SCAIM Seminar: Multilevel preconditioners for simulations and optimization on dynamic, adaptive meshes

For the efficient solution of large, sparse, linear systems of equations, Ax = b, we usually need a preconditioning matrix P, in an appropriate sense close to the inverse of A, such that solving PAx = Pb converges fast. If we need to solve a sequence of problems in which the matrix A changes slowly (and the right hand side b arbitrarily), we would like to adapt the preconditioner rather than compute a new one from scratch for each problem.

After a brief introduction to iterative linear solvers, we discuss adaptive preconditioners for time-dependent simulations and nonlinear optimization problems (topology optimization) with dynamic mesh adaptation. Adaptive meshing greatly reduces the computational cost of simulations and optimization. Unfortunately, it also carries a number of problems for preconditioning in iterative linear solvers, as changes in the mesh lead to structural changes in the linear systems we must solve. As a result, a new preconditioner must be computed after every change in the mesh, which might be prohibitively expensive. Here, we propose preconditioners that are cheap to update for dynamic changes to the mesh as well as for changes in the matrix due to nonlinearity of the underlying problem; more specifically, we propose preconditioners that require only local changes to the preconditioner for local changes in the mesh and nonlinear terms. Our preconditioners combine sparse approximate inverses with multilevel correction. For further information see [1,2].

[1] Shun Wang and Eric de Sturler, Multilevel sparse approximate inverse preconditioners for adaptive mesh refinement. Linear Algebra Appl., 431:409-426, 2009.

[2] Shun Wang, Krylov subspace methods for topology optimization on adaptive meshes. PhD thesis, University of Illinois at Urbana-Champaign, Department of Computer Science, September 2007. Advisor: Eric de Sturler, Co-Advisor: Glaucio H. Paulino.