Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation

Self Cite

In this paper, we are concerned with the stochastic time-fractional diffusion-wave equations in a Hilbert space. The main objective of this paper is to establish properties of the stochastic weak solutions of the initial-boundary value problem, such as the existence, uniqueness and regularity estimates. Moreover, we apply the obtained theories to an inverse source problem. The uniqueness of this inverse problem under the boundary measurements is proved.

Section: Background and Literaturementioning

confidence: 99%

Section: Literaturementioning

confidence: 99%

Well-posedness of the stochastic time-fractional diffusion and wave equations and inverse random source problems

Lassas

Zhang

2023

Self Cite

“…Similar results are obtained about the uniqueness and the stability of the inverse problem. A related inverse random source problem on the stochastic fractional diffusion equation for the time-dependent noise may be found in [27]. If the random sources are perturbed by a space-dependent noise, the mild solution approach is not available anymore since the spatial noise may not be regular enough to guarantee the well-posedness of the problem.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of an inverse random source problem for the time fractional diffusion equation via PhaseLift

Gong

Wang

et al. 2021

This paper is concerned with the inverse random source problem for a stochastic time fractional diffusion equation, where the source is assumed to be driven by a Gaussian random field. The direct problem is shown to be well-posed by examining the well-posedness and regularity of the solution for the equivalent stochastic two-point boundary value problem in the frequency domain. For the inverse problem, the Fourier modulus of the diffusion coefficient of the random source is proved to be uniquely determined by the variance of the Fourier transform of the boundary data. As a phase retrieval for the inverse problem, the phaselift method with random masks is applied to recover the diffusion coefficient from its Fourier modulus. Numerical experiments are reported to demonstrate the effectiveness of the proposed method.

“…fixed point iterations. (ii) is to recover a time-dependent potential q(t) from time-dependent observations [15] (or a time-dependent source from observation at one point [28]), which behaves similarly to (i) due to the directional alignment of the unknown and observations.…”

Section: Introductionmentioning

confidence: 99%

Recovering the potential and order in one-dimensional time-fractional diffusion with unknown initial condition and source ^*

Jin

2021

This paper is concerned with an inverse problem of recovering a potential term and fractional order in a one-dimensional subdiffusion problem, which involves a Djrbashian-Caputo fractional derivative of order α ∈ (0, 1) in time, from the lateral Cauchy data. In the model, we do not assume a full knowledge of the initial data and the source term, since they might be unavailable in some practical applications. We prove the unique recovery of the spatiallydependent potential coefficient and the order α of the derivation simultaneously from the measured trace data at one end point, when the model is equipped with a boundary excitation with a compact support away from t = 0. One of the initial data and the source can also be uniquely determined, provided that the other is known. The analysis employs a representation of the solution and the time analyticity of the associated function. Further, we discuss a two-stage procedure, directly inspired by the analysis, for the numerical identification of the order and potential coefficient, and illustrate the feasibility of the recovery with several numerical experiments.