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Regularization Of Inverse Problems

Parameter estimation for block‐oriented nonlinear systems using the key term separation

Summary This article is concerned with the parameter identification problem of nonlinear dynamic responses for the linear time‐invariant system by means of an impulse excitation signal and discrete...

Separable multi-innovation stochastic gradient estimation algorithm for the nonlinear dynamic responses of systems

When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model related to orders of the fractional derivatives, are often unknown and difficult to be directly measured, which requires one to discuss inverse problems of identifying these physical quantities from some indirectly observed information of solutions. Inverse problems in determining these unknown parameters of the model are not only theoretically interesting, but also necessary for finding solutions to initial-boundary value problems and studying properties of solutions. This chapter surveys works on such inverse problems for fractional diffusion equations.

Inverse Problems of Determining Parameters of the Fractional Partial Differential Equations

Decomposition‐based multiinnovation gradient identification algorithms for a special bilinear system based on its input‐output representation

In the present paper, we consider an inverse source problem for a fractional diffusion equation. This problem is ill-posed, i.e. the solution (if it exists) does not depend continuously on the data. Based on an a priori assumption, we give the optimal error bound analysis and a conditional stability result. Moreover, we use the Fourier regularization method to deal with this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Meanwhile, a new a posteriori parameter choice rule is also proposed. For the a priori and the a posteriori regularization parameters choice rules, we all obtain the convergence error estimates which are all order optimal. Numerical examples are presented to illustrate the validity and effectiveness of this method.

The inverse source problem for time-fractional diffusion equation: stability analysis and regularization

In the present paper, we consider an inverse source problem for a fractional diffusion equation. This problem is ill-posed, i.e. the solution (if it exists) does not depend continuously on the data. We give the mollification regularization method to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, a new a posteriori parameter choice rule is also proposed and a good error estimate is also obtained. Numerical examples are presented to illustrate the validity and effectiveness of this method.

A mollification regularization method for unknown source in time-fractional diffusion equation

The quasi‐reversibility method for a final value problem of the time‐fractional diffusion equation with inhomogeneous source

In this paper, an inverse source problem for the Helium Production–Diffusion Equation on a columnar symmetric domain is investigated. Based on an a priori assumption, the optimal error bound analys...

Landweber iteration regularization method for identifying unknown source on a columnar symmetric domain

We study in this paper the problem of determining time-dependent source functions in a parabolic equation with data given at some fixed locations in the domain. To solve this ill-posed inverse problem, we develop a mollification regularization method with a Gaussian kernel. We derive an a priori error estimate between the exact solution and its regularized approximation. Moreover, we propose an a posteriori parameter choice strategy for the selection of the regularization strength and derive an error estimate associated with the strategy. Numerical results are presented to illustrate the accuracy and efficiency of our method.

Xiao-Xiao Li

Papers

The inverse source problem for time-fractional diffusion equation: stability analysis and regularization

A mollification regularization method for unknown source in time-fractional diffusion equation

The quasi‐reversibility method for a final value problem of the time‐fractional diffusion equation with inhomogeneous source

Landweber iteration regularization method for identifying unknown source on a columnar symmetric domain

A mollification regularization method for identifying the time-dependent heat source problem