Identification of a time-dependent source term for a time fractional diffusion problem

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.     

Two regularization methods for a Cauchy problem for the Laplace equation

A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y=0, and boundary data are for x=0 and x=π. The solution for 0<y?1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

Spectral Galerkin method and its application to a Cauchy problem of Helmholtz equation

Based on a mathematical model of laser beams, we present a spectral Galerkin method for solving a Cauchy problem of the Helmholtz equation in a rectangle, where the Cauchy data pairs are given at y?=?0 and boundary data are for x?=?0 and x?=?π. The solution is sought in the interval 0?<?y?<?1. The spectral Galerkin method is considered as a regularization method. We then perform an analysis on the error bound for this method. For illustration, several numerical experiments are constructed to demonstrate the feasibility and efficiency of the proposed 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.     

A regularization method for solving the Cauchy problem for the Helmholtz equation

In this paper, we investigate a Cauchy problem associated with Helmholtz-type equation in an infinite “strip”. This problem is well known to be severely ill-posed. The optimal error bound for the problem with only nonhomogeneous Neumann data is deduced, which is independent of the selected regularization methods. A framework of a modified Tikhonov regularization in conjunction with the Morozov’s discrepancy principle is proposed, it may be useful to the other linear ill-posed problems and helpful for the other regularization methods. Some sharp error estimates between the exact solutions and their regularization approximation are given. Numerical tests are also provided to show that the modified Tikhonov method works well.     

双调和方程柯西问题的修正的Tikhonov正则化方法

本文研究了双调和方程柯西问题,这类是不适定的,即问题的解(如果存在)不连续依赖于测量数据.首先在精确解的先验假设下给出问题的条件稳定性结果.接着利用修正的Tikhonov正则化方法求解此不适定问题.在先验和后验正则化参数选取规则下,给出正则解和精确解之间的误差估计式.最后给出几个数值例子验证此正则化方法求解此类反问题的有效性.     

Sharp estimates for approximations to a nonlinear backward heat equation

A nonlinear backward heat problem for an infinite strip is considered. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we use the Fourier regularization method to solve the problem. Some sharp estimates of the error between the exact solution and its regularization approximation are given.     

Regularization parameter selection and an efficient algorithm for total variation-regularized positron emission tomography

In positron emission tomography, image data corresponds to measurements of emitted photons from a radioactive tracer in the subject. Such count data is typically modeled using a Poisson random variable, leading to the use of the negative-log Poisson likelihood fit-to-data function. Regularization is needed, however, in order to guarantee reconstructions with minimal artifacts. Given that tracer densities are primarily smoothly varying, but also contain sharp jumps (or edges), total variation regularization is a natural choice. However, the resulting computational problem is quite challenging. In this paper, we present an efficient computational method for this problem. Convergence of the method has been shown for quadratic regularization functions and here convergence is shown for total variation regularization. We also present three regularization parameter choice methods for use on total variation-regularized negative-log Poisson likelihood problems. We test the computational and regularization parameter selection methods on two synthetic data sets.     

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.     

Numerical solution of forward and backward problem for 2-D heat conduction equation

For a two-dimensional heat conduction problem, we consider its initial boundary value problem and the related inverse problem of determining the initial temperature distribution from transient temperature measurements. The conditional stability for this inverse problem and the error analysis for the Tikhonov regularization are presented. An implicit inversion method, which is based on the regularization technique and the successive over-relaxation (SOR) iteration process, is established. Due to the explicit difference scheme for a direct heat problem developed in this paper, the inversion process is very efficient, while the application of SOR technique makes our inversion convergent rapidly. Numerical results illustrating our method are also given.     

解带有二次约束非凸二次规划问题的一个分枝缩减方法

在这篇论文里,有机地把外逼近方法与分枝定界技术结合起来,提出了解带有二次约束非凸二次规划问题的一个分枝缩减方法;给出了原问题的一个新的线性规划松弛,以便确定它在超矩形上全局最优值的一个下界;利用超矩形的一个深度二级剖分方法,以及超矩形的缩减和删除技术,提高算法的收敛速度;证明了在知道原问题可行点的条件下,该算法在有限步里就可以获得原问题的一个全局最优化解,并且用一个例子说明了该算法是有效的.     

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.     

The regularization method for an obstacle problem

Summary. We give a relatively complete analysis for the regularization method, which is usually used in solving non-differentiable minimization problems. The model problem considered in the paper is an obstacle problem. In addition to the usual convergence result and a-priori error estimates, we provide a-posteriori error estimates which are highly desired for practical implementation of the regularization method. Received March 22, 1993 / Revised version received October 11, 1993     

Regularization of a backward problem for the inhomogeneous time-fractional wave equation

In this paper, we consider a backward problem for an inhomogeneous time-fractional wave equation in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The existence and regularity for the backward problem are investigated. The backward problem is ill-posed, and we propose a regularizing scheme by using a modified regularization method. We also prove the convergence rate for the regularized solution by using some a priori regularization parameter choice rule.     

Besov regularization,thresholding and wavelets for smoothing data

Smoothing of data is a problem very important for many applications ranging from 1-D signals (e.g., speech) to 2-D and 3-D signals (e.g., images). Many methods exist in the literature for facing the problem; in the present paper we point our attention on regularization. We shall treat regularization methods in a general framework which is well suited for wavelet analysis; in particular we shall investigate on the relation existing between thresholding methods and regularization. We shall also introduce a new regularization method (Besov regularization), which includes some known regularization and thresholding methods as particular cases. Numerical experiments based on some test problems will be performed in order to compare the performance of some methods of smoothing data.

AMS (MOS) Subject Classifications: 65R30, 41A60.     

Central difference regularization method for inverse source problem on the Poisson equation

In this paper, we consider an inverse problem of determining an unknown source for the Poisson equation. Since this problem is mildly ill-posed, we apply a central difference regularization method to solve this problem. Furthermore, the convergence estimate is established under a priori choice of the regularization parameter. Some numerical results verify that the proposed method is stable and effective.     

A comparison of some inverse methods for estimating the initial condition of the heat equation

In this work we analyze two explicit methods for the solution of an inverse heat conduction problem and we confront them with the least-squares method, using for the solution of the associated direct problem a classical finite difference method and a method based on an integral formulation. Finally, the Tikhonov regularization connected to the least-squares criterion is examined. We show that the explicit approaches to this inverse heat conduction problem will present disastrous results unless some kind of regularization is used.     

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

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogs and they are used to model anomalous diffusion, especially in physics. In this paper, we study a backward problem for an inhomogeneous time-fractional diffusion equation with variable coefficients in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The backward problem is ill-posed and we propose a regularizing scheme by using Tikhonov regularization method. We also prove the convergence rate for the regularized solution by using an a priori regularization parameter choice rule. Numerical examples illustrate applicability and high accuracy of the proposed method.     

A variational-type method of fundamental solutions for a Cauchy problem of Laplace’s equation

In this paper, we propose a new regularization method based on a finite-dimensional subspace generated from fundamental solutions for solving a Cauchy problem of Laplace’s equation in a simply-connected bounded domain. Based on a global conditional stability for the Cauchy problem of Laplace’s equation, the convergence analysis is given under a suitable choice for a regularization parameter and an a-priori bound assumption to the solution. Numerical experiments are provided to support the analysis and to show the effectiveness of the proposed method from both accuracy and stability.     

求解热源识别问题的修正吉洪诺夫正则化方法

该文研究了一个热源识别问题,通过引入修正吉洪诺夫方法来处理问题的不适定性,在一种先验和一种后验参数选取准则下,分别获得了问题的误差估计.数值例子进一步验证了方法的有效性和稳定性.