Text

Smoothing Analysis of Two Robust Multigrid Methods for Elliptic Optimal Control Problems

In this paper we study and compare two multigrid relaxation schemes with coarsening by two, three, and four for solving elliptic sparse optimal control problems with control constraints and combined
and cost functional. First, we perform a detailed local Fourier analysis (LFA) of a well-known collective Jacobi relaxation (CJR) scheme for the unconstrained case with only cost functional, where the optimal smoothing factors are derived. This insightful analysis reveals that the optimal relaxation parameters depend on both the mesh step size and the regularization parameter , which was not investigated in literature. Second, we propose and analyze a new mass-based Braess--Sarazin relaxation (BSR) scheme, which is proven to provide smaller smoothing factors than the CJR scheme when for a small constant . Finally, these schemes are successfully extended to control-constrained cases through the semismooth Newton method, where the corresponding Jacobian systems are treated by the proposed multigrid schemes. The nonstandard coarsening by three or four with BSR is competitive with the standard coarsening by two. Numerical examples are presented to validate our theoretical outcomes. The proposed inexact BSR (IBSR) scheme, where two preconditioned conjugate gradients (PCG) iterations are applied to solve the Schur complement system, yields a better computational efficiency than the CJR scheme in the conducted numerical comparison.

Ketersediaan

Informasi Detail

Judul Seri

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS; Vol.44 No.1 January-March 2023

No. Panggil

-

Penerbit

: .,

Deskripsi Fisik

p. 1-26

Bahasa

English

ISBN/ISSN

-

Klasifikasi

NONE

Tipe Isi

-

Tipe Media

-

Tipe Pembawa

-

Edisi

-

Subjek

SEMISMOOTH NEWTON METHOD
MULTIGRID
LOCAL FOURIER ANALYSIS
SMOOTHING FACTOR
COLLECTIVE JACOBI RELAXATION
BRAESS–SARAZIN RELAXATION

Info Detail Spesifik

https://doi.org/10.1137/22M1485759

Pernyataan Tanggungjawab

Yunhui He, Jun Liu

Versi lain/terkait

Lampiran Berkas

Komentar