Identification of the anti-diffusion coefficient for the linear Kuramoto-Sivashinsky equation
type: Item
instance of: scholarly article
title: Identification of the anti-diffusion coefficient for the linear Kuramoto-Sivashinsky equation
zbMATH Open document ID: 1465.49004
author: Yanyan Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2021-02-25T00:00:00Z
review text: The direct problem is written as \[ u_{t}+ (\sigma(x)u_{xx})_{xx} + \gamma(x) u_{xx} = f\ \ ((x,t) \in (0, L)\times (0,T)),\ \] \[ u(0,t) = 0,\ u(L, t) = 0,\ u_{x}(0, t) = 0, \ u_{x}(L, t) = 0,\ u(x, 0)=u_{0}(x). \] The corresponding inverse problem is to recover the unknown coefficient \(\gamma(x)\) using that data \(u(x,T)=m(x)\). This inverse problem is reduced to the optimization problem \[ \min_{\gamma\in M}\int_{0}^{L}|u(x,T)-m(x)|^{2}\,dx + \alpha\|\gamma\|_{W_{2}^{1}(0,L)}^{2}, \] where \(M=\{\gamma\in W_{2}^{1}(0,L): \ \|\gamma\|_{W_{2}^{1}(0,L)}\leq \eta \}\) and \(\alpha,\eta \) are positive parameters. It is proven that a solution to this optimization problem is unique and continuously depends on the data. The numerical algorithm for solving the optimization problem is described and the results of numerical experiments are presented.
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 49N45
zbMATH DE Number: 7315637
zbMATH Keywords: Kuramoto-Sivashinsky equation
zbMATH Keywords: inverse problem
zbMATH Keywords: nonlinear PDE optimization
reviewed by: S. G. Pyatkov
describes a project that uses: DOLFIN
describes a project that uses: UFL
describes a project that uses: L-BFGS-B
describes a project that uses: FEniCS
describes a project that uses: LBFGS-B
describes a project that uses: L-BFGS
MaRDI profile type: Publication
full work available at URL: j.jmaa.2020.124747
OpenAlex ID: W3097599097
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DOI: 10.1016/J.JMAA.2020.124747
description: scientific article; zbMATH DE number 7315637