KKT Preconditioners for PDE-Constrained Optimization with the Helmholtz Equation

Kouri, Drew P.; Ridzal, Denis; Tuminaro, Ray

doi:10.1137/20m1349199

KKT Preconditioners for PDE-Constrained Optimization with the Helmholtz Equation

Journal Article · Thu May 13 00:00:00 EDT 2021 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/20m1349199· OSTI ID:1784959

^[1]; Ridzal, Denis ^[1]; Tuminaro, Ray ^[2]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sandia National Lab. (SNL-CA), Livermore, CA (United States)

This paper considers preconditioners for the linear systems that arise from optimal control and inverse problems involving the Helmholtz equation. Specifically, we explore an all-at-once approach. The main contribution centers on the analysis of two block preconditioners. Variations of these preconditioners have been proposed and analyzed in prior works for optimal control problems where the underlying partial differential equation is a Laplace-like operator. In this paper, we extend some of the prior convergence results to Helmholtz-based optimization applications. Our analysis examines situations where control variables and observations are restricted to subregions of the computational domain. We prove that solver convergence rates do not deteriorate as the mesh is refined or as the wavenumber increases. More specifically, for one of the preconditioners we prove accelerated convergence as the wavenumber increases. Additionally, in situations where the control and observation subregions are disjoint, we observe that solver convergence rates have a weak dependence on the regularization parameter. We give a partial analysis of this behavior. We illustrate the performance of the preconditioners on control problems motivated by acoustic testing.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-CA), Livermore, CA (United States); Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1784959

Report Number(s):: SAND--2020-6750J; 687075

Journal Information:: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 5 Vol. 43; ISSN 1064-8275

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

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