic hazard analysis\\
\vspace{0.07cm}
$IM$: intensity measure, choice to be made (!)  } 

\end{itemize}

\end{frame}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% \begin{frame}
% \frametitle{Intensity Measure (IM)}
% IM serves as the proxy of damaging ground motions
% 
% \vspace{0.3cm}
% 
% \begin{itemize}
% \item[-] Does a single IM, e.g., $Sa(T_0)$, represent all uncertainty? 
% %%  influencing EDP? 
% %\begin{itemize}
% %\item[--] \small Structure nonlinearity 
% %% \item[--] \small Liquefaction: PGA and duration
% %\item[--] \small Higher mode response
% %\end{itemize}
% 
% \vspace{3mm}
% 
% %\item[-] Practically difficult, contentious to choose 
% \item[-] IM choice sometimes contentious 
% 
% % \begin{itemize}
% % % \item[--] \small Geo-hazard: Liquefaction, slope deformation
% % % \item[--] \small PGA v.s. AI v.s. RMS for liquefaction
% % \item[--] \small AI v.s. PGV v.s. CAV for dam embankment
% % \end{itemize}
% 
% \vspace{3mm}
% 
% % \item Additional effort for new GMPEs 
% 
% % \begin{itemize}
% % \item[--] \small vector hazard: GMPE with covariance of IMs, fragility as function of IMs, rarely used  
% % \end{itemize}
% 
% % \item[-] Miscommunication: seismologists and engineers
% % 
% % \begin{itemize}
% % \item[] \small $Sa(T_0)$ not compatible with time domain nonlinear analysis
% % \end{itemize}
% 
% \end{itemize}
% \end{frame}
% 
% 







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\subsection{Risk Analysis Example}
%  %\subsection{TDNIPSRA Formulation}






















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% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame} 
% \frametitle{Current State of Art Seismic Risk Analysis (SRA)}
% 
% 
% \begin{itemize}
% %\vspace{2mm}
% 
% \item[-] Intensity measure (IM) selected as a proxy for ground motions, 
% usually Spectral acceleration $Sa(T_0)$
% 
% \vspace{4mm}
% \item[-] Ground Motion Prediction Equations (GMPEs) need development, ergodic or site specific 
% 
% \vspace{4mm}
% \item[-] Probabilistic seismic hazard analysis (PSHA)
% % for ground motion $\lambda(Sa>z)$
% % \begin{equation*}
% % \resizebox{0.85\hsize}{!}{%
% % $\lambda(Sa>z) = \sum_{i=1}^{NFL} \underbrace{N_i \int\int f_{mi}(M) f_{ri}(R|M)}_\text{seismic source characterization (SSC)} \underbrace{P(Sa>z|M, R)}_\text{GMPE} dM dR$}
% % \end{equation*}
% 
% \vspace{4mm}
% \item[-] Fragility analysis $P(EDP>x|IM=z)$, deterministic time domain FEM,
% perhaps using Monte Carlo (MC)
% 
% %  \begin{itemize}
% % 
% %     \item[-] Records selection: Spectrum-matching technique UHS, etc   
% % 
% %     \item[-] Incremental dynamic analysis: Monte Carlo 
% % 
% %   \end{itemize}
% 
% 
% 
% \end{itemize}
% 
% % \begin{textblock}{15}(2.2, 9.2)
% % \begin{figure}[H]
% % \flushleft
% % % \includegraphics[width=0.38\linewidth]{pic/hazard_curve.png}
% % \includegraphics[width=0.38\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/hazard_curve.pdf}
% % \enspace
% % \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/design_spectra.png}
% % \end{figure}
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% \subsection{Issues in State-of-the-art SRA}

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%  \frametitle{Seismic  Risk Analysis Challenges}
%  
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%  \begin{itemize}
%  
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%  \item[-] IM serves as the proxy of damaging ground motions
%  \vspace{2mm}
%  \item[-] Does a single IM, e.g., $Sa(T_0)$, represent all uncertainty? 
%  %\item[-] Practically difficult/contentious to choose 
%  
%  
%  %\vspace{3mm}
%  \vspace{2mm}
%  \item[-] IMs difficult to choose, Spectral Acc, PGA, PGV...
%  
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%  %%\vspace{3mm}
%  %\item[-] Single IM does not contain all/most uncertainty
%  
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%  \vspace{2mm}
%  \item[-] Fragility analysis: incremental dynamic analysis (IDA)
%  % using Monte Carlo method
%  
%  \vspace{2mm}
%  \item[-] Use of Monte Carlo method, accuracy, efficiency...
%  
%  %\vspace{3mm}
%  \vspace{2mm}
%  \item[-] Monte Carlo, computationally expensive, CyberShake for LA, 20,000
%  cases, 100Y runtime, (Maechling et al. 2007)
%  
%  % 
%  % 
%  % \vspace{3mm}
%  % \item[-] Miscommunication between seismologists and struct/geotech engineers,
%  % $Sa(T_0)$ not compatible with nonlinear FEM
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%  \frametitle{Time Domain Intrusive PSRA Framework}
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%  \begin{itemize}
%  
%  %\vspace*{2mm}
%  \item[-] Stochastic Elastic-Plastic Finite Element Method, SEPFEM, 
%   ${M}  \ddot{u_i}  +  {C}  \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$, 
%   (Sett et al. 2011)
%   
%  
%  \vspace*{4mm}
%  \item[-] Uncertain elastic-plastic material
%  %stress and stiffness solution using
%  %Forward Kolmogorov, Fokker-Planck equation
%  
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%  \vspace*{4mm}
%  \item[-] Uncertain seismic loads/motions
%  % using Domain Reduction Method
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%  \vspace*{4mm}
%  \item[-] Results, probability distribution functions for  $\sigma_{ij}$, 
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%  \begin{itemize}
%  
%  %\item[-] Material uncertainties:  expanded along stochastic shape functions: 
%  \item[-] Material uncertainties: stochastic shape functions: 
%  $E^{ep}(x,t,\theta) = \sum_{i=0}^{P_d} E_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
%  
%  \vspace*{1mm}
%  \item[-] Loading uncertainties: stochastic shape functions
%  $F(x,t,\theta) = \sum_{i=0}^{P_f} F_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
%  
%  \vspace*{1mm} 
%  \item[-] Displacement expanded: stochastic shape functions:  
%  $u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
%  
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%  \vspace*{1mm} 
%  \item[-]
%  Stochastic system of equations 
%  \vspace*{-2mm}
%  \begin{tiny}
%  \[
%  \begin{bmatrix}
%      \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)}    & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
%      \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)}    & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\  \\
%      \vdots & \vdots & \vdots & \vdots\\
%      \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)}     & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
%  \end{bmatrix}
%  \begin{bmatrix}
%      u_{10} \\
%      \vdots \\
%      u_{N0}\\
%      \vdots \\
%      u_{1P_u}\\
%      \vdots \\
%      u_{NP_u}
%  \end{bmatrix}
%  =
%  %\]
%  %\[
%  \begin{bmatrix}
%       \sum_{i=0}^{P_f} f_i  <\Psi_0\zeta_i> \\
%      \sum_{i=0}^{P_f} f_i   <\Psi_1\zeta_i> \\
%     \sum_{i=0}^{P_f} f_i    <\Psi_2\zeta_i> \\
%      \vdots \\
%       \sum_{i=0}^{P_f} f_i  <\Psi_{P_u}\zeta_i>\\
%  \end{bmatrix}
%  \]
%  \end{tiny}
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%  % \begin{frame}
%  % \frametitle{Stochastic Elastic-Plastic Finite Element Method}
%  % %\frametitle{SEPFEM : Formulation}
%  % 
%  % Stochastic system of equations 
%  % 
%  % \begin{tiny}
%  % \[
%  % \begin{bmatrix}
%  %     \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)}    & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
%  %     \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)}    & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\  \\
%  %     \vdots & \vdots & \vdots & \vdots\\
%  %     \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)}     & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
%  % \end{bmatrix}
%  % \begin{bmatrix}
%  %     u_{10} \\
%  %     \vdots \\
%  %     u_{N0}\\
%  %     \vdots \\
%  %     u_{1P_u}\\
%  %     \vdots \\
%  %     u_{NP_u}
%  % \end{bmatrix}
%  % =
%  % %\]
%  % %\[
%  % \begin{bmatrix}
%  %      \sum_{i=0}^{P_f} f_i  <\Psi_0\zeta_i> \\
%  %     \sum_{i=0}^{P_f} f_i   <\Psi_1\zeta_i> \\
%  %    \sum_{i=0}^{P_f} f_i    <\Psi_2\zeta_i> \\
%  %     \vdots \\
%  %      \sum_{i=0}^{P_f} f_i  <\Psi_{P_u}\zeta_i>\\
%  % \end{bmatrix}
%  % \]
%  % \end{tiny}
%  % 
%  % 
%  % 
%  % %   \normalsize{Typical number of terms required for a SEPFEM problem} \vspace{1cm}\\
%  %   \scalebox{0.7}{
%  %   \begin{tabular}{ c c c c}
%  %    \# KL terms material & \# KL terms load & PC order displacement& Total \# terms per DoF\\ \hline
%  %    4 & 4 & 10 & 43758 \\  
%  %    4 & 4 & 20 & 3 108 105 \\    
%  % %   4 & 4 & 30 & 48 903 492 \\  
%  %    6 & 6 & 10 & 646 646 \\    
%  % %   6 & 6 & 20 & 225 792 840 \\  
%  % %   6 & 6 & 30 & 1.1058 $10^{10}$ \\    
%  % %   8 & 8 & 10 & 5 311 735 \\  
%  % %   8 & 8 & 20 & 7.3079 $10^{9}$ \\    
%  % %   8 & 8 & 30 & 9.9149 $10^{11}$\\  
%  %    
%  %    ...  & ... & ... & ...\\    \hline
%  %   \end{tabular}}
%  % 
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%  %  \begin{frame}
%  %  
%  %  \frametitle{Monte Carlo (MC)}
%  %  
%  %  \begin{itemize}
%  %  
%  %  \item Monte Carlo simulations: non-intrusive approach 
%  %  
%  %  \begin{itemize}
%  %  \item [--] \small Slow convergence rate $1/\sqrt{N}$
%  %  \item [--] \small Hard for stable tail distribution toward low-risk level 
%  %  \end{itemize}
%  %  
%  %  \item Fragility curve: incremental dynamic analysis (IDA)  
%  %  
%  %  \begin{itemize}
%  %  \item [--] \small Impractical for large $3D$ nonlinear ESSI system 
%  %  \end{itemize}
%  %  
%  %  \item Uncertain seismic wave propagation over regional geology  
%  %  
%  %  \begin{itemize}
%  %  \item [--] \small CyberShake from SCEC 
%  %  \item [--] \small Los Angeles, over 20,000 scenarios within 200 km, \textbf{300 million CPU-hours and over 100 years} (Maechling et al. 2007)
%  %  \end{itemize}
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% 
% %\subsection{TDNIPSRA Example}
% \subsection[TDNIPSRA Example]{Example}






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\begingroup

\setbeamertemplate{footline}{}

\begin{frame}

%\frametitle{TDNIPSRA Framework}
\frametitle{Application: Seismic Hazard}

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

\begin{textblock}{15}(0, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
\end{textblock}

\begin{textblock}{15}(0.3, 3.5)
\scriptsize{Seismic source characterization}
\end{textblock}

\begin{textblock}{15}(2.9, 5.2)
\tiny{UCERF3 (2014)}
\end{textblock}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(5.1, 6.5)
%$\Rightarrow$
{\Large $\rightarrow$}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{textblock}{15}(5.8, 3.9)
\vspace*{1mm}
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
\end{textblock}


\begin{textblock}{15}(7.1, 6.2)
\scalebox{.9}{\tiny{Fourier spectra}}
\\
\vspace*{-0.2cm}
\scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
\end{textblock}

\begin{textblock}{15}(6.1, 3.5)
\scriptsize{Stochastic ground motion}
\end{textblock}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(9.9, 6.5)
%{\bf $\Rightarrow$}
{\Large $\rightarrow$}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{textblock}{15}(10.5, 4.2)
% \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
\end{textblock}

\begin{textblock}{15}(11.1, 9.6)
\scriptsize{Uncertainty characterization \\
\hspace{0.1cm} Hermite polynomial chaos}
\end{textblock}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(10.2, 13.2)
{\Large $\leftarrow$}
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\begin{textblock}{15}(11, 11.2)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
\end{textblock}

\begin{textblock}{15}(5.3, 10.75)
\includegraphics[width=0.33\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/probabilsitc_evolution.png}
\end{textblock}

\begin{textblock}{15}(5.4, 9.6)
\scriptsize{\quad \quad Uncertainty propagation \\ 
\quad \quad \quad \quad SEPFEM}
\end{textblock}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(4.6, 13.2)
%$\Leftarrow$
{\Large $\leftarrow$}
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\begin{textblock}{15}(0.3, 11.0)
\includegraphics[width=0.29\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/seismic_risk_result_framework.png}
\end{textblock}

\begin{tikzpicture}[remember picture, overlay]
\draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
([xshift=-25pt,yshift=-55pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=-18pt]$(pic cs:b)+(0pt,-2pt)$);
\end{tikzpicture}


\begin{textblock}{15}(-0.1, 9.3)
\scriptsize
\quad \quad \quad \quad $\lambda(EDP>z)=$

$\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
\end{textblock}

\begin{textblock}{15}(1.6, 10.7)
\scriptsize{EDP hazard/risk}
\end{textblock}

\end{frame}

\endgroup
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% \begingroup
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% \setbeamertemplate{footline}{}
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% \frametitle{Time Domain Intrusive PSRA Framework}
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% 
% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
% 
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
% 
% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
% 
% 
% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
% \end{textblock}
% 
% 
% \begin{textblock}{15}(7.1, 6.2)
% \scalebox{.9}{\tiny{Fourier spectra}}
% \\
% \vspace*{-0.2cm}
% \scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
% \end{textblock}
% 
% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
% 
% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \end{textblock}
% 
% \begin{textblock}{15}(10.5, 4.2)
% % \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(11.1, 9.6)
% \scriptsize{Uncertainty characterization \\
% \hspace{0.1cm} Hermite polynomial chaos}
% \end{textblock}
% 
% \begin{textblock}{15}(11, 11.2)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
% \end{textblock}
% 
% % \begin{textblock}{15}(10.2, 13.2)
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% % \end{textblock}
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% % \begin{textblock}{15}(5.4, 9.6)
% % \scriptsize{\quad \quad Uncertainty propagation \\ 
% % \quad \quad \quad \quad stochastic FEM}
% % \end{textblock}
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% % \begin{textblock}{15}(4.6, 13.2)
% % $\Leftarrow$
% % \end{textblock}
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% % \begin{textblock}{15}(0.3, 11.0)
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% % \draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
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% 
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% % \begin{textblock}{15}(-0.1, 9.3)
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% % \quad \quad \quad \quad $\lambda(EDP>z)=$
% 
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% % \end{textblock}
% 
% % \begin{textblock}{15}(1.6, 10.7)
% % \scriptsize{EDP hazard/risk}
% % \end{textblock}
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% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
% 
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
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% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
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% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
% \end{textblock}
% 
% 
% \begin{textblock}{15}(7.1, 6.2)
% \scalebox{.9}{\tiny{Fourier spectra}}
% \\
% \vspace*{-0.2cm}
% \scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
% \end{textblock}
% 
% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
% 
% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \end{textblock}
% 
% \begin{textblock}{15}(10.5, 4.2)
% % \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(11.1, 9.6)
% \scriptsize{Uncertainty characterization \\
% \hspace{0.1cm} Hermite polynomial chaos}
% \end{textblock}
% 
% 
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% $\Leftarrow$
% \end{textblock}
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% \begin{textblock}{15}(11, 11.2)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
% \end{textblock}
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% \begin{textblock}{15}(5.3, 10.75)
% \includegraphics[width=0.33\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/probabilsitc_evolution.png}
% \end{textblock}
% 
% \begin{textblock}{15}(5.4, 9.6)
% \scriptsize{\quad \quad Uncertainty propagation \\ 
% \quad \quad \quad \quad stochastic FEM}
% \end{textblock}
% 
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% % \end{textblock}
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% % \includegraphics[width=0.29\linewidth]{pic/seismic_risk_result_framework.png}
% % \end{textblock}
% 
% % \begin{tikzpicture}[remember picture, overlay]
% % \draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
% % ([xshift=-25pt,yshift=-52pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=-18pt]$(pic cs:b)+(0pt,-2pt)$);
% % \end{tikzpicture}
% 
% 
% % \begin{textblock}{15}(-0.1, 9.3)
% % \scriptsize
% % \quad \quad \quad \quad $\lambda(EDP>z)=$
% 
% % $\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
% % \end{textblock}
% 
% % \begin{textblock}{15}(1.6, 10.7)
% % \scriptsize{EDP hazard/risk}
% % \end{textblock}
% 
% \end{frame}
% 
% \endgroup
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% 
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begingroup
% 
% \setbeamertemplate{footline}{}
% 
% \begin{frame}
% 
% \frametitle{Time Domain Intrusive PSRA Framework}
% 
% 
% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
% 
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
% 
% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
% 
% 
% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
% \end{textblock}
% 
% 
% \begin{textblock}{15}(7.1, 6.2)
% \scalebox{.9}{\tiny{Fourier spectra}}
% \\
% \vspace*{-0.2cm}
% \scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
% \end{textblock}
% 
% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
% 
% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \end{textblock}
% 
% \begin{textblock}{15}(10.5, 4.2)
% % \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(11.1, 9.6)
% \scriptsize{Uncertainty characterization \\
% \hspace{0.1cm} Hermite polynomial chaos}
% \end{textblock}
% 
% 
% \begin{textblock}{15}(10.2, 13.2)
% $\Leftarrow$
% \end{textblock}
% 
% \begin{textblock}{15}(11, 11.2)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(5.3, 10.75)
% \includegraphics[width=0.33\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/probabilsitc_evolution.png}
% \end{textblock}
% 
% \begin{textblock}{15}(5.4, 9.6)
% \scriptsize{\quad \quad Uncertainty propagation \\ 
% \quad \quad \quad \quad stochastic FEM}
% \end{textblock}
% 
% \begin{textblock}{15}(4.6, 13.2)
% $\Leftarrow$
% \end{textblock}
% 
% \begin{textblock}{15}(0.3, 11.0)
% \includegraphics[width=0.29\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/seismic_risk_result_framework.png}
% \end{textblock}
% 
% \begin{tikzpicture}[remember picture, overlay]
% \draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
% ([xshift=-25pt,yshift=-52pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=-18pt]$(pic cs:b)+(0pt,-2pt)$);
% \end{tikzpicture}
% 
% 
% \begin{textblock}{15}(-0.1, 9.25)
% \scriptsize
% \quad \quad \quad \quad $\lambda(EDP>z)=$
% 
% $\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
% \end{textblock}
% 
% \begin{textblock}{15}(1.6, 10.7)
% \scriptsize{EDP hazard/risk}
% \end{textblock}
% 
% \end{frame}
% 
% \endgroup
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 


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% 
% 
% \begin{frame}
% 
% \frametitle{Stochastic Ground Motion Modeling}
% 
% % \vspace{-0.4cm}
% 
% \begin{itemize}
% 
% %\item[-] \normalsize Shift from modeling specific IM to fundamental characteristics of ground motions
% \item[-] \normalsize Shift from modeling specific IM to fundamental characteristics of ground motions
% 
% \begin{itemize}
% 
% \item[-] \normalsize Uncertain Fourier amplitude spectra (FAS)  
% 
% \item[-] \normalsize Uncertain Fourier phase spectra (FPS)
% 
% \end{itemize}
% 
% % \item \scriptsize Mean behavior of stochastic FAS  
% 
% % \begin{itemize}
% 
% % \item[--] \scriptsize $w^2$ source radiation spectrum by \textit{Brune(1970)}
% 
% % \item[--] \scriptsize Systematic studies by \textit{ \textbf{Boore}(1983, \textbf{2003}, 2015)}.  
% 
% % \end{itemize}
% 
% \vspace{0.05cm}
% 
% %\item[-] \normalsize Recent GMPE study of FAS,
% %(FAS marginal median \& variability GMPEs by \textit{{Bora et al.
% %(2018)}} and {\textit{Bayless \& Abrahamson (2019)}} ; 
% %FAS Inter-frequency correlation GMPE by \textit{Stafford(2017)} and
% %{\textit{Bayless \& Abrahamson (2018)}})
% 
% 
% \vspace*{1mm}
% \item[-] \normalsize GMPE studies of FAS,
% (
% \textit{{Bora et al. (2018)}},
% \textit{Bayless \& Abrahamson (2018,2019)}, 
% \textit{Stafford(2017)}, 
% %{\textit{Bayless \& Abrahamson (2018)}
% )
% 
% % \begin{itemize}
% % 
% % \item[-] \scriptsize FAS marginal median \& variability GMPEs by \textit{\textbf{Bora et al. (2018)}} and \textbf{\textit{Bayless \& Abrahamson (2019)}}
% % 
% % %\vspace{0.1cm}
% % 
% % \item[-] \scriptsize FAS Inter-frequency correlation GMPE by \textit{Stafford(2017)} and \textbf{\textit{Bayless \& Abrahamson (2018)}}.  
% % 
% % \end{itemize}
% 
% 
% %\vspace{0.05cm}
% 
% %\item[-] \normalsize Stochastic FPS by phase derivative (Boore,2005)
% %(Logistic phase derivative model by {\textit{Baglio \& Abrahamson (2017)}})
% \vspace*{1mm}
% \item[-] \normalsize Stochastic FPS by phase derivative (Boore,2005)
% (Logistic phase derivative model by {\textit{Baglio \& Abrahamson (2017)}})
% 
% \vspace*{1mm}
% \item[-] \normalsize Near future change from \textbf{ $\boldsymbol{Sa(T_0)}$} to \textbf{FAS} and \textbf{FPS}   
% %next five years
% % as envisioned by Abrahamson (2018)
% 
% \end{itemize}
% 
% \end{frame}
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% %\subsection{Illustrative Example}
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 
%\frametitle{TDNIPSRA Example Object}
\frametitle{Example Object}

\begin{textblock}{15}(0.5, 4.0)
\includegraphics[width=0.47\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/faults_configuration_new.pdf}
\end{textblock}

\begin{textblock}{15}(7.5, 3.4)
\includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SSC_legend.pdf}
\end{textblock}

\begin{textblock}{15}(0.8, 11.6)
\scriptsize
\begin{itemize}
\item Fault 1: San Gregorio fault 
\item Fault 2: Calaveras fault
\item Uncertainty: Segmentation, \\ slip rate, rupture geometry, etc.
\end{itemize} 
\end{textblock}

\begin{textblock}{15}(8.5, 11.6)
\scriptsize
\begin{itemize}
\item 371 total seismic scenarios 
\item $M \ 5 \sim 5.5$ and $6.5 \sim 7.0$
\item $R_{jb} \ 20km \sim 40km$
\end{itemize}
\end{textblock}

\end{frame}
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\begin{frame}

\frametitle{Stochastic Ground Motion Modeling}


\begin{textblock}{15}(1.0, 3.7)
\small Realizations of simulated uncertain motions for scenario $M=7$, $R=15km$: 
\end{textblock}

\begin{textblock}{15}(0.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series100.pdf}
\end{textblock}

\begin{textblock}{15}(5.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series343.pdf} \enspace
\end{textblock}

\begin{textblock}{15}(10.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series439.pdf}
\end{textblock}

\begin{textblock}{15}(1.0, 9.2)
\small Verification with GMPE: 
\end{textblock}

\begin{textblock}{15}(0.3, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SA_GMPE_verification_std_08_no_smooth.pdf}
\end{textblock}

\begin{textblock}{15}(5.6, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Goodness_fit_std_08_no_smooth.pdf}
\end{textblock}

\begin{textblock}{15}(10.8, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Standard_deviation_std_08_no_smooth_new.pdf}
\end{textblock}

% \begin{textblock}{15}(0.5, 11.0)
% \begin{itemize}
% \item $\Delta \sigma= 84bar$, $\kappa=0.03s$ with total $\sigma=0.8ln$. 
% \item Simulated median is not biased. 
% \item Consistent total uncertainties with GMPE.  
% \end{itemize}
% \end{textblock}

\end{frame}
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\begin{frame} 
\frametitle{Stochastic Ground Motion Characterization}

{\begin{textblock}{15}(-0.1, 3.62)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_mean_acc_from_acc.pdf}

\quad \quad \quad Acc. marginal mean
\end{textblock}

\begin{textblock}{15}(3.7, 3.62)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_var_acc_from_acc.pdf}

\quad \quad \quad Acc. marginal S.D. 
\end{textblock}

\begin{textblock}{15}(7.6, 3.8)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_exact_acc_correlation_from_acc.pdf}

\quad \quad \quad Acc. realization Cov.
\end{textblock}

\begin{textblock}{15}(11.8, 3.9)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_simulated_acc_correlation_from_acc.pdf}

\quad \quad Acc. synthesized Cov.
\end{textblock}}

\begin{textblock}{15}(-0.1, 9.3)
\scriptsize
\includegraphics[width=0.31\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_mean_dis_from_dis.pdf}
\end{textblock}

\begin{textblock}{15}(0.9, 13.75)
\scriptsize
Dis. marginal mean
\end{textblock}

\begin{textblock}{15}(4.2, 9.4)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_var_dis_from_dis.pdf}
\end{textblock}

\begin{textblock}{15}(5.1, 13.75)
\scriptsize
Dis. marginal S.D. 
\end{textblock}

\begin{textblock}{15}(8.2, 9.5)
\scriptsize
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_exact_dis_correlation_from_dis.pdf}

\quad \quad Dis. realization Cov.
\end{textblock}

\begin{textblock}{15}(12.2, 9.6)
\scriptsize
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_simulated_dis_correlation_from_dis.pdf}

\quad Dis. synthesized Cov.
\end{textblock}
\end{frame}
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\begin{frame}

\frametitle{Stochastic Soil and Structure Modeling}

%Uncertain 1D shear response 



\begin{figure}[!htbp]
\centering
%\subfloat[Uncertain $H_a$]{
%\hspace{-0.8cm}
%\includegraphics[width=0.53\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/constitutive_relation_uncertainHa_certainCr_MC_verification.pdf}}
%\subfloat[Uncertain $H_a$ and $C_r$]{
%\hspace{-0.2cm}
%\includegraphics[width=0.53\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/constitutive_relation_uncertainHa_uncertainCr_MC_verification.pdf}}
%\vspace{-2mm}
%\caption{\label{figure_probabilisitc_constitutive_relation} Intrusive probabilistic modeling of Armstrong-Frederick hysteretic behavior and  verification  with Monte Carlo simulation: (a) Gaussian distributed  $Ha$  with  mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), $C_r = 17.6$. (b) Gaussian distributed  $Ha$  with  mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), Gaussian distributed $C_r$ with mean 17.6 and 15\% COV.}
\subfloat[Frame]{
\hspace{-0.8cm}
\includegraphics[width=2.5cm]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/Shear-Frame-8-levels.jpg}}
\subfloat[Interstory response]{
\hspace{10mm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Papers/2019/1D_risk/version6/Figures/constitutive_relation_uncertainHa_uncertainCr_MC_verification.pdf}}
%\vspace{-2mm}
%\caption{\label{figure_probabilisitc_constitutive_relation} Intrusive probabilistic modeling of Armstrong-Frederick hysteretic behavior and  verification  with Monte Carlo simulation: (a) Gaussian distributed  $Ha$  with  mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), $C_r = 17.6$. (b) Gaussian distributed  $Ha$  with  mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), Gaussian distributed $C_r$ with mean 17.6 and 15\% COV.}
\end{figure}
%



\end{frame}
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\begin{frame} 

\frametitle{Probabilistic Dynamic Structural Response}

\begin{textblock}{15}(0.7, 4.5)
\scriptsize
\includegraphics[width=0.46\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/shear_frame_illustration_update.pdf}
\end{textblock}

\begin{textblock}{15}(0.2, 10.8)
\begin{itemize}
\scriptsize \item Coefficient of variation 15$\%$ for $H_a$ and $C_r$
%\scriptsize \item Exponential correlation with correlation  \\
%length $l_c = 10$ floors 
\scriptsize \item Time domain stochastic \\
                  El-Pl FEM analysis (SEPFEM)
% : uncertain \\ structure with uncertain excitations
\end{itemize}
\end{textblock}

\begin{textblock}{15}(7.7, 5)
\scriptsize
\includegraphics[width=0.52\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Probabilistic_Response_Node_1_new.pdf}
\end{textblock}

\begin{textblock}{15}(8.3, 4.2)
\scriptsize Probabilistic response of top floor from SFEM
\end{textblock}
\end{frame}

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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begingroup
% 
% \setbeamertemplate{footline}{}
% 
% \begin{frame} 
% 
% \frametitle{Seismic Risk Analysis}
% 
% \begin{textblock}{15}(1.9,3.8)
% \scriptsize
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/MIDR_PDF_evolution.pdf}
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/PDF_MIDR_combine.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(1.9,9.5)
% \scriptsize
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/MIDR_distribution_different_floors.pdf}
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Risk_MIDR.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(0.8, 3.8)
% \scriptsize Engineering demand parameter (EDP): Maximum inter-story drift ratio (MIDR) 
% \end{textblock}
% \end{frame}
% \endgroup
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begingroup
% \setbeamertemplate{footline}{}
% \begin{frame} 
% 
% \frametitle{Seismic Risk Analysis}
% 
% \begin{textblock}{15}(1.9,3.8)
% \scriptsize
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/PFA_distribution.pdf}
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Risk_PFA.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(1.9,9.3)
% \scriptsize
% \includegraphics[width=0.41\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/2D_EDP_PDF_1e5.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(8.7,9.4)
% \scriptsize
% \includegraphics[width=0.40\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/2D_EDP_PDF_downview_1e5.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(0.8, 3.8)
% \scriptsize Engineering demand parameter (EDP): Peak floor acceleration (PFA) 
% \end{textblock}
% \end{frame}
% \endgroup
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
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\begin{frame} 

\frametitle{Seismic Risk, Forward Analysis}

%\vspace{-0.5cm}
\vspace*{2mm}

\begin{itemize}

%  \item[-] \small Damage measure (DM) defined on multiple EDPs: 
%  \item[-] \small Damage measure (DM) defined on single EDP:
  \item[-] Damage measure defined on single EDP:

\vspace*{-3mm}

%\begin{textblock}{15}(0.7,8.9)
\begin{table}[!htbp]
\small
\resizebox{0.98\hsize}{!}{
\begin{tabular}{ccccccc}
%\hline
\textbf{DM} & MIDR\textgreater{}0.5\% & \textbf{MIDR\textgreater{}1\%} & MIDR\textgreater{}2\% & PFA\textgreater{}0.5${\rm m/s^2}$ & \textbf{PFA\textgreater{}1\boldsymbol{${\rm m/s^2}$}} & PFA\textgreater{}1.5${\rm m/s^2}$ \\ 
\hline
\textbf{Risk [/yr]} & 6.66$\times 10^{-3}$                    & \textbf{3.83\boldsymbol{$\times 10^{-3}$}}                  & 9.97$\times 10^{-5}$                   & 6.65$\times 10^{-3}$                 & \textbf{1.92 \boldsymbol{$\times 10^{-3}$}}               & 9.45$\times 10^{-5}$                 \\ 
%\hline
\end{tabular}}
\end{table}
%\end{textblock}


\vspace{4mm}
  \item[-] Damage measure (DM) defined on multiple EDPs: 

%    \vspace{2mm}
    {\scriptsize $DM: \{\text{MIDR}>1\%\,  \cup \,\text{PFA}>1{\rm m/s^2} \}$, seismic risk is \boldsymbol{$4.2 \times 10^{-3}/yr$} }

    \vspace{1mm}
    {\scriptsize $DM: \{\text{MIDR}>1\%\,  \cap \,\text{PFA}>1{\rm m/s^2} \}$, seismic risk is \boldsymbol{$1.71 \times 10^{-3}/yr$}}

    \vspace{3mm}


%\vspace{20mm}

\vspace{4mm}
\item[-] \small Seismic risk for DM defined on multiple EDPs can be quite
different from that defined on single EDP

\end{itemize}
\end{frame}
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%  \item[-] Incremental el--pl constitutive equation
%  %  
%      \begin{eqnarray}
%      \nonumber
%    \Delta \sigma_{ij} 
%    = 
%  %  E^{EP}_{ijkl} 
%       E^{EP}_{ijkl} \; \Delta \epsilon_{kl}  
%       =  
%       \left[
%       E^{el}_{ijkl}
%         -
%         \frac{\displaystyle  E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
%             {\displaystyle n_{rs} E^{el}_{rstu} m_{tu} - \xi_* h_*}
%             \right]
%      \Delta \epsilon_{kl}
%      \end{eqnarray}
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%  \item[-] Dynamic Finite Elements
%  %
%  \begin{equation}
%  { M} \ddot{ u_i} +
%  { C} \dot{ u_i} +
%  { K}^{ep} { u_i} =
%  { F(t)}
%      \nonumber
%  \end{equation}
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%  \item[-] Material and loads are uncertain
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%  \includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0-m.pdf}
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%  Dynamic Finite Elements 
%  $
%  { M} \ddot{ u_i} +
%  { C} \dot{ u_i} +
%  { K}^{ep} { u_i} =
%  { F(t)}$
%  
%  
%  \begin{itemize}
%  \vspace*{2mm}
%  \item[-] Input random field/process{\normalsize{(non-Gaussian, heterogeneous/ non-stationary)}}:
%            Multi-dimensional Hermite Polynomial Chaos (PC) with {known coefficients}
%  %\vspace{0.05in}
%  \vspace*{2mm}
%  \item[-] Output response process: Multi-dimensional Hermite PC with {unknown coefficients}
%  % \vspace{0.05in}
%  \vspace*{2mm}
%   \item[-] Galerkin projection: minimize the error to compute unknown coefficients of response process
%  % %\vspace{0.05in}
%  % \vspace*{2mm}
%  %  \item[-] Time integration using Newmark's method
%  % % : Update coefficients following
%  % % an elastic-plastic constitutive law at each time step
%   
%  \end{itemize}
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%  %\scriptsize 
%  %Note: PC = Polynomial Chaos
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%  % \begin{frame}{Discretization of Input Random Process/Field $\beta(x,\theta)$}
%  % \begin{center}
%  % \includegraphics[scale=0.35]{/home/jeremic/tex/works/Thesis/FangboWang/slides_13Mar2019/Fangbo_slides/figs/PC_KL_explanation.PNG} \\
%  % \end{center}
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%  % \footnotesize{Note: $\beta(x,\theta)$ is an input random process with any
%  % marginal distribution,  \\ \hspace{21mm} with any covariance structure;} \\
%  % \footnotesize{\hspace{8mm} $\gamma(x,\theta)$ is a zero-mean unit-variance Gaussian random process.} \\
%  % 
%  % \end{frame}
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%  \begin{frame}{Polynomial Chaos Representation}
%  
%  %\scriptsize{
%  Material random field: \\
%  %\vspace{-0.3cm}
%  %\begin{equation*} 
%  $D(x, \theta)= \sum_{i=1}^{P1} a_i(x) \Psi_i(\left\{\xi_r(\theta)\right\})$      
%  %\end{equation*}
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%  \vspace{4mm}
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%  Seismic loads/motions random process: \\
%  %\vspace{-0.3cm}
%  %\begin{equation*} 
%  $f_m(t, \theta)=\sum_{j=1}^{P_2} f_{mj}(t) \Psi_j(\{\xi_k(\theta)\})$         
%  %\end{equation*}
%  
%  \vspace{4mm}
%  
%  Displacement response:  \\
%  %\vspace{-0.3cm}
%  %\begin{equation*} 
%  $u_n(t, \theta)=\sum_{k=1}^{P_3} d_{nk}(t) \Psi_k(\{\xi_l(\theta)\})$         
%  %\end{equation*}
%  
%  \vspace{3mm}
%  
%  %Acceleration response:
%  %%\vspace{-0.3cm}
%  %%\begin{equation*} 
%  %$\ddot u_n(t, \theta)=\sum_{k=1}^{P_3} \ddot d_{nk}(t) \Psi_k(\{\xi_l(\theta)\})$         
%  %%\end{equation*}
%  
%  %\vspace{3mm}
%  \vspace{5mm}
%  
%  where $a_i(x), f_{mj}(t)$ are {known PC coefficients}, while $d_{nk}(t)$
%  are {unknown PC coefficients}.
%  %}
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%  % \subsection{Direct Solution for Probabilistic Stiffness and Stress in 1D}
%  % 
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%    BEGGINING PEP    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  \begin{frame}{Direct Probabilistic Constitutive Solution in 1D}
%  
%  
%  % \begin{itemize}
%  %  
%  %  \vspace{0.5cm}
%  % 
%  % \item<1-> Probabilistic constitutive modeling : \vspace{0.5cm}
%  
%  \begin{itemize}
%  
%  
%  \vspace*{4mm}
%  \item[-]  Zero elastic region elasto-plasticity with stochastic Armstrong-Frederick
%  kinematic hardening
%  
%  $ \Delta\sigma =\ H_a  \Delta \epsilon  -  c_r \sigma |\Delta \epsilon| ;
%  \hspace{0.5cm} 
%  E_t = {d\sigma}/{d\epsilon} = H_a \pm c_r \sigma $
%  
%  \vspace*{4mm}
%  \item[-] Uncertain: 
%        init. stiff. $H_a$, 
%        shear strength $H_a/c_r$, 
%        strain $\Delta \epsilon$: 
%  
%  $ H_a = \Sigma h_i \Phi_i; \;\;\;  
%    C_r = \Sigma c_i \Phi_i; \;\;\; 
%    \Delta\epsilon =  \Sigma \Delta\epsilon_i \Phi_i $
%  
%  
%  
%  \vspace*{4mm}
%  \item[-] Resulting stress and stiffness are also uncertain 
%  
%  % - 
%  % - $ \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i \Phi_i = \sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i  \Delta \epsilon_k \Phi_i \Phi_k - \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_i \Delta \epsilon_k \sigma_l \Phi_j \Phi_k \Phi_l$
%  % - 
%  % - $ \sum_{l=1}^{P_{E_t}} \Delta E_{t_i} \Phi_i = \sum_{i=1}^{P_h}  h_i  \Phi_i \pm \sum_{i=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_i  \sigma_l  \Phi_i \Phi_l$
%  % - 
%  
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%  \end{itemize}
%  
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%  % \vspace{0.5cm}
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%  
%  %\item<1-> Time integration is done via Newmark algorithm
%  
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%  % \end{itemize}
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%  \begin{frame}{Direct Probabilistic Stiffness Solution}
%   
%   \begin{itemize}
%    
%  
%  \item[-] Analytic product for all the components, 
%  
%  $ E^{EP}_{ijkl} 
%       =  
%       \left[
%       E^{el}_{ijkl}
%         -
%         \frac{\displaystyle  E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
%             {\displaystyle n_{rs} E^{el}_{rstu} m_{tu} - \xi_* h_*}
%             \right]
%  $
%  
%  
%  
%  
%  \vspace*{2mm}
%  \item[-] Stiffness: each Polynomial Chaos component is updated incrementally
%  % at each Gauss Point via stochastic Galerkin projection
%  
%  
%  
%    \small{$E_{t_1}^{n+1}  =  \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \ h_i   <\Phi_i \Phi_1> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j  \sigma_l^{n+1} <\Phi_j  \Phi_l \Phi_1>\}$}
%    \\
%    . . . 
%  %  
%  %
%  %  $\large{\vdots}$
%  \\  
%       \small{$E_{t_P}^{n+1}  =  \frac{1}{<\Phi_1\Phi_P> }\{\sum_{i=1}^{P_h} \ h_i   <\Phi_i \Phi_P> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j  \sigma_l^{n+1} <\Phi_j  \Phi_l \Phi_P>\}$}
%    
%    
%  \vspace*{2mm}
%  \item[-] Total stiffness is : 
%    
%    $ E_{t}^{n+1} = \sum_{l=1}^{P_{E}} E_{t_i}^{n+1} \Phi_i $
%  
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%  \begin{frame}{Direct Probabilistic Stress Solution}
%   
%   \begin{itemize}
%    
%    
%  
%  \item[-] Analytic product, for each stress component, 
%  
%  $ \Delta \sigma_{ij} =  E^{EP}_{ijkl} \; \Delta \epsilon_{kl} $ 
%  %      =  
%  %      \left[
%  %      E^{el}_{ijkl}
%  %        -
%  %        \frac{\displaystyle  E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
%  %            {\displaystyle n_{rs} E^{el}_{rstu} m_{tu} - \xi_* h_*}
%  %            \right]
%  %     \Delta \epsilon_{kl}
%  % 
%  
%  
%  \vspace*{2mm}
%  \item[-] Incremental stress: each  Polynomial  Chaos  component  is updated
%  incrementally 
%  % via stochastic Galerkin projection
%  
%  
%   
%   
%    {$\Delta\sigma_1^{n+1}  = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i  \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_1>- \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_1>\}$}
%  \\
%  . . . 
%  \\
%  % ${\vdots}$ 
%  {$\Delta\sigma_P^{n+1}  = \frac{1}{<\Phi_P\Phi_P> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i  \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_P>- \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_P>\}$}
%  
%  
%  \vspace*{2mm}
%  \item[-] Stress  update: 
%  
%  $ \sum_{l=1}^{P_{\sigma}} \sigma_i^{n+1} \Phi_i = \sum_{l=1}^{P_{\sigma}} \sigma_i^{n} \Phi_i + \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i^{n+1} \Phi_i$
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%  %\section[Formulation]{Stochastic Dynamic Finite Element Formulation}
%  %\subsection[Time domain stochastic Galerkin method]{Time domain stochastic Galerkin method}
%  %\frame{\tableofcontents[currentsubsection,sectionstyle=show/shaded]}
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%  \frametitle{Stochastic Elastic-Plastic Finite Element Method}
%  
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%  \begin{itemize}
%  
%  \item[-] Material uncertainty expanded into stochastic shape funcs.
%  %$E(x,t,\theta) = \sum_{i=0}^{P_d} r_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
%  
%  \vspace*{1mm}
%  \item[-] Loading uncertainty expanded into stochastic shape funcs.
%  %$f(x,t,\theta) = \sum_{i=0}^{P_f} f_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
%  
%  \vspace*{1mm} 
%  \item[-] Displacement expanded into stochastic shape funcs.
%  %$u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
%  
%  %\item 
%  %Stochastic system of equation resulting from Galerkin approach (static example):
%  %
%  %\item  Time domain integration  using Newmark and/or HHT, in probabilistic spaces
%  
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%  \item[-] Jeremi{\'c} et al. 2011
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%  \begin{tiny}
%  \[
%  %$
%  \begin{bmatrix}
%      \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)}    & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
%      \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)}    & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\  \\
%      \vdots & \vdots & \vdots & \vdots\\
%      \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)}     & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
%  \end{bmatrix}
%  \begin{bmatrix}
%      \Delta u_{10} \\
%      \vdots \\
%      \Delta u_{N0}\\
%      \vdots \\
%      \Delta u_{1P_u}\\
%      \vdots \\
%      \Delta u_{NP_u}
%  \end{bmatrix}
%  =
%  %\]
%  %\[
%  \begin{bmatrix}
%       \sum_{i=0}^{P_f} f_i  <\Psi_0\zeta_i> \\
%      \sum_{i=0}^{P_f} f_i   <\Psi_1\zeta_i> \\
%     \sum_{i=0}^{P_f} f_i    <\Psi_2\zeta_i> \\
%      \vdots \\
%       \sum_{i=0}^{P_f} f_i  <\Psi_{P_u}\zeta_i>\\
%  \end{bmatrix}
%  %$
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%  
%    \item[-] SEPFEM offers a complete probabilistic solution 
%  
%    \item[-]  It is NOT based on Monte Carlo approach
%  
%    \item[-]  System of equations grows (!)
%  
%  \end{itemize}
%  
%  
%  %   \normalsize{Typical number of terms required for a SEPFEM problem} \vspace{1cm}\\
%    \scalebox{0.7}{
%    \begin{tabular}{ c c c c}
%     \# KL terms material & \# KL terms load & PC order displacement& Total \# terms per DoF\\ \hline
%     4 & 4 & 10 & 43758 \\  
%     4 & 4 & 20 & 3 108 105 \\    
%     4 & 4 & 30 & 48 903 492 \\  
%     6 & 6 & 10 & 646 646 \\    
%     6 & 6 & 20 & 225 792 840 \\  
%     6 & 6 & 30 & 1.1058 $10^{10}$ \\    
%     8 & 8 & 10 & 5 311 735 \\  
%     8 & 8 & 20 & 7.3079 $10^{9}$ \\    
%     8 & 8 & 30 & 9.9149 $10^{11}$\\  
%     
%     ...  & ... & ... & ...\\   
%  %    \hline
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%    
%    
%    \subsection{Backward  Uncertainty Propagation, Sensitivities}
%    %\subsection{Sobol Sensitivity Analysis}
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%    \frametitle{Backward Uncertainty Propagation, Sensitivities}
%    
%    \begin{itemize}
%    
%    \vspace*{2mm}
%      \item[-] Given forward uncertain response, PDFs, CDFs...
%    
%    \vspace*{6mm}
%      \item[-] Contributions of uncertain input to forward uncertainties
%    
%    \vspace*{6mm}
%      \item[-] Sensitivity of forward uncertain response to input uncertainties
%    
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%    \end{itemize}
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%    \frametitle{ANOVA Representation}
%    
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%    
%    %\vspace*{10mm}
%    
%    Model with $n$ uncertain inputs ($\boldsymbol{x}$) and scalar output $y$:
%    
%    
%    \vspace*{-6mm}
%    \begin{equation}
%    y = f(\boldsymbol{x}) \mbox{;} \ \  \boldsymbol{x} \in I^{n}
%    \nonumber
%    \end{equation}
%    
%    % input parameters $\boldsymbol{x}$ are defined in $n$ dimensional unit
%    % cube $I^{n}$
%    %
%    
%    \vspace*{5mm}
%     
%    The ANalysis Of VAriance representation
%    % of $f(x)$ 
%    (Sobol 2001):
%    
%    \vspace*{-6mm}
%    \begin{eqnarray*}
%    f(x_1, ... x_n) = f_0 + \sum_{i=1}^{n} f_i(x_i) + 
%    \sum_{1\leq i<j\leq n} 
%    f_{ij}(x_i, x_j) 
%    +...
%    %\\
%    %+ 
%    f_{1,...n}(x_1,...x_n) 
%    \nonumber
%    \label{eq_ANOVA_representation}
%    \end{eqnarray*}
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%    
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%    Total of $2^n$ summands
%    
%    \vspace*{4mm}
%    %
%    Mean value $f_0 = \int_{I^{n}} f(\boldsymbol{x}) d \boldsymbol{x}$ 
%    %
%    
%    \vspace*{4mm}
%    Integral of each summand 
%    $\int_{0}^{1} f_{i_1,... i_s}({x_{i_1},... x_{i_s}}) dx_{i_k} = 0 \mbox{;~}
%     1 \leq k \leq s$ 
%    %in Equation \ref{eq_ANOVA_representation} over any of its independent variable is zero: 
%    
%    %\begin{equation}
%    %\int_{0}^{1} f_{i_1, ..., i_s}({x_{i_1}, x_{i_2}, ..., x_{i_s}}) \ dx_{i_k} = 0 \ \ \ \mathrm{for} \ \ \ 1 \leq k \leq s 
%    %\label{eq_summand_integral_zero}
%    %\end{equation}
%    % 
%    % 
%    % 
%    
%    
%    \vspace*{4mm}
%    Summands are orthogonal to each other
%    
%    \vspace*{-6mm}
%    %
%    \begin{eqnarray*}
%    \int_{I^{n}} f_{i_1,...i_s}({x_{i_1}, x_{i_2},...x_{i_s}}) 
%    f_{j_1,...j_t}({x_{j_1}, x_{j_2},...x_{j_t}}) d \boldsymbol{x} = 0 
%    \mbox{;}
%    %\mathrm{for} \ \ \
%    \\
%    \{i_1,...i_s\} \neq \{j_1,...j_t\}  
%    \nonumber
%    \label{eq_orthogonal}
%    \end{eqnarray*}
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%    \frametitle{Sensitivity Analysis, ANOVA Representation}
%    
%    ANOVA representation is unique!
%    % and derived analytically
%    
%    \vspace*{5mm}
%     
%    Univariate terms: 
%    %
%    \begin{eqnarray*}
%    f_i(x_i) = \int_{I^{n-1}} f(\boldsymbol{x}) \ d\boldsymbol{x}_{\sim i} - f_0 
%    \end{eqnarray*}
%    
%    %$\int_{I^{n-1}} (\cdot) \ d\boldsymbol{x}_{\sim i}$ is integral
%    %over all dimensions except $x_i$.
%    %
%    
%    \vspace*{5mm}
%    
%    
%    Bivariate terms:
%    % 
%    \begin{eqnarray*}
%    f_{ij}(x_i, x_j) = \int_{I^{n-2}} f(\boldsymbol{x}) \ d\boldsymbol{x}_{\sim [ij]} - f_i(x_i) - f_j(x_j) - f_0 
%    \end{eqnarray*}
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%    \frametitle{Sensitivity Analysis, Variance}
%    
%    \vspace*{3mm}
%    
%    Total variance of the probabilistic model response
%    $y=f(\boldsymbol{X})$ is
%    %
%    \begin{eqnarray*}
%    %D = \mathrm{Var}[f(\boldsymbol{X})] = \int_{I^{n}} f^2(\boldsymbol{x}) \ d\boldsymbol{x} - f_0^2
%    D = {Var}[f(\boldsymbol{X})] = \int_{I^{n}} f^2(\boldsymbol{x}) \ d\boldsymbol{x} - f_0^2
%    \end{eqnarray*}
%    
%    %Using Equations~\ref{eq_ANOVA_representation} and \ref{eq_orthogonal}, 
%    Total variance $D$, decomposed 
%    %
%    \begin{eqnarray*}
%    D = \sum_{i=1}^{n} D_i + \sum_{1\leq i <j \leq n} D_{ij} + ... + D_{1,2,...,n} = \sum_{s=1}^{n}\sum_{i_1<...<i_s}D_{i_1...i_s}
%    \label{eq_variance_decomposition}
%    \end{eqnarray*}
%    %
%    
%    Variance contribution from individual summand
%    %
%    \begin{eqnarray*}
%    D_{i_1..i_s} = \int_{I^s} f^2_{i_1...i_s}(x_{i_1}, ..., x_{i_s}) \ dx_{i_1}, ...dx_{i_s}, 
%    \\
%    1\leq i_1 < ... < i_s \leq n \mbox{;~} s= 1,...n
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%    \frametitle{Sobol Indices}
%    
%    
%    \begin{itemize}
%    
%    \item[-] Sobol' indices $S_{i_1...i_s}$,
%    fractional contributions from 
%    random inputs $\{X_{i_1}, ..., X_{i_s}\}$ to the total variance $D$:
%    $S_{i_1...i_s} = D_{i_1...i_s} / D$
%    % %The Sobol indices $S_{i_1...i_s}$: 
%    % %
%    % \begin{equation*}
%    % S_{i_1...i_s} = D_{i_1...i_s} / D
%    % \end{equation*}
%    % %
%    
%    
%    \vspace*{2mm}
%    
%    \item[-] First order indices $S_i$ $\rightarrow$ individual influence of each uncertain input
%    parameter 
%    %$X_i$
%    %
%    
%    \vspace*{2mm}
%    \item[-] Higher order indices $S_{i_1...i_s}$ $\rightarrow$ mixed influence from groups
%    of uncertain input parameters
%    %
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%    \vspace*{2mm}
%    \item[-] Total sensitivity indices,
%    % $S^{\mathrm{total}}_i$
%    %$\rightarrow$ 
%    influence of input parameter $X_i$
%    %
%    \begin{equation*}
%    %S^{\mathrm{total}}_i = \sum_{\mathscr{S}_i}D_{i_1...i_s} 
%    S^{{total}}_i = \sum_{\mathscr{S}_i}D_{i_1...i_s} 
%    \end{equation*}
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%    % \begin{frame}
%    % \frametitle{Sensitivity Analysis}
%    % 
%    %        
%    % The Sobol indices for global sensitivity analysis can be used for
%    % 
%    % Ranking of input uncertain parameters $\boldsymbol{X} = [X_1, ..., X_n]$. 
%    % 
%    % Neglect of inessential uncertain parameters and high order terms in Equation~\ref{eq_ANOVA_representation}.   
%    % %
%    % 
%    % 
%    % 
%    % The total sensitivity indices $S^{\mathrm{total}}_i$ is defined to evaluate the total influence of a certain input parameter $X_i$ as: 
%    % %
%    % \begin{equation*}
%    % S^{\mathrm{total}}_i = \sum_{\mathscr{S}_i}D_{i_1...i_s} 
%    % \end{equation*}
%    % %
%    % 
%    % set $\mathscr{S}_i$ contains all the indexes involving index $i$ 
%    % and Sobol' indices set $\{D_{i_1...i_s}\}$ is the
%    % collection of all partial sensitivity indices that are related to parameter
%    % $X_i$.
%    % %
%    % \begin{equation*}
%    % \mathscr{S}_i = \{(i_1, ..., i_s): \exists k, 1\leq k \leq s, i_k = i\}
%    % \label{eq_index_set}
%    % \end{equation*}
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%    \begin{frame}
%    \frametitle{Sobol Indices and Polynomial Chaos}
%    
%    
%    \vspace*{3mm}
%    
%    PC expansion of response, ANOVA form (Sudret 2008)
%    
%    %
%    %\vspace*{1mm}
%    
%    Multi-dimensional PC bases $\{\Psi_j(\boldsymbol{\xi})\}$ 
%    decomposed into products of single dimension PC chaos bases of different orders
%    %
%    \vspace*{-1mm}
%    \begin{equation*}
%    \Psi_j(\boldsymbol{\xi}) = \prod_{i=1}^{n} \phi_{\alpha_i}(\xi_i) 
%    \nonumber
%    \end{equation*}
%    %
%    $\phi_{\alpha_i}(\xi_i)$ is the single dimensional, order $\alpha_i$, polynomial function of underlying basic random variable $\xi_i$. 
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%    % \begin{frame}
%    % \frametitle{Sobol Sensitivity Analysis}
%    % 
%                
%    From ANOVA representation of probabilistic model response, the PC-based Sobol'
%    indices $S_{i_1...i_s}^{PC}$ are
%    %
%    
%    \vspace*{-1mm}
%    \begin{equation*}
%    %S_{i_1...i_s}^{PC} = \sum_{\boldsymbol{\alpha} \in \mathscr{S}_{{i_1},...,i_s}} y^2_{\boldsymbol{\alpha}} \boldsymbol{E} \left[\Psi_{\boldsymbol{\alpha}}^2\right] / D^{PC}
%    S_{i_1...i_s}^{PC} = \sum_{\boldsymbol{\alpha} \in {S}_{{i_1},...,i_s}} y^2_{\boldsymbol{\alpha}} \boldsymbol{E} \left[\Psi_{\boldsymbol{\alpha}}^2\right] / D^{PC}
%    \end{equation*}
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%    \frametitle{Sobol Sensitivity Analysis}
%    
%        
%    %Total Sobol' indices $S_{j_1...j_t}^{PC, \mathrm{total}}$
%    Total Sobol' indices $S_{j_1...j_t}^{PC, {total}}$
%    % for any group of
%    %parameters $\{\xi_{j_1}, ..., \xi_{j_t}\}$ is
%    %
%    \begin{equation*}
%    %S_{j_1...j_t}^{PC, \mathrm{total}} = \sum_{(i_1, ..., i_s) \in \mathscr{S}_{j_1,...,j_t}} S_{i_1...i_s}^{PC}   
%    S_{j_1...j_t}^{PC, {total}} = \sum_{(i_1, ..., i_s) \in {S}_{j_1,...,j_t}} S_{i_1...i_s}^{PC}   
%    \end{equation*}
%    %
%    %where $\mathscr{S}_{j_1,...,j_t} = \{(i_1, ..., i_s): (j_1,...,j_t) \subset (i_1, ..., i_s)\}$
%    where ${S}_{j_1,...,j_t} = \{(i_1, ..., i_s): (j_1,...,j_t) \subset (i_1, ..., i_s)\}$
%    
%    % $\mathscr{S}_{j_1,...,j_t}$ is defined as: 
%    % %
%    % \begin{equation}
%    % \mathscr{S}_{j_1,...,j_t} = \{(i_1, ..., i_s): (j_1,...,j_t) \subset (i_1, ..., i_s)\}
%    % \end{equation}
%    % %
%    
%    
%    \vspace*{10mm}
%    Using PC representation of probabilistic model response,
%    Sobol' sensitivity indices are \underline{analytic} and \underline{inexpensive}
%    %evaluated with very small computational expense. 
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%    \frametitle{Application: Stochastic Site Response}
%    
%    \vspace*{5mm}
%    \begin{itemize}
%    % \item[-] Three-layered ground with \\
%    %       uncertain material properties and \\
%    %       uncertain seismic rock motions
%    
%     \item[-] Uncertain material: \\
%           uncertain random field, \\
%           marginally lognormal \\
%           distribution, \\
%           exponential correlation \\
%           length 10m
%    
%    \vspace*{2mm}
%    \item[-]  Uncertain  seismic \\
%              rock motions:  \\
%              seismic  scenario \\
%               M=7,  R=50km \\
%    %       Stochastic Fourier amplitude spectra, \\
%    %       Stochastic Fourier phase derivative
%    
%    \end{itemize}
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%    \begin{figure}[!hbpt]
%    \begin{flushright}
%    \includegraphics[width=5.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Three-Layer-Model.jpg}
%    \\
%    \includegraphics[width=5.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Input-motions.jpg}
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%    Lognormal distributed random field with PC Dim. 3 Order 2
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%    \begin{itemize}
%     \item[-] UCERF3 (Field et al. 2014)
%     \item[-] Stochastic motions (Boore 2003) 
%     \item[-] Polynomial Chaos Karhunen-Lo{\`e}ve expansion
%     \item[-] Probabilistic DRM (Bielak et al. 2003, Wang et al. 2021)
%    \end{itemize}
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%    \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/UCERF3.pdf}
%    \includegraphics[width=4.0cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/SMSIM.pdf}
%    %\includegraphics[width=3.5cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_exact_dis_correlation_from_dis.pdf}
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%    
%    %\vspace*{1mm}
%    \item[-] Modeling fundamental characteristics of uncertain ground motions,
%    Stochastic Fourier amplitude spectra (FAS). and Stochastic Fourier phase spectra
%    (FPS) and not specific IM
%    
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%    
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%     FAS GMPEs by Bora et al. (2015, 2018), Bayless \&
%     Abrahamson (2019), 
%    % for marginal median \& variability, 
%    %Inter-frequency correlation structure by 
%    Stafford(2017) and Bayless \& Abrahamson (2018).  
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%    \item[-] Stochastic FPS by phase derivative (Boore,2005), Logistic phase derivative
%    model by Baglio \& Abrahamson (2017)
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%    %\item Upcoming: a major change from \textbf{ $\boldsymbol{Sa(T_0)}$ to FAS} in next five years as envisioned by Abrahamson (2018)
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%    % \frametitle{Stochastic Seismic Motions}
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%    % 
%    % %Lognormal distributed random field with PC Dim. 3 Order 2
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%    % %\vspace*{-50mm}
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%    % \hspace*{-10mm}
%    % \includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic_seismic_motions.jpg}
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% \includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic-motions-DISP.jpg}
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% \includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic-motions-ACC.jpg}
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% \frametitle{Stochastic Site Response}
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% \hspace*{-10mm}
% \includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic-site-response.jpg}
% \end{center}
% \end{figure}
% 
% 
% \vspace*{-2mm}
% PC dimensions $\xi_1$, - $\xi_3$ are from uncertain soil
% 
% PC dimensions $\xi_4$, - $\xi_{153}$ are from uncertain rock motions
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% 
% \frametitle{Sensitivity of PGA from Uncertain Soil}
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% %  
% 
% 
% \vspace*{10mm}
% \begin{itemize}
%   \item[-] First 10 terms \\
%         from soil uncertainty
% 
% 
% 
%   \item[-] Total Sobol     \\
%         sensitivity index  \\
%         $S_{1-3}^{PC,total} = 0.51$
% 
% 
% \end{itemize}
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% \vspace*{-40mm}
% \begin{small}
% %\begin{center}
% \begin{table}[!htb]
% \hspace*{50mm}
% \begin{tabular}{l l}
% Sobol Index           &  Value  \\ \hline 
% $S_{1,123}^{PC}$      &   0,04389  \\
% $S_{1,118}^{PC}$      &   0,02605  \\
% $S_{1,127}^{PC}$      &   0,02370  \\
% $S_{1,100}^{PC}$      &   0,01759  \\
% $S_{1,103}^{PC}$      &   0,01700  \\
% $S_{1,134}^{PC}$      &   0,01680  \\
% $S_{1,141}^{PC}$      &   0,01611  \\
% $S_{1,110}^{PC}$      &   0,01358  \\
% $S_{1,130}^{PC}$      &   0,01303  \\
% $S_{1,132}^{PC}$      &   0,01068 \\
% . . .                &   . . .     \\ \hline  
%     \end{tabular}
% \end{table}
% %\end{center}
% \end{small}
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% %
% %
% %PC dimensions $\xi_1$, - $\xi_3$ are from uncertain soil
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% %PC dimensions $\xi_4$, - $\xi_{153}$ are from uncertain motions
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% \begin{frame}
% 
% \frametitle{Sensitivity of  PGA from Uncertain Rock Motions}
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% %  
% 
% 
% \vspace*{10mm}
% \begin{itemize}
%   \item[-] First 10 terms \\
%         from motions uncertainty
% 
% 
%   \item[-] Total Sobol     \\
%         sensitivity index \\
%         $S_{1-153}^{PC,total} = 0.98$
% 
% 
% \end{itemize}
% 
% 
% 
% \vspace*{-40mm}
% \begin{small}
% %\begin{center}
% \begin{table}[!htb]
% \hspace*{50mm}
% \begin{tabular}{l l}
% Sobol Index           &  Value  \\ \hline 
% $S_{1,123}^{PC}$      &   0,04389  \\
% $S_{137}^{PC}$      &   0,03459  \\
% $S_{110}^{PC}$      &   0,03061  \\
% $S_{118}^{PC}$      &   0,02698  \\
% $S_{1,118}^{PC}$      &   0,02605  \\
% $S_{108}^{PC}$      &   0,02482   \\
% $S_{141}^{PC}$      &   0,02373  \\
% $S_{1,127}^{PC}$      &   0,02370  \\
% $S_{1,100}^{PC}$      &   0,01759  \\
% $S_{1,103}^{PC}$      &   0,01700  \\
% . . .                &   . . .     \\ \hline  
%     \end{tabular}
% \end{table}
% %\end{center}
% \end{small}
% 
% 
% 
% 
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% %
% %PC dimensions $\xi_1$, - $\xi_3$ are from uncertain soil
% %
% %PC dimensions $\xi_4$, - $\xi_{153}$ are from uncertain motions
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\begin{frame}

\frametitle{Sensitivity,  Backward Analysis}

%  


%\begin{itemize}
%
%
\vspace*{4mm}
%  \item[-] 
Total variance in PGA, in this particular case (!), 
dominated by uncertain ground motions
    \begin{itemize}
\vspace*{4mm}
      \item[] $49$\% from uncertain rock motions at depth
\vspace*{4mm}
      \item[] $2$\% from uncertain soil
\vspace*{4mm}
      \item[] $49$\% from interaction of uncertain rock motions and  uncertain soil 


    \end{itemize}


%  \item 
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\section{Summary}



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%    \frametitle{Appropriate Science and Engineering Quotes}
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%   \begin{small}
%   
%   \vspace*{2mm}
%   \item[]  Fran{\c c}ois-Marie Arouet,  Voltaire: \\
%   "Le doute n'est pas une condition agr{\'e}able, 
%   mais la certitude est absurde"
%   
%   
%   % \vspace*{2mm}
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%   % 
%   %  \vspace*{1mm}
%   % %1895   
%   % \item[-] William Thomson, Lord Kelvin:   
%   % "Heavier-than-air flying machines are impossible."
%   % 
%   % 
%   
%   % 
%   % 
%   %  \vspace*{1mm}
%   % %1943   
%   % \item[-] Thomas Watson, IBM Chairman:   
%   % "I think there is a world market for maybe five computers."
%   % 
%   
%    \vspace*{3mm}
%   \item[] Max  Planck:  \\
%   "A    new    scientific  truth does not triumph by convincing  its opponents and
%   making  them see the light, but rather because its opponents eventually die, and
%   a  new  generation  grows  up  that  is familiar with it" 
%   %(Science advances one funeral at a time)
%   
%   
%   % 
%   %  \vspace*{3mm}
%   % \item[-]  Niklaus  Wirth:   
%   % "Software is getting slower more rapidly than hardware becomes faster."
%   
%   
%   
%    \vspace*{3mm}
%   \item[] Theodore Von K{\'a}rm{\'a}n: \\
%   "The Scientist studies what is,
%    the engineer creates what has never been"
%   
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\begin{frame}
  \frametitle{Summary}

  \begin{itemize}

%    \item Importance   of   using  proper  models  correctly  (verification,
%    validation, level of sophistication)
%     
 
%    \item Reduction of modeling uncertainty 
%

\vspace*{2mm}
    \item[-] Analysis of uncertainties and sensitivities

\vspace*{4mm}
    \item[-] Full, direct, intrusive probabilistic modeling
%    \item[-] Predict and Inform

\vspace*{4mm}
    \item[-] No need to define IMs
%    \item[-] Predict and Inform

%  \vspace*{4mm}
%  %  \item[-]  
%  %    \url{http://real-essi.us/}
%  %     \url{http://real-essi.info/}
%    \item[-] Engineer Needs to Know!
%  

 
%\vspace*{4mm}
%  \item[-]  Real-ESSI Simulator Systems
%% {\includegraphics[width=12mm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/Real_ESSI_in_different_langauges/Real_ESSI_Chinese.jpeg}}
%%    \url{http://real-essi.us/}
%

\vspace*{4mm}
  \item[-]  
%    \url{http://real-essi.us/}
    \url{http://real-essi.us/}
      
%
%\vspace*{1mm}
%    \item[-] Sensitivity analysis


%\vspace*{1mm}
%   \item[-]     Collaborators:    Feng,  Yang, Wang,  Yang.
   
   

% \vspace*{1mm}
%   \item[-]  Funding  from  and  collaboration  with the ATC/US-FEMA, US-DOE, US-NRC, US-NSF, 
%   CNSC-CCSN, UN-IAEA, and Shimizu Corp. is greatly appreciated,

% 
%\vspace*{1mm}
%  \item[-]  
%    \url{http://sokocalo.engr.ucdavis.edu/~jeremic}
%      

\end{itemize}



\end{frame}
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% \begin{frame}
%   \frametitle{Dedication}
% 
%  RIP:  Neboj{\v s}a Orbovi{\' c}  (1962-2021) \\
% %  who pushed hard for sensitivity analysis \\
% \vspace*{5mm}
% \hspace*{30mm}
% % \begin{center}
%   \includegraphics[width=4cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Dedication_Photos/Nebojsa_Orbovic/Nebojsa_Orbovic_26Feb2016.jpg}
% % \end{center}
% 
%   
% % 
% %\vspace*{1mm}
% %  \item[-]  
% %    \url{http://sokocalo.engr.ucdavis.edu/~jeremic}
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