Heuristic Parameter Choice Rules for Tikhonov Regularization with Weakly Bounded Noise

We study the choice of the regularization parameter for linear ill-posed problems in the presence of noise that is possibly unbounded but only finite in a weaker norm, and when the noise-level is unknown. For this task, we analyze several heuristic parameter choice rules, such as the quasi-optimality, heuristic discrepancy, and Hanke-Raus rules and adapt the latter two to the weakly bounded noise case. We prove convergence and convergence rates under certain noise conditions. Moreover, we analyze and provide conditions for the convergence of the parameter choice by the generalized cross-validation and predictive mean-square error rules.     

Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces

The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a sequential variant of the discrepancy principle is analysed. In many cases, such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here, we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.     

Multi-penalty regularization in learning theory

In this paper we establish the error estimates for multi-penalty regularization under the general smoothness assumption in the context of learning theory. One of the motivation for this work is to study the convergence analysis of two-parameter regularization theoretically in the manifold learning setting. In this spirit, we obtain the error bounds for the manifold learning problem using more general framework of multi-penalty regularization. We propose a new parameter choice rule “the balanced-discrepancy principle” and analyze the convergence of the scheme with the help of estimated error bounds. We show that multi-penalty regularization with the proposed parameter choice exhibits the convergence rates similar to single-penalty regularization. Finally on a series of test samples we demonstrate the superiority of multi-parameter regularization over single-penalty regularization.     

THE QUASI-BOUNDARY VALUE METHOD FOR IDENTIFYING THE INITIAL VALUE OF THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION

In this article, we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation. This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem. We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule. Some numerical results in one-dimensional case and two-dimensional case show that our method is effcient and stable.     

Necessary and sufficient conditions for power convergence rate of approximations in Tikhonov’s scheme for solving ill-posed optimization problems

We investigate a rate of convergence of estimates for approximations generated by Tikhonov’s scheme for solving ill-posed optimization problems with smooth functionals under a structural nonlinearity condition in a Hilbert space, in the cases of exact and noisy input data. In the noise-free case, we prove that the power source representation of the desired solution is close to a necessary and sufficient condition for the power convergence estimate having the same exponent with respect to the regularization parameter. In the presence of a noise, we give a parameter choice rule that leads for Tikhonov’s scheme to a power accuracy estimate with respect to the noise level.     

On the quasi-optimal rules for the choice of the regularization parameter in case of a noisy operator

A usual way to characterize the quality of different a posteriori parameter choices is to prove their order-optimality on the different sets of solutions. In paper by Raus and H?marik (J Inverse Ill-Posed Probl 15(4):419–439, 2007) we introduced the property of the quasi-optimality to characterize the quality of the rule of the a posteriori choice of the regularization parameter for concrete problem Au = f in case of exact operator and discussed the quasi-optimality of different well-known rules for the a posteriori parameter choice as the discrepancy principle, the modification of the discrepancy principle, balancing principle and monotone error rule. In this paper we generalize the concept of the quasi-optimality for the case of a noisy operator and concretize results for the mentioned parameter choice rules.     

Source Identification for the Time-Fractional Diffusion Equation With Robin Boundary Conditions北大核心CSCD

对Robin边界条件时间分数阶扩散方程的源项辨识问题进行了研究。这类问题是不适定的,因此提出了一种迭代型正则化方法,得到了源项辨识问题的正则近似解。给出了先验和后验参数选取规则下正则近似解和精确解之间的误差估计,数值算例验证了该方法的有效性。     

On a terminal value problem for a generalization of the fractional diffusion equation with hyper-Bessel operator

In this paper, we consider an inverse problem of recovering the initial value for a generalization of time-fractional diffusion equation, where the time derivative is replaced by a regularized hyper-Bessel operator. First, we investigate the existence and regularity of our terminal value problem. Then we show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Tikhonov regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.     

Identifying an unknown source in the Poisson equation by the method of Tikhonov regularization in Hilbert scales

In this paper, we consider the problem for identifying the unknown source in the Poisson equation. The Tikhonov regularization method in Hilbert scales is extended to deal with illposedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. The user does not need to estimate the smoothness parameter and the a priori bound of the exact solution when the a posteriori choice rule is used. Numerical examples show that the proposed method is effective and stable.     

Adaptive Choice of the Regularization Parameter in Numerical Differentiation

We investigate a novel adaptive choice rule of the Tikhonov regularization parameter in numerical differentiation which is a classic ill-posed problem. By assuming a general unknown Hölder type error estimate derived for numerical differentiation, we choose a regularization parameter in a geometric set providing a nearly optimal convergence rate with very limited a-priori information. Numerical simulation in image edge detection verifies reliability and efficiency of the new adaptive approach.     

Balancing principle in supervised learning for a general regularization scheme

We discuss the problem of parameter choice in learning algorithms generated by a general regularization scheme. Such a scheme covers well-known algorithms as regularized least squares and gradient descent learning. It is known that in contrast to classical deterministic regularization methods, the performance of regularized learning algorithms is influenced not only by the smoothness of a target function, but also by the capacity of a space, where regularization is performed. In the infinite dimensional case the latter one is usually measured in terms of the effective dimension. In the context of supervised learning both the smoothness and effective dimension are intrinsically unknown a priori. Therefore we are interested in a posteriori regularization parameter choice, and we propose a new form of the balancing principle. An advantage of this strategy over the known rules such as cross-validation based adaptation is that it does not require any data splitting and allows the use of all available labeled data in the construction of regularized approximants. We provide the analysis of the proposed rule and demonstrate its advantage in simulations.     

A MODIFIED TIKHONOV REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF LAPLACE EQUATION

In this paper,we consider the Cauchy problem for the Laplace equation,which is severely ill-posed in the sense that the solution does not depend continuously on the data.A modified Tikhonov regularization method is proposed to solve this problem.An error estimate for the a priori parameter choice between the exact solution and its regularized approximation is obtained.Moreover,an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained.Numerical examples illustrate the validity and effectiveness of this method.     

Numerical differentiation by radial basis functions approximation

Based on radial basis functions approximation, we develop in this paper a new com-putational algorithm for numerical differentiation. Under an a priori and an a posteriori choice rules for the regularization parameter, we also give a proof on the convergence error estimate in reconstructing the unknown partial derivatives from scattered noisy data in multi-dimension. Numerical examples verify that the proposed regularization strategy with the a posteriori choice rule is effective and stable to solve the numerical differential problem. *The work described in this paper was partially supported by a grant from CityU (Project No. 7001646) and partially supported by the National Natural Science Foundation of China (No. 10571079).     

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.     

AIR Tools — A MATLAB package of algebraic iterative reconstruction methods

We present a MATLAB package with implementations of several algebraic iterative reconstruction methods for discretizations of inverse problems. These so-called row action methods rely on semi-convergence for achieving the necessary regularization of the problem. Two classes of methods are implemented: Algebraic Reconstruction Techniques (ART) and Simultaneous Iterative Reconstruction Techniques (SIRT). In addition we provide a few simplified test problems from medical and seismic tomography. For each iterative method, a number of strategies are available for choosing the relaxation parameter and the stopping rule. The relaxation parameter can be fixed, or chosen adaptively in each iteration; in the former case we provide a new “training” algorithm that finds the optimal parameter for a given test problem. The stopping rules provided are the discrepancy principle, the monotone error rule, and the NCP criterion; for the first two methods “training” can be used to find the optimal discrepancy parameter.     

A posteriori regularization parameter choice rule for the quasi-boundary value method for the backward time-fractional diffusion problem

In this paper, we consider a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine the initial data from a noisy final data. We propose a quasi-boundary value regularization method combined with an a posteriori regularization parameter choice rule to deal with the backward problem and give the corresponding convergence estimate.     

Old and new parameter choice rules for discrete ill-posed problems

Linear discrete ill-posed problems are difficult to solve numerically because their solution is very sensitive to perturbations, which may stem from errors in the data and from round-off errors introduced during the solution process. The computation of a meaningful approximate solution requires that the given problem be replaced by a nearby problem that is less sensitive to disturbances. This replacement is known as regularization. A regularization parameter determines how much the regularized problem differs from the original one. The proper choice of this parameter is important for the quality of the computed solution. This paper studies the performance of known and new approaches to choosing a suitable value of the regularization parameter for the truncated singular value decomposition method and for the LSQR iterative Krylov subspace method in the situation when no accurate estimate of the norm of the error in the data is available. The regularization parameter choice rules considered include several L-curve methods, Regińska’s method and a modification thereof, extrapolation methods, the quasi-optimality criterion, rules designed for use with LSQR, as well as hybrid methods.     

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.     

Adaptive Kernel Methods Using the Balancing Principle

The regularization parameter choice is a fundamental problem in Learning Theory since the performance of most supervised algorithms crucially depends on the choice of one or more of such parameters. In particular a main theoretical issue regards the amount of prior knowledge needed to choose the regularization parameter in order to obtain good learning rates. In this paper we present a parameter choice strategy, called the balancing principle, to choose the regularization parameter without knowledge of the regularity of the target function. Such a choice adaptively achieves the best error rate. Our main result applies to regularization algorithms in reproducing kernel Hilbert space with the square loss, though we also study how a similar principle can be used in other situations. As a straightforward corollary we can immediately derive adaptive parameter choices for various kernel methods recently studied. Numerical experiments with the proposed parameter choice rules are also presented.     

IDENTIFYING AN UNKNOWN SOURCE IN SPACE-FRACTIONAL DIFFUSION EQUATION

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