20130916-CitcomUQ.tex

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\title{Inverse problems and uncertainty quantification}
\author{{\bf Jed Brown} \\
\texttt{jedbrown@mcs.anl.gov}
}

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{
  Mathematics and Computer Science Division \\ Argonne National Laboratory
}

\date{Citcom workshop, 2013-09-16}

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\subject{Talks}


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\begin{frame}
  \titlepage
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\begin{frame}{Forward models}
  \begin{block}{Deterministic forward modeling with transient PDE}
    \begin{equation*}
      u(t,x; p)
    \end{equation*}
    known parameters $p$: initial/boundary data, coefficients, material models, \ldots
  \end{block}
  \begin{block}{Forward model with uncertainty (posterior)}
    \begin{equation*}
      u(t,x,\mathfrak z; p)
    \end{equation*}
    noise $\mathfrak z$ is stochastic representation of incompletely-modeled processes
  \end{block}
  \begin{block}{Parameter optimization with uncertainty}
    \begin{equation*}
      \min_p \int_{\mathfrak z} f\big(u(t,x,\mathfrak z,p) \big)
    \end{equation*}
  \end{block}
\end{frame}

\begin{frame}{Inversion and design}
  \begin{block}{Data Assimilation}
    \begin{equation*}
      \hat p(t,x,d,d_0) = \argmin_p \int_{\mathfrak y} \int_{\mathfrak z} \norm{d(u(t,x,\mathfrak z,p) + \mathfrak y) - d_0}^2 + \mathrm{Prior}(p)
    \end{equation*}
    infer $p$ from sparse observations $d_0$ with noise $\mathfrak y$
  \end{block}
  \begin{block}{Optimal experimental design}
    \begin{equation*}
      \hat d = \argmin_d \int_p \norm{\hat p(t,x,d,d_0(p)) - p} + \mathrm{cost}(d)
    \end{equation*}    
    choose ``affordable'' sparse observations $d$ to minimize risk in the data assimilation problem over a region of parameter/model space
  \end{block}
\end{frame}

\begin{frame}{Methods for inversion and UQ}
  \begin{itemize}
  \item Stochastic Galerkin/collocation
    \begin{itemize}
    \item Sparse grids, Smolyak quadrature
    \item Accurate in low-dimensional spaces
    \end{itemize}
  \item Bayesian approach
    \begin{equation*}
      \pi_{\text{post}}(x | y_{\text{obs}}) \sim \pi_{\text{prior}} (x) \pi_{\text{like}}(y_{\text{obs}}| x)
    \end{equation*}
    \begin{itemize}
    \item Gaussian prior and linear model: Gaussian posterior
    \item Maximum a posteriori (MAP) point (scalable)
      \begin{equation*}
        \min_{x} \ \ - \log \pi_{\text{post}}(x | y_{\text{obs}})
      \end{equation*}
      like regularized deterministic inversion
    \item MAP point efficient to compute using adjoints
    \item Markov-chain Monte Carlo: high dimensions, nonlinear, but converges slowly
    \end{itemize}
  \end{itemize}
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\end{document}