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Multigrid
Preconditioners
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Research

Multilevel solvers in general, and algebraic multigrid (AMG) in particular, have been active areas of research in the last decade or so. These methods have the potential to provide nearly h-independent solvers for a variety of problems.

One area that has been the focus of my recent works is that of AMG solvers for Maxwell's equations. Specifically, we consider problem reformulations that can make the system significantly easier for an AMG code to solve.

Collaborators

Saddle-Point Preconditioners

If 90% of physical problems one could possibly want to solve can be reduced to a linear system, then 90% of those are challenging to solve using an iterative method unless a good preconditioner is used. My thesis work at UIUC focused precisely on addressing this issue for (generalized) saddle-point problems.

Saddle-point problems arise from a number of applications, ranging from fluid flow (incompressible Navier-Stokes) and metal deformation all the way to constrained optimization and optimal control problems. There has been a great deal of work on preconditioners for specific problems (the incompressible Navier-Stokes equations come to mind), but the preconditioners we consider are general in scope and make no assumptions of symmetry or definiteness.

Collaborators

Monte Carlo for Particle Simulations

Monte Carlo algorithms for particle simulation generally work by moving particles (either one by one or all at once) and then deciding on whether or not to accept this move based on the ratio of some quantity (usually energy) between the new and old positions. These methods sample the space of allowable configurations with the goal of providing aggregate statistics about the underlying distribution in the end.

One of the most expensive steps in this process involves calculating the probability of accepting (or rejecting) a proposed particle move. Classical methods can take up to O(n^3) time to move n particles. We aim to cut that calculation by a factor of n.

Collaborators

Derivative-Free Optimization

It is not infrequent that someone might want to minimize (or maximize) some function where derivative information is not available (or might be too expensive to acquire). In these cases, the function evaluations are often expensive, so building a surrogate model using that to drive the optimization process is a reasonable thing to do. In this context, we consider Model-Assisted Pattern Search (MAPS), a locally convergent method that uses a series of models (creating using kriging) to guide a pattern search process.

Collaborators

Chris Siefert

Chris Siefert


Contact
E-mail: csiefer@sandia.gov
(505)844-4696 (Phone)
(505)284-2518 (Fax)


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Sandia National Laboratories
P.O. Box 5800, MS 1323
Albuquerque, NM 87185-1323


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William and Mary
UIUC