An inverse backward problem for degenerate two-dimensional parabolic equation

Opuscula Mathematica

article

Khalid Atifi (corresponding author)¹  El-Hassan Essoufi¹  Bouchra Khouiti¹ 

[1] Laboratoire de Mathématiques, Informatique et Sciences de l'ingénieur ,(MISI), Université Hassan 1

关键词: data assimilation;    adjoint method;    regularization;    heat equation;    inverse problem;    degenerate equations;    optimization.;   

DOI  :  10.7494/OpMath.2020.40.4.427

学科分类：环境科学（综合）

来源: AGH University of Science and Technology Press

【 摘 要 】

This paper deals with the determination of an initial condition in the degenerate two-dimensional parabolic equation \[\partial_{t}u-\mathrm{div}\left(a(x,y)I_2\nabla u\right)=f,\quad (x,y)\in\Omega,\; t\in(0,T),\] where \(\Omega\) is an open, bounded subset of \(\mathbb{R}^2\), \(a \in C^1(\bar{\Omega})\) with \(a\geqslant 0\) everywhere, and \(f\in L^{2}(\Omega \times (0,T))\), with initial and boundary conditions \[u(x,y,0)=u_0(x,y), \quad u\mid_{\partial\Omega}=0,\] from final observations. This inverse problem is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. To show the convergence of the descent method, we prove the Lipschitz continuity of the gradient of the Tikhonov functional. Also we present some numerical experiments to show the performance and stability of the proposed approach.

【 授权许可】

CC BY-NC