TY  - JOUR
ID  - Atifi2020
LB  - Atifi2020
AU  - Atifi, Khalid
AU  - Essoufi, El-Hassan
AU  - Khouiti, Bouchra
TI  - An inverse backward problem for degenerate two-dimensional parabolic equation
ST  - An inverse backward problem for degenerate two-dimensional parabolic equation
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2020
VL  - 40
IS  - 4
SP  - 427
EP  - 449
DO  - https://doi.org/10.7494/OpMath.2020.40.4.427
UR  - https://doi.org/10.7494/OpMath.2020.40.4.427
AB  - 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.
KW  - data assimilation
KW  - adjoint method
KW  - regularization
KW  - heat equation
KW  - inverse problem
KW  - degenerate equations
KW  - optimization
SN  - 1232-9274
ER  -