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.     

Iterated Tikhonov regularization with a general penalty term

Tikhonov regularization is one of the most popular approaches to solving linear discrete ill‐posed problems. The choice of the regularization matrix may significantly affect the quality of the computed solution. When the regularization matrix is the identity, iterated Tikhonov regularization can yield computed approximate solutions of higher quality than (standard) Tikhonov regularization. This paper provides an analysis of iterated Tikhonov regularization with a regularization matrix different from the identity. Computed examples illustrate the performance of this method.     

Regularization for a class of quasi-variational-hemivariational inequalities

The aim of the present paper is to study the solvability and regularization for a class of multivalued quasi-variational–hemivariational inequalities in reflexive Banach spaces. By applying the Kluge fixed point theorem and the Minty technique, we prove the solvability of the considered multivalued quasi-variational–hemivariational inequality, based on which some convergence results are obtained by introducing its regularization problem with the help of regularization operator. The applicability of the obtained abstract results is established by a mathematical model of a frictional contact problem with a class of elastic material, where the existence and stability results for the weak solution of contact problem are studied.     

Total variation regularization for the reconstruction of a mountain topography

The aim of this paper is to reconstruct a paleo mountain topography using a total variation (TV) regularization. A coupled system integrates the tectonic process with the surface process to simulate the evolution of a paleo mountain topography. The tectonic process and the surface process are described by a 3D convection-diffusion equation and a 2D convection-diffusion equation, respectively. We recover a piecewise smooth velocity field for the tectonic process as well as reconstruct a piecewise smooth mountain topography for the surface process using a TV regularization in an iterative fashion. The effects of the number of samples and of wavelengths on inversions are investigated. In our numerical experiments, we shall experience three difficulties: (I) recovering a large quantity of information from the limited amount of measurement data; (II) detecting sharp features; (III) choosing a properly initial guess value for a TV regularization. The numerical experiments show that a TV regularization is an efficient and stable algorithm.     

Stability estimates for a class of semi-linear ill-posed problems

We study a final value problem for a nonlinear parabolic equation with positive self-adjoint unbounded operator coefficients. The problem is ill-posed. The regularized equation is given by a modified quasi-reversibility method. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained.     

On an a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems

Summary. In the study of the choice of the regularization parameter for Tikhonov regularization of nonlinear ill-posed problems, Scherzer, Engl and Kunisch proposed an a posteriori strategy in 1993. To prove the optimality of the strategy, they imposed many very restrictive conditions on the problem under consideration. Their results are difficult to apply to concrete problems since one can not make sure whether their assumptions are valid. In this paper we give a further study on this strategy, and show that Tikhonov regularization is order optimal for each with the regularization parameter chosen according to this strategy under some simple and easy-checking assumptions. This paper weakens the conditions needed in the existing results, and provides a theoretical guidance to numerical experiments. Received August 8, 1997 / Revised version received January 26, 1998     

Convergence analysis of a regularized approximation for solving Fredholm integral equations of the first kind

In this paper, we suggest a convergence analysis for solving Fredholm integral equations of the first kind using Tikhonov regularization under supremum norm. We also provide an a priori parameter choice strategy for choosing the regularization parameter and obtain an error estimate.     

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.     

Optimal order results for a class of regularization methods using unbounded operators

A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an a priori parameter choice strategy, it is shown that the method yields the optimal order. Error estimates have also been obtained under stronger assumptions on the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper include, as particular cases for Tikhonov regularization, the main result of Mair (1994) with an a priori parameter choice, and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also some recent results of Tautenhahn (1996) in the setting of Hilbert scales.     

Two iterative methods for a Cauchy problem of the elliptic equation with variable coefficients in a strip region

We investigate a Cauchy problem for the elliptic equation with variable coefficients. This problem is severely ill-posed in the sense of Hadamard and the regularization techniques are required to stabilize numerical computations. We give two iterative methods to deal with it. Under an a-priori and an a-posteriori selection rule for the regularization parameter, the convergence rates of two algorithms are obtained. Numerical results show that two methods work well.     

Fourier regularization for a backward heat equation

In this paper a simple and convenient new regularization method for solving backward heat equation—Fourier regularization method is given. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.     

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.     

Semi-discretization Difference Approximation for a Cauchy Problem of Heat Equation in Two-dimensional Space

In this paper we consider a semi-descretization difference scheme for solving a Cauchy problem of heat equation in two-dimensional setting. Some error estimates are proved for the semi-descretization difference regularization method which cannot be fitted into the framework of regularization theory presented by Engl, Hanke and Neubauer. Numerical results show that the proposed method works well.     

Regularized Adaptation of Fuzzy Inference Systems. Modelling the Opinion of a Medical Expert about Physical Fitness: An Application

This study presents a new approach to adaptation of Sugeno type fuzzy inference systems using regularization, since regularization improves the robustness of standard parameter estimation algorithms leading to stable fuzzy approximation. The proposed method can be used for modelling, identification and control of physical processes. A recursive method for on-line identification of fuzzy parameters employing Tikhonov regularization is suggested. The power of approach was shown by applying it to the modelling, identification, and adaptive control problems of dynamic processes. The proposed approach was used for modelling of human-decisions (experience) with a fuzzy inference system and for the fuzzy approximation of physical fitness with real world medical data.     

Discretized Tikhonov–Phillips regularization for a naturally linearized parameter identification problem

The problem of identification of the diffusion coefficient in the partial differential equation is considered. We discuss a natural linearization of this problem and application of discretized Tikhonov–Phillips regularization to its linear version. Using recent results of regularization theory, we propose a strategy for the choice of regularization and discretization parameters which automatically adapts to unknown smoothness of the coefficient. The estimation of the accuracy will be given and various numerical test supporting theoretical results will be presented.     

On the convergence of a regularizing Levenberg–Marquardt scheme for nonlinear ill-posed problems

In this note, we study the convergence of the Levenberg–Marquardt regularization scheme for nonlinear ill-posed problems. We consider the case that the initial error satisfies a source condition. Our main result shows that if the regularization parameter does not grow too fast (not faster than a geometric sequence), then the scheme converges with optimal convergence rates. Our analysis is based on our recent work on the convergence of the exponential Euler regularization scheme (Hochbruck et al. in Inverse Probl 25(7):075009, 2009).     

一类非线性不适定问题的Tikhonov正则化

本文在B anach空间中,研究一类非线性不适定问题的正则化,所研究的算子是多值的,且是近拟的,假定原始问题是可解的,利用带双参数的T ikhonov正则化方法构造出逼近步骤.     

Numerical solutions of a two-phase membrane problem

In this paper different numerical methods for a two-phase free boundary problem are discussed. In the first method a novel iterative scheme for the two-phase membrane is considered. We study the regularization method and give an a posteriori error estimate which is needed for the implementation of the regularization method. Moreover, an efficient algorithm based on the finite element method is presented. It is shown that the sequence constructed by the algorithm is monotone and converges to the solution of the given free boundary problem. These methods can be applied for the one-phase obstacle problem as well.     

Duality-based regularization in a linear convex mathematical programming problem

For a linear convex mathematical programming (MP) problem with equality and inequality constraints in a Hilbert space, a dual-type algorithm is constructed that is stable with respect to input data errors. In the algorithm, the dual of the original optimization problem is solved directly on the basis of Tikhonov regularization. It is shown that the necessary optimality conditions in the original MP problem are derived in a natural manner by using dual regularization in conjunction with the constructive generation of a minimizing sequence. An iterative regularization of the dual algorithm is considered. A stopping rule for the iteration process is presented in the case of a finite fixed error in the input data.     

Holomorphic Regularization of Singular Perturbations in a Banach Space

The holomorphic regularization method, which is a natural extension of Lomov’s regularization method, is used to solve strongly nonlinear singularly perturbed equations in Banach spaces. The existence of pseudoholomorphic solutions of such equations is proved, and the analytic properties of their Galerkin approximations are studied.