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A Direct Parallel-in-Time Quasi-Boundary Value Method for Inverse Space-Dependent Source Problems

Simultaneous identification of initial value and source strength in a transmission problem for a parabolic equation

In this paper, we consider an inverse problem of identifying a pair of function {u(x,t),r(t)} in the time fractional diffusion equation from two kinds of observations. By virtue of the high-precision Legendre spectral collocation and mollification method in the spatial and time direction severally, we present an efficient algorithm to solve the inverse problem. Finally, two numerical examples explicate the effectiveness and stability of the proposed method. 

Numerical inversion of reaction parameter for a time-fractional diffusion equation by Legendre spectral collocation and mollification method

Iterated fractional Tikhonov method for recovering the source term and initial data simultaneously in a two-dimensional diffusion equation

We introduce a regularization method with two differential operators for solving a linear ill-posed operator equation system. The existence and uniqueness of regularized solutions to the problem are derived. With an a priori as well as an a posteriori parameter choice strategy, convergence and convergence rates of the regularized solution are also obtained. As an application, we apply the regularization to a simultaneous inversion of the source term and the initial value problem for a heat conduction equation, numerical experiments are presented to demonstrate the effectiveness of the proposed method. 

Regularization with two differential operators and its application to inverse problems

Abstract This paper is concerned with the inverse problem for determining the space-dependent source and the initial value simultaneously in a parabolic equation from two over-specified measurements. By means of transforming information of the initial value into the source term and obtaining a combined source term, the parabolic equation problem is converted into a parabolic problem with homogeneous conditions. Then the considered inverse problem is formulated into a regularized minimization problem, which is implemented by the finite element method based on solving a sequence of well-posed direct problems. The uniqueness of inverse solutions are proved by the solvability of the corresponding variational problem, and the conditional stability as well as the convergence rate of regularized solutions are also provided. Then the error estimate of approximate regularization solutions is presented in the finite-dimensional space. The proposed method is a very fast non-iterative algorithm, and it can successfully solve the multi-dimensional inverse problem for recovering the space-dependent source and the initial value simultaneously. Numerical results of five examples including one- and two-dimensional cases show that the proposed method is efficient and robust with respect to data noise.

A non-iterative method for recovering the space-dependent source and the initial value simultaneously in a parabolic equation

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A non-iterative method for recovering the space-dependent source and the initial value simultaneously in a parabolic equation