MULTIGRID PRECONDITIONERS FOR BI-CGSTAB FOR THE SPARSE-GRID SOLUTION OF HIGH-DIMENSIONAL ANISOTROPIC DIFFUSION EQUATION

Robust and effcient solution techniques are developed for high-dimensional parabolic partial differential equations (PDEs). Presented is a solver based on the Krylov subspace method Bi-CGSTAB pre- conditioned with d-multigrid. Developing the perfect multigrid method, as a stand-alone solver for a single problem discretized on a particular grid, often requires a lot of optimal tuning and expert insight; on the other hand Krylov-subspace based methods are robust but much less effcient unless used in combination with a very suitable preconditioner. The precondi- tioner that we employ is d-multigrid. We aim for a robust combination of the two so that it results in a solver that converges well for a wide class of discrete problems arising from discretization on various anisotropic grids. This is exactly what we encounter during the sparse grid solution of a high-dimensional problem. Different multigrid components are discussed and presented with operator construction formulae )in abstract d dimen- sions( We also present convergence diagrams for various multigrid )solvers and preconditioners ( that we develop in this work, and explain their ap- plicability.