Convergence analysis and proof of acceleration for NGMRES applied to the Picard iteration for Navier-Stokes equations
Provides the first convergence proof for NGMRES, identifying the acceleration mechanism, benefiting computational fluid dynamics practitioners using iterative solvers.
The authors prove convergence of NGMRES for accelerating Picard iteration in Navier-Stokes equations, showing it scales the Lipschitz constant by the optimization gain, with numerical experiments confirming sharp estimates and improved performance even for divergent cases.
We consider nonlinear GMRES (NGMRES) as an acceleration technique for the Navier-Stokes Picard iteration, a direction that has not previously been explored. We identify the optimal norm for the least squares optimization problem arising in the NGMRES algorithm, and establish a convergence analysis for NGMRES with general depth that proves NGMRES scales the Picard Lipschitz constant by the gain of the optimization problem. To our knowledge, this is the first convergence proof for NGMRES that identifies the mechanism responsible for convergence acceleration. Numerical experiments demonstrate that the convergence estimates are remarkably sharp. In addition, NGMRES greatly improves the performance of the Picard iteration, even in cases where the unaccelerated iteration diverges.