20120628-IGS.tex

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\title{A parallel unstructured implicit 3D polythermal model for outlet glaciers}
\author{Jed Brown,\\
Iulian Grindeanu, Dmitry Karpeev, Barry Smith, Tim Tautges}


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{
  Mathematics and Computer Science Division, Argonne National Laboratory
}

\date{IGS 2012-06-28}

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\subject{Talks}


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  \titlepage
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\section{Conservative models}
\input{slides/GroundingLine/Steepness.tex}
\begin{frame}
  \includegraphics[width=0.8\textwidth]{figures/VHT/VHTJakoContourStream}
\end{frame}
\input{slides/VHTMotivation.tex}
\input{slides/VHTEquationsSimple.tex}

\section{Conforming boundaries and model non-smoothness}
\begin{frame}{Why care about conforming and non-smoothness?}
  \begin{itemize}
  \item Subshelf ocean circulation is physically richer and more difficult to solve than ice flow
    \begin{itemize}
    \item Heat transfer is sensitive to boundary layer processes with thickness $< \SI{1}{\metre}$, $\Reynolds \sim 10^6$
    \item Countless engineering studies: wall modeling is limited, significant normal resolution still necessary
    \item Unaligned interface anisotropy is bad for Eulerian AMR methods
    \end{itemize}
  \item We care about high-dimensional sensitivity analysis and inversion
    \begin{itemize}
    \item Forward model evaluations alone are an inefficient way to explore a high-dimensional space
    \item Each source of model non-smoothness requires a lot of analysis to use adjoint methods
    \item Conforming moving mesh methods eliminate all but ``essential'' non-smoothness (like contact)
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Mesh motion via Inverse Beltrami formulation}
  \begin{itemize}
  \item nonlinear elliptic (or parabolic) equation for mesh location
  \item prescribe resolution and anisotropy using target metric tensor
  \item efficient solution using Newton-Krylov multigrid or nonlinear multigrid
  \item conservative slip boundary conditions at most surfaces
  \item ALE transport scheme corrected to satisfy geometric conservation law
  \end{itemize}
\end{frame}

\begin{frame}{Transport}
  \begin{theorem}[Godunov 1954]
    Non-oscillatory linear spatial discretizations for transport are at most first order accurate.
  \end{theorem}

  \begin{itemize}
  \item First order accurate discretizations have unacceptably high numerical diffusion
  \item Discretization choices:
    \begin{itemize}
    \item Limited or reconstructed finite volume or finite difference
      \begin{itemize}
      \item Second order TVD limiters have corners
      \item Weighted Essential Non-Oscillatory (WENO) smooth, but very nonlinear
      \item Central or upwind
      \item Inconvenient for unstructured grids
      \end{itemize}
    \item Nonlinearly stabilized continuous finite element
      \begin{itemize}
      \item Currently used, but fragile and messy
      \end{itemize}
    \item Discontinuous Galerkin
      \begin{itemize}
      \item Cell-wise entropy stability without limiters
      \item Improved robustness with smooth limiters
      \item Transitioning to this
      \end{itemize}
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{}
  \begin{center}
    \color{red}{\Huge Joke}
  \end{center}
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\section{Multilevel Solvers}
\begin{frame}{Why do we need multilevel solvers?}
  \begin{itemize}
  \item Elliptic problems are globally coupled
  \item Without a coarse level, number of iterations proportional to inverse mesh size
  \item High-volume local communication is an inefficient way to communicate long-range information, bad for parallel models
  \item Most important with 3D flow features and/or slippery beds
  \item Nested/split multilevel methods
    \begin{itemize}
    \item Decompose problem into simpler sub-problems, use multilevel methods on each
    \item Good reuse of existing software
    \item More synchronization due to nesting, more suitable after linearization
    \end{itemize}
  \item Monolithic/coupled multilevel methods
    \begin{itemize}
    \item Better convergence and lower synchronization, but harder to get right
    \item Internal nonlinearities resolved locally
    \item More discretization-specific, less software reuse
    \end{itemize}
  \end{itemize}
\end{frame}
\input{slides/VHTSolvers.tex}
\begin{frame}{Full Approximation Scheme}
  \begin{align*}
    \tilde u^h &\gets S^h_{\text{pre}} u^h_0 & \text{pre-smooth} \\
    L^H u^H &= I_h^H f^h + \underbrace{L^H \hat I_h^H \tilde u^h - I_h^H L^h \tilde u^h}_{\tau_h^H} & \text{solve coarse problem for $u^H$} \\
    u^h & \gets S^h_{\text{post}} \Big[ \tilde u^h + I_H^h (u^H - \hat I_h^H \tilde u^h) \Big] & \text{apply correction and post-smooth}
  \end{align*}
  \begin{itemize}
  \item Nonlinearities from spatial discretization fixed locally
  \item No assembled matrices so better floating point utilization, less memory
  \item Makes progress on all physical components at once
  \item FD and DG good, less efficient for continuous finite element methods
  \item Influence of surface evolution is low rank, no need to visit finest level on each iteration
  \end{itemize}
\end{frame}

\begin{frame}{Outlook}
  \begin{itemize}
  \item Basal hydrology model
    \begin{itemize}
    \item Need mesh-independent statistics
    \item Numerical homogenization?
    \end{itemize}
  \item True inverse and sensitivity support, but need to invert for the right thing
  \item Dynamic remeshing after large topology changes
  \item Finish FAS multigrid for full coupled system including geometry
  \item User-friendliness of process
    \begin{itemize}
    \item Currently using georeferenced initial/boundary data via GDAL
    \item But meshing process is not fully automatic
    \item Better: multiresolution database (Mark Fahnestock)
    \end{itemize}
  \end{itemize}
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\end{document}