That the linear systems arising upon the discretization of elliptic PDEs can be solved efficiently is well-known, and iterative solvers that often attain linear complexity (multigrid, Krylov methods, etc) have proven very successful. Interestingly, it has recently been demonstrated that it is often possible to directly compute an approximate inverse to the coefficient matrix in linear (or close to linear) time. The talk will argue that such direct solvers have several compelling qualities, including improved stability and robustness, the ability to solve certain problems that have remained intractable to iterative methods, and dramatic improvements in speed in certain environments.
After a general introduction to the field, particular attention will be paid to a set of recently developed randomized algorithms that construct data sparse representations of large dense matrices that arise in scientific computations. These algorithms are entirely black box, and interact with the linear operator to be compressed only via the matrix-vector multiplication.