Multilevel preconditioners for simulations and optimization on dynamic, adaptive meshes

We present adaptive preconditioners for parallel, time-dependent simulations and nonlinear optimization problems 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 [1] and [2].