On a backward problem for multidimensional Ginzburg-Landau equation with random data

TL;DR

This paper addresses the ill-posed backward problem for the multidimensional Ginzburg-Landau equation with random data, introducing a new regularization method and analyzing its convergence properties.

Contribution

The authors develop a novel regularization approach combined with statistical methods for solving the backward Ginzburg-Landau problem with random data.

Findings

Established an upper bound on the convergence rate of the mean integrated squared error.

Proved convergence in both $L^2$ and $H^1$ norms.

Provided theoretical guarantees for the regularization method.

Abstract

In this paper, we consider a backward in time problem for Ginzburg-Landau equation in multidimensional domain associated with some random data. The problem is ill-posed in the sense of Hadamard. To regularize the instable solution, we develop a new regularized method combined with statistical approach to solve this problem. We prove a upper bound, on the rate of convergence of the mean integrated squared error in $L^2 $ norm and $H^1$ norm.