Convergence rates in expectation for a nonlinear backward parabolic equation with Gaussian white noise

TL;DR

This paper investigates how quickly solutions to a nonlinear backward parabolic equation with Gaussian white noise converge in expectation when estimating initial conditions from noisy final observations, introducing a regularization approach.

Contribution

It presents a novel regularized method for estimating initial conditions and establishes convergence rates for the mean squared error in this nonlinear stochastic setting.

Findings

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

Proposed a regularization technique for solving the inverse problem.

Demonstrated the effectiveness of the method through theoretical analysis.

Abstract

The main purpose of this paper is to study the problem of determining initial condition of nonlinear parabolic equation from noisy observations of the final condition. We introduce a regularized method to establish an approximate solution. We prove an upper bound on the rate of convergence of the mean integrated squared error.