Lecture 20 Summary

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Discussed preconditioning: finding an easy-to-invert M such that M-1A has clustered eigenvalues. Derived the preconditioned conjugate gradient method (showing how the apparent non-Hermitian-ness of M-1A is not actually a problem as long as M is Hermitian positive-definite).

Via a simple analysis of the discretized Poisson's equation, then generalized to any discretized grid/mesh with nearest-neighbor interactions, argued that the number of steps in unpreconditioned CG is (at least) proportional to the diameter of the grid for sparse matrices of this type, and that this exactly corresponds to the square root of the condition number in the Poisson case. Hence, an ideal preconditioner really needs some kind of long-range interaction.

Discussed multigrid. (Briefly) explained why the naive approach of simple using a courser grid as a preconditioner is not enough, because the course-grid solutions necessarily live in a subspace of the fine-grid solutions. Hence, some form of "smoothing", typically combination with another iterative scheme (typically a stationary scheme) is needed.