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.
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.
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.
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.