CompMath Seminar - Nicolás Barnarfi, University of Santiago, Chili

Title: Accelerated Quasi-Newton schemes in multiphysics problems

Abstract:

Quasi-Newton methods is a rather loose term, used to refer to any variant of a Newton method in which the Jacobian matrix is modified. This is done in order to obtain a method that converges in more iterations, but where the inversion of the (approximated) Jacobian is much faster. In this presentation, we will show some block-partitioned strategies where the components of the Jacobian can be lagged to the residual through the iterations. This yields, on one hand, an approximated Jacobian whose preconditioner is much better than the one known for the original one, but on the other hand the iteration count will be much higher. This increase in nonlinear iterations can be alleviated by using Anderson acceleration, which enjoys many theoretical properties that make it a desirable accelerator.

This approach is best understood in the framework of nonlinear preconditioners. The resulting scheme will be simply an Anderson accelerated Richardson scheme, where each iteration is preconditioned by the action of the inverse of the approximate Jacobian. We highlight this step, because the action of the Jacobian inverse can be represented by the solution of a linear system by means of, i.e. a preconditioned GMRES iterative method, but it can also be approximated simply by the action of the preconditioner. This shows that the action of the iterative linear solver can be sometimes \emph{discarded}, in favor of reducing computational time, as well as keeping reducing the overall complexity in parameters that our methods require.

The resulting nonlinear scheme will be balanced in terms of who does what: the preconditioner is optimal for the approximate Jacobian, and the nonlinear terms that are lagged to the residual are adequately handled by the acceleration scheme. We will show how this can yield not only significant reductions in computational time, but also a more robust solver. These claims will be validated by several numerical tests in different physical contexts.