Numerical study on the quasi-solution for some nonlinear inverse stochastic parabolic problems

Abstract

Abstract In this paper, we prove the existence of quasi-solution for a class of nonlinear inverse stochastic parabolic partial differential equation (NISPDE) with additive noise. This is a parabolic problem of the inverse stochastic nonlinear heat equation type. The proofs are based on minimization method and stochastic variational formulation. For this purpose, we obtain a stability estimation from a stochastic variational formulation. Then, by constructing a compact subset of admissible functions set, the continuity of minimization functional is proved. These results provide the existence of the quasi-solution for this proposed problem. In order to numerical approximation of the quasi-solution, we consider Lie–Trotter splitting method. Due to this, the NISPDE may be split into two subproblems such that one of them is linear. The nonlinear subproblem is solved exactly and the Chebyshev wavelets and B-spline functions basis have been used for approximating solution of the corresponding linear subproblem. Then for finding a stable this solution, the Levenberg–Marquardt regularization technique is used. Finally, numerical example confirms the efficiency and accuracy of this method.