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Per-Gunnar Martinsson : Fast numerical methods for solving linear PDEs

Linear boundary value problems occur ubiquitously in many areas of science and engineering, and the cost of computing approximate solutions to such equations is often what determines which problems can, and which cannot, be modelled computationally. Due to advances in the last few decades (multigrid, FFT, fast multipole methods, etc), we today have at our disposal numerical methods for most linear boundary value problems that are "fast" in the sense that their computational cost grows almost linearly with problem size. Most existing "fast" schemes are based on iterative techniques in which a sequence of incrementally more accurate solutions is constructed. In contrast, we propose the use of recently developed methods that are capable of directly inverting large systems of linear equations in almost linear time. Such "fast direct methods" have several advantages over existing iterative methods: (1) Dramatic speed-ups in applications involving the repeated solution of similar problems (e.g. optimal design, molecular dynamics). (2) The ability to solve inherently ill-conditioned problems (such as scattering problems) without the use of custom designed preconditioners. (3) The ability to construct spectral decompositions of differential and integral operators. (4) Improved robustness and stability. In the talk, we will also describe how randomized sampling can be used to rapidly and accurately construct low rank approximations to matrices. The cost of constructing a rank k approximation to an m x n matrix A for which an O(m+n) matrix-vector multiplication scheme is available is O((m+n)*k). This cost is the same as that of Lanczos, but the randomized scheme is significantly more robust. For a general matrix A, the cost of the randomized scheme is O(m*n*log(k)), which should be compared to the O(m*n*k) cost of existing deterministic methods.

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