A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation

In this paper, we consider an inverse source problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine a space-dependent source term in the time-fractional diffusion equation from a noisy final data. Based on a series expression of the solution, we can transform the original inverse problem into a first kind integral equation. The uniqueness and a conditional stability for the space-dependent source term can be obtained. Further, we propose a modified quasi-boundary value regularization method to deal with the inverse source problem and obtain two kinds of convergence rates by using an a priori and an a posteriori regularization parameter choice rule, respectively. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.     

Tikhonov regularization method for a backward problem for the time-fractional diffusion equation

This paper is devoted to solve a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain by the Tikhonov regularization method. Based on the eigenfunction expansion of the solution, the backward problem for searching the initial data is changed to solve a Fredholm integral equation of the first kind. The conditional stability for the backward problem is obtained. We use the Tikhonov regularization method to deal with the integral equation and obtain the series expression of solution. Furthermore, the convergence rates for the Tikhonov regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Two numerical examples in one-dimensional and two-dimensional cases respectively are investigated. Numerical results show that the proposed method is effective and stable.     

The fractional Tikhonov regularization method for simultaneous inversion of the source term and initial value in a space-fractional Allen-Cahn equation

In this paper, we consider the inverse problem for identifying the source term and initial value simultaneously in a space-fractional Allen-Cahn equation. This problem is ill-posed, i.e., the solution of this problem does not depend continuously on the data. The fractional Tikhonov method is used to solve this problem. Under the a priori and the a posteriori regularization parameter choice rules, the error estimates between the regularization solutions and the exact solutions are obtained, respectively. Different numerical examples are presented to illustrate the validity and effectiveness of our method.     

Recovering a space-dependent source term in a time-fractional diffusion wave equation

This work is concerned with identifying a space-dependent source function from noisy final time measured data in a time-fractional diffusion wave equation by a variational regularization approach. We provide a regularity of direct problem as well as the existence and uniqueness of adjoint problem. The uniqueness of the inverse source problem is discussed. Using the Tikhonov regularization method, the inverse source problem is formulated into a variational problem and a conjugate gradient algorithm is proposed to solve it. The efficiency and robust of the proposed method are supported by some numerical experiments.     

Exponential Tikhonov Regularization Method for Solving an Inverse Source Problem of Time Fractional Diffusion Equation

In this paper, we mainly study an inverse source problem of time fractional diffusion equation in a bounded domain with an over-specified terminal condition at a fixed time. A novel regularization method, which we call the exponential Tikhonov regularization method with a parameter $\gamma$, is proposed to solve the inverse source problem, and the corresponding convergence analysis is given under a-priori and a-posteriori regularization parameter choice rules. When $\gamma$ is less than or equal to zero, the optimal convergence rate can be achieved and it is independent of the value of $\gamma$. However, when $\gamma$ is greater than zero, the optimal convergence rate depends on the value of $\gamma$ which is related to the regularity of the unknown source. Finally, numerical experiments are conducted for showing the effectiveness of the proposed exponential regularization method.     

Two Tikhonov‐type regularization methods for inverse source problem on the Poisson equation

In this paper, we investigate a problem of the identification of an unknown source on Poisson equation from some fixed location. A conditional stability estimate for an inverse heat source problem is proved. We show that such a problem is mildly ill‐posed and further present two Tikhonov‐type regularization methods (a generalized Tikhonov regularization method and a simplified generalized Tikhonov regularization method) to deal with this problem. Convergence estimates are presented under the a priori choice of the regularization parameter. Numerical results are presented to illustrate the accuracy and efficiency of our methods. Copyright © 2012 John Wiley & Sons, Ltd.     

The revised generalized Tikhonov regularization for the inverse time-dependent heat source problem

This paper investigates the inverse problem of finding a time-dependent heat source in a parabolic equation where the data is given at a fixed location. A conditional stability result is given, and a revised generalized Tikhonov regularization method with error estimate is also provided. Numerical examples show that the regularization method is effective and stable.     

偏微分方程边值反问题的数值方法研究

本文研究了奇异积分方程在反边值问题中的应用问题.利用圆周上的自然积分方程及其反演公式,把Laplace方程的边值反问题转化为一对超奇异积分方程和弱奇异积分方程的组合,通过选取三角插值近似奇异积分的计算并构造相应的配置格式,并使用Tikhonov正则化方法求解所得到的线性方程组.数值实验表明了该方法的有效性.     

An inverse problem of identifying the time-dependent potential in a fourth-order pseudo-parabolic equation from additional condition

The aim of this work is to identify numerically, for the first time, the time-dependent potential coefficient in a fourth-order pseudo-parabolic equation with nonlocal initial data, nonlocal boundary conditions, and the boundary data as overdetermination condition. This problem emerges significantly in the modeling of various phenomena in physics and engineering. From literature we already know that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data. For the numerical realization, we apply the quintic B-spline (QB-spline) collocation method for discretizing the pseudo-parabolic problem and the Tikhonov regularization for finding a stable and accurate solution. The resulting nonlinear minimization problem is solved using the MATLAB subroutine lsqnonlin. Moreover, the von Neumann stability analysis is also discussed.     

A backward problem for the time‐fractional diffusion equation

In this paper, we are concerned with the backward problem of reconstructing the initial condition of a time‐fractional diffusion equation from interior measurements. We establish uniqueness results and provide stability analysis. Our method is based on the eigenfunction expansion of the forward solution and the Tikhonov regularization to tackle the ill‐posedness issue of the underlying inverse problem. Some numerical examples are included to illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

On Regularization of a Source Identification Problem in a Parabolic PDE and its Finite Dimensional Analysis

We consider the inverse problem of identifying a general source term, which is a function of both time variable and the spatial variable, in a parabolic PDE from the knowledge of boundary measurements of the solution on some portion of the lateral boundary. We transform this inverse problem into a problem of solving a compact linear operator equation. For the regularization of the operator equation with noisy data, we employ the standard Tikhonov regularization, and its finite dimensional realization is done using a discretization procedure involving the space $L^2(0,\tau;L^2(Ω))$. For illustrating the specification of an a priori source condition, we have explicitly obtained the range space of the adjoint of the operator involved in the operator equation.     

Numerical solution of the inverse electrocardiography problem with the use of the Tikhonov regularization method

The inverse electrocardiography problem related to medical diagnostics is considered in terms of potentials. Within the framework of the quasi-stationary model of the electric field of the heart, the solution of the problem is reduced to the solution of the Cauchy problem for the Laplace equation in R ³. A numerical algorithm based on the Tikhonov regularization method is proposed for the solution of this problem. The Cauchy problem for the Laplace equation is reduced to an operator equation of the first kind, which is solved via minimization of the Tikhonov functional with the regularization parameter chosen according to the discrepancy principle. In addition, an algorithm based on numerical solution of the corresponding Euler equation is proposed for minimization of the Tikhonov functional. The Euler equation is solved using an iteration method that involves solution of mixed boundary value problems for the Laplace equation. An individual mixed problem is solved by means of the method of boundary integral equations of the potential theory. In the study, the inverse electrocardiography problem is solved in region Ω close to the real geometry of the torso and heart.     

Numerical methods for some inverse problems of heart electrophysiology

We consider numerical methods for solving inverse problems that arise in heart electrophysiology. The first inverse problem is the Cauchy problem for the Laplace equation. Its solution algorithm is based on the Tikhonov regularization method and the method of boundary integral equations. The second inverse problem is the problem of finding the discontinuity surface of the coefficient of conductivity of a medium on the basis of the potential and its normal derivative given on the exterior surface. For its numerical solution, we suggest a method based on the method of boundary integral equations and the assumption on a special representation of the unknown surface.     

The Sinc‐Galerkin method for solving an inverse parabolic problem with unknown source term

The inverse problem of determining an unknown source term depending on space variable in a parabolic equation is considered. A numerical algorithm is presented for recovering the unknown function and obtaining a solution of the problem. As this inverse problem is ill‐posed, Tikhonov regularization is used for finding a stable solution. For solving the direct problem, a Galerkin method with the Sinc basis functions in both the space and time domains is presented. This approximate solution displays an exponential convergence rate and is valid on the infinite time interval. Finally, some examples are presented to illustrate the ability and efficiency of this numerical method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

A Tikhonov-type method for solving a multidimensional inverse heat source problem in an unbounded domain

In this study we prove a stability estimate for an inverse heat source problem in the n-dimensional case. We present a revised generalized Tikhonov regularization and obtain an error estimate. Numerical experiments for the one-dimensional and two-dimensional cases show that the revised generalized Tikhonov regularization works well.     

An approximate solution of a Fredholm integral equation of the first kind by the residual method

A Tikhonov finite-dimensional approximation is applied to a Fredholm integral equation of the first kind. This allows using a variational regularization method with a regularization parameter from the residual principle and reducing the problem to a system of linear algebraic equations. The accuracy of the approximate solution is estimated with allowance for the error of the finitedimensional approximation of the problem. The use of this approach is illustrated by solving an inverse boundary value problem for the heat conduction equation.     

LOCAL REGULARIZATION METHODS FOR THE STABILIZATION OF LINEAR ILL-POSED EQUATIONS OF VOLTERRA TYPE

Inverse problems based on first-kind Volterra integral equations appear naturally in the study of many applications, from geophysical problems to the inverse heat conduction problem. The ill-posedness of such problems means that a regularization technique is required, but classical regularization schemes like Tikhonov regularization destroy the causal nature of the underlying Volterra problem and, in general, can produce oversmoothed results. In this paper we investigate a class of local regularization methods in which the original (unstable) problem is approximated by a parameterized family of well-posed, second-kind Volterra equations. Being Volterra, these approximating second-kind equations retain the causality of the original problem and allow for quick sequential solution techniques. In addition, the regularizing method we develop is based on the use of a regularization parameter which is a function (rather than a single constant), allowing for more or less smoothing at localized points in the domain. We take this approach even further by adopting the flexibility of an additional penalty term (with variable penalty function) and illustrate the sequential selection of the penalty function in a numerical example.     

A fractional Tikhonov method for solving a Cauchy problem of Helmholtz equation

In this paper, we study a fractional Tikhonov regularization method (FTRM) for solving a Cauchy problem of Helmholtz equation in the frequency domain. On the one hand, the FTRM retains the advantage of classical Tikhonov method. On the other hand, our method can prevent the effect of oversmoothing of classical Tikhonov method and conveniently control the amount of damping. The convergence error estimates between the exact solution and its regularization approximation are constructed. Several interesting numerical examples are provided, which validate the effectiveness of the proposed method.     

一维热传导方程移动边界识别的计算方法

构造了一种正则化的积分方程方法来由Cauchy数据确定一维热传导方程的移动边界．在将区域延拓至规则区域后,通过Fourier方法将问题转化为一个第一类Volterra积分方程．然后分别用Lavrentiev正则化方法以及Tikhonov正则化方法将不稳定的第一类Volterra积分方程转化为适定的第二类积分方程,并分别将积分方程转化为常微分方程组,并用Runge—Kutta方法数值求解,以及直接离散来求解．最后通过自由边界上的条件得到数值的移动边界．通过一些数值试验表明此方法是有效可行的,并且给出的方法无需迭代,数值计算较简单．     

On a radially symmetric inverse heat conduction problem

In this paper, we treat an inverse problem for a radially symmetric heat equation, which arises from non-destructive evaluation by thermal imaging. The problem can also be considered as an inverse heat conduction problem. Based on a weighted energy method, we give a conditional stability estimate. A feasible regularization method is provided for numerical simulation. The reconstruction experiment is done for verifying the efficiency of the regularization method.