Determination of the initial density in nonlocal diffusion from final time measurements

Abstract

<p style='text-indent:20px;'>This paper is concerned with an inverse problem related to a fractional parabolic equation. We aim to reconstruct an unknown initial condition from noise measurement of the final time solution. It is a typical nonlinear and ill-posed inverse problem related to a nonlocal operator. The considered problem is motivated by a probabilistic framework when the initial condition represents the initial probability distribution of the position of a particle. We show the identifiability of this inverse problem by proving the existence of its unique solution with respect to the final observed data. The inverse problem is formulated as a regularized optimization one minimizing a least-squares type cost functional. In this work, we have discussed some theoretical and practical issues related to the considered problem. The existence, uniqueness, and stability of the optimization problem solution have been proved. The conjugate gradient method combined with Morozov's discrepancy principle are exploited for building an iterative reconstruction process. Some numerical examples are carried out showing the accuracy and efficiency of the proposed method.</p>

Inverse initial value problem for space-time fractional diffusion equations

“…Pertaining to inverse problems related to time-fractional diffusion equations, the interested reader may refer to [13,14,15,16,17,18]. On the other hand, for inverse problems related to space-fractional diffusion equations, [19,20,21,22] are valuable resources. In general, for differential equations involving fractional derivatives in both time and space variables, many researchers have addressed inverse problems in recent years.…”