Modern algorithms for the fast solution of elliptic PDEs will be studied. Possible topics include (the class will be adapted to the background and interests of the students each term):
Linear Algebra: Matrix factorizations and low-rank approximations; randomized methods for low-rank approximation; fast algorithms for rank-structured matrices
Solution of multi-body problems: Ewald summation, Barnes-Hutt, Fast Multiple Method
Introduction to integral equations to solve acoustic and electromagnetic problems. Discretization of integral equations (Nyström, Galerkin).
Introduction to iterative solvers and operator preconditioning for integral equations.
Fast direct solvers for integral equations (hierarchical block separable matrices).
Fast direct solvers for elliptic PDEs (Laplace or Helmholtz equation): fast direct sparse solvers, sweeping schemes.