Multi-Resolution Approximate Inverses

I hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. ii The University of Waterloo requires the signatures of all persons using or photocopying this thesis. Please sign below, and give address and date. This thesis presents a new preconditioner for elliptic PDE problems on unstructured meshes. Using ideas from second generation wavelets, a multi-resolution basis is constructed to effectively compress the inverse of the matrix, resolving the sparsity vs. quality problem of standard approximate inverses. This finally allows the approximate inverse approach to scale well, giving fast convergence for Krylov subspace accelerators on a wide variety of large unstructured problems. Implementation details are discussed, including ordering and construction of fac-tored approximate inverses, discretization and basis construction in one and two dimensions, and possibilities for parallelism. The numerical experiments in one and two dimensions confirm the capabilities of the scheme. Along the way I highlight many new avenues for research, including the connections to multigrid and other multi-resolution schemes. iv Acknowledgements I would like to first thank Wei-Pai Tang for his excellent ideas, support, and guidance. I'm also indebted to Peter Forsyth for advice on discretization and multigrid, and very helpful suggestions for revisions; to Sivabal Sivaloganathan for introducing me to Green's functions and reading the drafts; to Rob Zvan for getting me started on irregular meshes; to Justin Wan for some fruitful conversations and sample meshes; to David Pooley for the barrier option pricing problem; and of course to the Natural Sciences and Engineering Research Council of Canada for their financial support. I especially want to thank my family for their much needed love and encouragement throughout the last year and particularly the final hectic weeks.