Speaker
Description
Algebraic multigrid (AMG) is a popular and effective solver for sparse linear systems arising from discretized partial differential equations (PDEs). The optimality and efficiency of AMG rests on the complementary relationship between relaxation (e.g., Gauss-Seidel) and interpolation, which when effective, results in optimal O($n$) scaling in the number of degrees-of-freedom $n$. Relaxation removes high-energy error, while interpolation maps low-energy error to a coarser space where it is reduced. While AMG is a relatively mature field for scalar PDEs that result in symmetric positive definite discretizations (SPD), such as the Poisson equation, there are a number of important matrix classes (e.g., nonsymmetric, indefinite, and discretizations of PDE systems) that remain problematic for AMG. In this talk, we review a number of developments that have extended the state-of-the-art for AMG beyond standard SPD problems. This includes (i) energy-minimization based interpolation and advanced algebraic coarsening strategies for convection-diffusion problems, highly anisotropic diffusion, discontinuous Galerkin discretizations, and indefinite problems, (ii) approximate ideal restriction (AIR) based methods for purely convective and highly nonsymmetric problems, (iii) adaptive AMG for detecting the near kernel of especially challenging matrices, and (iv) AMG approaches for PDE systems such as curl-curl and Stokes equations.