KKT Preconditioners for PDE-Constrained Optimization with the Helmholtz Equation
- 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.
- Research Organization:
- Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Sandia National Laboratories (SNL-CA), Livermore, CA (United States)
- Sponsoring Organization:
- USDOE National Nuclear Security Administration (NNSA)
- Grant/Contract Number:
- AC04-94AL85000
- OSTI ID:
- 1784959
- Report Number(s):
- SAND--2020-6750J; {"Journal ID: ISSN 1064-8275",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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