Graduate Student / Postdoc Seminar

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Fast direct solvers: Foundations and Challenges

Speaker: Mike O'Neil, Courant Institute, New York University

Location: TBA

Date: Friday, May 1, 2026, 1 p.m.

Synopsis:

Fast Direct Solvers (FDS) address the problem of solving a system of linear equations Ax=b arising from the discretization of either an elliptic PDE or of an associated integral equation. The matrix A will be sparse when the PDE is discretized directly, and dense when an integral equation formulation is used. For decades, industry practice for large scale problems has been to use iterative solvers such as multigrid, GMRES, or conjugate gradients. In contrast, a direct solver builds an approximation to the inverse of A or an easily invertible factorization of A (e.g. LU or Cholesky). A major development in numerical analysis in the last couple of decades has been the emergence of algorithms for constructing such factorizations or performing such inversions in linear or close to linear time. Such methods must necessarily exploit that the inverse of A is ``data-sparse,'' e.g. that it can be tessellated into blocks that have low numerical rank. This talk will cover the development of FDS's for both sparse and dense matrices, recent developments in the field, as well as future challenges and opportunities.