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

Abstract

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.

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

“…By the properties and the Itô isometry of Brownian motions, the authors showed the uniqueness and stability of the inverse random source problem. The articles [12,14,36,48] provided the well-posedness of the direct problem and established the stability of the inverse problem for determining the time-dependent sources. The main idea is to use the Duhamel principle and strong maximum principle.…”

“…The main idea is to use the Duhamel principle and strong maximum principle. Moreover, [14,36,48] developed several effective reconstruction algorithms.…”

“…Unlike most of the existing works, which solve the stochastic inverse problems by the final overdetermination data [11,46] or the interior point data [12,14,36,48], this work uses the moments of the partial boundary measurements. As a result, we need to overcome some novel difficulties in the proofs.…”

Estimating the memory parameter for potentially non-linear and non-Gaussian time series with wavelets

“…The mean of the LRMW estimates is denoted by ᾱw and the mean of the GPH estimates is denoted by ᾱgph . The linear processes are generated according to (23), where { t } are i.i.d. Normal(0, 0.1 2 ).…”

“…Our work would be beneficial for other research in the field of inverse problems. First, the LRMW estimator, which is constructed with wavelet decompositions-a popular tool in inverse problems ( [31,39,44])-can be applied ( [1]) to estimate the Hurst exponent H of a fractional Brownian motion (FBM) (see definition 4 in appendix D), which is a process model that is studied and intensively used in the field of inverse problems (e.g., [10,19,23,27]). In that field, Hurst exponent H is usually assumed to be given in the considered model.…”