An inverse potential problem for the stochastic diffusion equation with a multiplicative white noise

TL;DR

This paper investigates the direct and inverse problems for a stochastic diffusion equation with multiplicative white noise, providing conditions for well-posedness, explicit reconstruction formulas, and regularization techniques for potential recovery.

Contribution

It introduces explicit formulas for reconstructing the potential from solution expectations and applies regularization methods to improve numerical stability.

Findings

Unique positive mild solution for nonnegative initial data

Explicit reconstruction formula for the potential squared

Effective regularization methods for numerical potential reconstruction

Abstract

This work concerns the direct and inverse potential problems for the stochastic diffusion equation driven by a multiplicative time-dependent white noise. The direct problem is to examine the well-posedness of the stochastic diffusion equation for a given potential, while the inverse problem is to determine the potential from the expectation of the solution at a fixed observation point inside the spatial domain. The direct problem is shown to admit a unique and positive mild solution if the initial value is nonnegative. Moreover, an explicit formula is deduced to reconstruct the square of the potential, which leads to the uniqueness of the inverse problem for nonnegative potential functions. Two regularization methods are utilized to overcome the instability of the numerical differentiation in the reconstruction formula. Numerical results show that the methods are effective to reconstruct both smooth and nonsmooth potential functions.