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\textsc{Numerical Analysis and Scientific Computing}
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\textsc{Seminar}
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\textit{Physics-based preconditioners for solving PDEs in highly heterogeneous media}
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\begin{center}\Large
Burat Aksoylu\par
Department of Mathematics and Center for Computation and Technology, Louisiana State University
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\textbf{Abstract:} Eigenvalues of smallest magnitude become a major bottleneck for
iterative solvers especially when the underlying physical properties
have severe contrasts. These contrasts are commonly found in many
applications such as composite materials, geological rock properties
and thermal and electrical conductivity.

The main objective of this work is to construct a method as algebraic
as possible that could efficiently exploit the connectivity of highly
heterogeneous media in the solution of diffusion operators.  We
propose an algebraic way of separating binary-like systems according
to a given threshold into high- and low-conductivity regimes of
coefficient size $O(m)$ and $O(1)$, respectively where $m \gg 1$.  The
condition number of the linear system depends both on the mesh size
$\Delta x$ and the coefficient size $m$. For our purposes, we address
only the $m$ dependence since the condition number of the linear
system is mainly governed by the high-conductivity subblock. Thus, the
proposed strategy is inspired by capturing the relevant physics
governing the problem. Based on the algebraic construction, a
two-stage preconditioning strategy is developed as follows: (1) a
first stage that comprises approximation to the components of the
solution associated to small eigenvalues and, (2) a second stage that
deals with the remaining solution components with a deflation strategy
(if ever needed). The deflation strategies are based on computing near
invariant subspaces corresponding to smallest and deflating them
by the use of recycled the Krylov subspaces.

Due to its algebraic nature, the proposed approach can support a wide
range of realistic geometries (e.g., layered and channelized
media). Numerical examples show that the proposed class of
physics-based preconditioners are more effective and robust compared
to a class of Krylov-based deflation methods on highly heterogeneous
media. We also report singular perturbation analysis of the stiffness
matrix and the impact of  the number of high-conductive regions
on various matrices.\vspace{\fill}
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Tuesday, September 25, 2007, 4:00 pm\par
Mathematics and Science Center: W304
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\textsc{Mathematics and Computer Science}\par
\textsc{Emory University}
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