Local Regularization Method Research Articles

Continuous future polynomial regularization of 1-smoothing Volterra problems

We examine a continuous, local regularization method for 1-smoothing, ill-posed, first-kind Volterra problems. We show that the method converges to the true solution as the noise in the data decreases to zero. Numerical examples are provided to demonstrate the performance of the method in comparison to earlier techniques.

Sparse modeling using orthogonal forward regression with PRESS statistic and regularization.

The paper introduces an efficient construction algorithm for obtaining sparse linear-in-the-weights regression models based on an approach of directly optimizing model generalization capability. This is achieved by utilizing the delete-1 cross validation concept and the associated leave-one-out test error also known as the predicted residual sums of squares (PRESS) statistic, without resorting to any other validation data set for model evaluation in the model construction process. Computational efficiency is ensured using an orthogonal forward regression, but the algorithm incrementally minimizes the PRESS statistic instead of the usual sum of the squared training errors. A local regularization method can naturally be incorporated into the model selection procedure to further enforce model sparsity. The proposed algorithm is fully automatic, and the user is not required to specify any criterion to terminate the model construction procedure. Comparisons with some of the existing state-of-art modeling methods are given, and several examples are included to demonstrate the ability of the proposed algorithm to effectively construct sparse models that generalize well.

Variable-smoothing local regularization methods for first-kind integralequations

We consider the local regularization problem for integral equations of the firstkind, generalizing previous work which applied only to problems of Volterra type.Our approach allows for local control of the regularization process, allowing forresolution of fine/sharp features of solutions without having to resort tonondifferentiable optimization techniques. In addition, we present examplesillustrating the numerical implementation of one version of the resulting localregularization algorithm and show that, under quite reasonable assumptions, theoperation count of the local method compares well with that of standardTikhonov regularization.

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.

On a Suboptimal Filtration Method for Solving Convolution-Type Integral Equations of the First Kind

In this paper a so-called local regularization method (Arsenin et al.,USSR Comput. Math. Math. Phys.28(3) (1988), 113–125; Arsenin and Timinov,Soviet Math. Dokl.32(2) (1985), 566–570; Tikhonov et al., “Mathematical Problems of Computer Tomography,” Nauka, Moscow, 1987) for solving convolution-type integral equations of the first kind is analyzed. It is shown that to increase the precision of the solution, it is appropriate to do only one approximation in the iterative scheme of this method. It is also shown that this method should not be interpreted as a method with variable regularization parameter α; one should view it as a method of suboptimal filtration. A new variant of a suboptimal filtration method is formulated.

Adaptive local regularization methods for the inverse ECG problem.

One of the fundamental problems in theoretical electrocardiography can be characterized by an inverse problem. We present new methods for achieving better estimates of heart surface potential distributions in terms of torso potentials through an inverse procedure. First, we outline an automatic adaptive refinement algorithm that minimizes the spatial discretization error in the transfer matrix, increasing the accuracy of the inverse solution. Second, we introduce a new local regularization procedure, which works by partitioning the global transfer matrix into sub-matrices, allowing for varying amounts of smoothing. Each submatrix represents a region within the underlying geometric model in which regularization can be specifically 'tuned' using an a priori scheme based on the L-curve method. This local regularization method can provide a substantial increase in accuracy compared to global regularization schemes. Within this context of local regularization, we show that a generalized version of the singular value decomposition (GSVD) can further improve the accuracy of ECG inverse solutions compared to standard SVD and Tikhonov approaches. We conclude with specific examples of these techniques using geometric models of the human thorax derived from MRI data.

Approximation of Ill-Posed Volterra Problems via Predictor–Corrector Regularization Methods

First-kind Volterra problems arise in numerous applications, from inverse problems in mathematical biology to inverse heat conduction problems. Unfortunately, such problems are also ill posed due to lack of continuous dependence of solutions on data. Consequently, numerical methods to solve first-kind Volterra equations are effective only when regularizing features are built into the algorithms or used to control stepsize. Classical methods often combine numerical discretization with Tikhonov regularization, but in doing so the underlying Volterra (or causal) nature of the original problem is often destroyed. Instead, a “predictor–corrector” type of numerical method is proposed which combines at each step “local regularization” ideas with the use of small intervals of future data. The result is a regularized numerical method which retains much of the causal nature of the Volterra problem and may be solved in fast sequential steps, often improving upon the performance of classical algorithms such as those based on standard Tikhonov regularization. In this paper, the discretized local regularization method is described, and proofs are given of convergence of the method, with rate of convergence being “best possible” with regard to the amount of error in the data. Further, by linking the regularization parameter of the stabilizing method (i.e., the length of the “future interval”in the future-sequential method) to the approximation stepsize, great simplification of the resulting numerical algorithm is obtained. Relevant numerical examples are included.

A local regularization method for linear operator equations of the first kind and its applications

A method is proposed for constructing a sequence of regularized solutions of a linear operator equation of the first kind in Hilbert spaces, based on the use of a priori information of spectral nature about the error in the initial data. The rigorous basis of the method is established. Applications of the method to linear operator equations with a completely continuous operator and a convolution-type integral operator are considered; an application to the fundamental equation of linear computer tomography is also discussed. Results of numerical experiments are presented.