Low-Rank Solution Algorithms for Parameter-Dependent Partial Differential Equations

Prof. Howard C. Elman, Department of Computer Science and UMIACS, University of Maryland

The collection of solutions of discrete parameter-dependent partial
differential equations often takes the form of a low-rank object.
We show that in this scenario, iterative algorithms for computing
these solutions can take advantage of this structure to reduce both
computational effort and memory requirements.  Implementation of such
solvers requires that explicit rank-compression computations be done
to truncate the ranks of intermediate quantities that must be computed.
We prove that when truncation strategies are used as part of a multigrid
solver, the resulting algorithms retain "textbook" (grid-independent)
convergence rates and we demonstrate how the truncation criteria affect
convergence behavior.  In addition, we show that these techniques can be
used to construct efficient solution algorithms for computing the
eigenvalues of parameter-dependent operators.

This is joint work with Tengfei Su.

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