Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2203.13066

Smoothing analysis of two robust multigrid methods for elliptic optimal control problems

Yunhui He·Jun Liu
View original source ↗Read source document ↗Export metadata ↓

Abstract

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. First, we perform a detailed local Fourier analysis (LFA) of a well-known collective Jacobi relaxation (CJR) scheme, where the optimal smoothing factors are derived. This insightful analysis reveals that the optimal relaxation parameters depend on mesh size and regularization parameters, 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 $α\ge ch^4$ for a small constant $c$. Here $α$ is the regularization parameter and $h$ is the spatial mesh step size. These schemes are successfully extended to control-constrained cases through the semi-smooth Newton method. Coarsening by three or four with BSR is competitive with coarsening by two. Numerical examples are presented to validate our theoretical outcomes. The proposed inexact BSR (IBSR) scheme, where two preconditioned conjugate gradients iterations are applied to the Schur complement system, yields better computational efficiency than the CJR scheme.

Explore related subjects

math.NAcs.NAmath.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yunhui He, Jun Liu. 2022-03-22. Smoothing analysis of two robust multigrid methods for elliptic optimal control problems. https://arxiv.org/abs/2203.13066

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fully spectral scheme for the linear BGK equation on the whole space

In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $Φ$, even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight $e^{-$Φ$}$ in space. The potential $Φ$ is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.

math.NA↗
arXiv2026

Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds

This paper establishes inverse inequalities for kernel-based approximation spaces defined on bounded Lipschitz domains in $\mathbb{R}^d$ and compact Riemannian manifolds. While inverse inequalities are well-studied for polynomial spaces, their extension to kernel-based trial spaces poses significant challenges. For bounded Lipschitz domains, we extend prior Bernstein inequalities, which only apply to a limited range of Sobolev orders, to the full range of lower and upper orders, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. For compact Riemannian manifolds, we focus on restricted kernels, which are defined as the restriction of positive definite kernels from the ambient Euclidean space to the manifold, and prove their counterparts.

math.NA↗
arXiv2026

Error Estimates for Hyperbolic Scaling Limits of Linear Kinetic Models on Networks

This paper studies linear discrete kinetic models on networks and their asymptotic behavior in the small Knudsen number limit. For coupling conditions at an n-edge junction under a symmetric formulation, we introduce a change of variables that reformulates the system into n independent initial-boundary value problems. The asymptotic expansions are then constructed and rigorously justified by deriving an error estimate based on the energy method.

math.NA↗