Combining approximate solutions for linear discrete ill-posed problems

An extrapolated TSVD method for linear discrete ill-posed problems with Kronecker structure

Cascadic multilevel methods for the solution of linear discrete ill-posed problems with noise-reducing restriction and prolongation operators recently have been developed for the restoration of blur- and noise-contaminated images. This is a particular ill-posed problem. The multilevel methods were found to determine accurate restorations with fairly little computational work. This paper describes noise-reducing multilevel methods for the solution of general linear discrete ill-posed problems.

Invertible smoothing preconditioners for linear discrete ill-posed problems

Tikhonov regularization is a popular method to approximate solutions of linear discrete ill-posed problems when the observed or measured data is contaminated by noise. Multiparameter Tikhonov regularization may improve the quality of the computed approximate solutions. We propose a new iterative method for large-scale multiparameter Tikhonov regularization with general regularization operators based on a multidirectional subspace expansion. The multidirectional subspace expansion may be combined with subspace truncation to avoid excessive growth of the search space. Furthermore, we introduce a simple and effective parameter selection strategy based on the discrepancy principle and related to perturbation results.

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.

The Newsletter is a key element in further enhancing the value of the journal to the inverse problems community. So why not be a part of this exciting forum by sending to our Bristol office material suitable for inclusion under any of the categories mentioned above. Your contributions will be very welcome. Book review Introduction to Inverse Problems in Imaging M Bertero and P Boccacci 1998 Bristol: Institute of Physics 362 pp ISBN 0-7503-0435-9 (pbk) £25.00, $49.00 This book shows that several problems in imaging are in fact linear inverse problems. In general inverse problems are ill-posed in the sense that small perturbations in the (measured) data have a significant influence on the output. Consequently, numerical methods developed in a general setting of inverse problems which cure the ill-posedness (such methods are called regularization methods) can be applied to solve problems in imaging. This is an excellent textbook on the principle of linear inverse problems, methods of their numerical solution and practical applications in imaging. It introduces basic ideas and methods for the solution of inverse and ill-posed problems while avoiding the mathematics of functional treatment of operator equations. This goal is achieved by a bottom-up presentation: the authors focus on the two problems of image deconvolution and tomography (bottom) and develop regularization theory specifically for these two problems. Most other books which are concerned with the numerical solution of inverse problems present a top-down approach. That is, a general regularization theory (top) is developed which is then applied to particular problems. The distinguished presentation makes the book very useful to students in applied mathematics, practitioners in engineering science and image processing. This book has the best prerequisites to establish a close link between the mathematical fields of imaging and inverse problems. Mainly the book focuses on two topics: image deconvolution and linear imaging systems. 1. In image deconvolution the goal is to restore a space invariant blurred image. This part of the book contains mathematical tools in image deconvolution, examples of space invariant imaging systems, a comparison of image deconvolution techniques and low-pass filtering techniques, the analysis of the ill-posedness of the deconvolution problem, regularization methods for deconvolution such as constrained least squares regularization, Tikhonov regularization, iterative regularization and statistical methods. Since the authors focus on particular problems they are able to analyse various methods for deconvolution on a very concrete level. As an example I would like to mention that the authors analyse Fourier methods for deconvolution. Thus they provide an optimal starting point to the theory of inverse problems for people working in the area of signal and image processing. 2. The second part of the book considers linear imaging systems that are not of a convolution type such as tomography problems, diffraction problems and inverse scattering problems. Efficient numerical methods for the solution of this class of problems such as singular value decomposition, Tikhonov regularization methods and Fourier-based methods are presented. O Scherzer Universität Linz

Geophysical Inverse Theory and Regularization Problems

This chapter discusses iterative methods for the solution of very large severely ill-conditioned linear systems of equations that arise from the discretization of linear ill-posed problems. The right-hand side vector represents the given data and is assumed to be contaminated by errors. Solution methods proposed in the literature employ some form of filtering to reduce the influence of the error in the right-hand side on the computed approximate solution. The amount of filtering is determined by a parameter often referred to as the regularization parameter. We discuss how the filtering affects the computed approximate solution and consider the selection of regularization parameter. Methods in which a suitable value of the regularization parameter is determined during the computation without user intervention are emphasized. New iterative solution methods based on expanding explicitly chosen filter functions in terms of Chebyshev polynomials are presented. The properties of these methods are illustrated with applications to image restoration.

Iterative Tikhonov regularization of tensor equations based on the Arnoldi process and some of its generalizations

A stopping criterion for iterative regularization methods

Approximation-free running SVD and its application to motion detection

A new approach with piecewise-constant arguments to approximate and numerical solutions of oscillatory problems

An approximate solution for linear boundary-value problems with slowly varying coefficients

Linear-regularization methods provide a simple technique for deter­mining stable approximate solutions of linear ill-posed problems such as Fredholm equations of the first kind, Cauchy problems for elliptic equations and backward solution of forward parabolic equations. In most of these problems the solution must be positive to satisfy physical plausibility. In this paper we consider ill-posed first-kind convolution equations and related problems such as numerical differentiation, Radon transform and Laplace-transform inversion. We investigate several linear regularization algorithms which provide positive approximate solutions for these problems at least in the absence of errors on the data. For noisy data the solution is not necessarily positive. Because the appearance of negative values can then only be an effect of the noise, the negative part of the solution should be negligible with a suitable choice of the regularization parameter. A price to pay for ensuring positivity is always, however, a reduction in resolution.

A pressure deconvolution technique based on ridge regression with existing convolution features was explored and reinterpreted to frame a new optimization problem structure. We found a way to split the feature matrix into a deconvolution part and a smoothing part that handle each effect independently. The deconvolution part is comprised of a convolution matrix (Toeplitz matrix) for discrete-time convolution while the smoothing part is composed of a basis function of the impulse response. Hence, this formulation preserves linearity (a superposition of pressure solution) and the kernel for nonlinear mapping (e.g., polynomial kernel, radial basis function kernel, etc.) is effective only on the smoothing part (a basis function of impulse response). Importantly, the key idea of existing convolution feature is the modelling of the impulse response with a linear combination of a basis functions.For single-well pressure behavior, ridge regression often struggles with high-frequency measurements that contain multistage responses. An impulse response might share similar functional characteristics, for example wellbore storage and pseudosteady state are both linear responses but there is only one linear basis function in the ridge regression which cannot describe them both. In traditional well test interpretation, the engineer is typically focused on certain parts of the impulse response on a log-scale, to identify reservoir behavior (e.g., early-time for wellbore storage, transient-time for infinite acting radial flow, late-time for boundary effect, etc.). Analogously, a basis function could be formed from multiple pressure responses in each log-interval (inherent from a solution of the diffusion equation) then bridged together with a constraint on function value and its derivative at each interval boundary. This is the basic principle of spline regression that has more flexibility in terms of function expression. In addition, it also subsumes ridge regression with Laplacian regularization.For multiwell pressure behavior, additional basis functions are included to handle interference responses. Typically, the exponential integral is approximated with a logarithm function for single-well pressure response which is also one of the basis functions in previous convolution features. Nonetheless, interference pressure response is more subtle and its solution could not be easily approximated, which previously resulted in a poor fitting using earlier methods. By inspection of the convergent series of the exponential integral, its higher-order terms are associated with 1/tm that can be easily included as a basis function to enhance the capability of the model and capture the detail of the interference test that is essential for multiwell problems. A full extension to multiwell problems is also presented with the additional derivative constraint for the impulse response and symmetry constraint. Finally, we can quantify the sensitivity of the deconvolution process using singular value decomposition (SVD) and Monte Carlo simulation (MCS). Using singular value decomposition, we could reveal the orthogonal basis of the feature matrix and its corresponding singular value that helped us identify the effectiveness of the basis functions as well as their susceptibility to noise of certain pattern. Nevertheless, singular value decomposition is only suitable for noisy pressure measurement problems without inequality constraint. Thus, Monte Carlo simulation is required for full quantification of noisy measurements in both pressure and flow rate. We can observe a propagation of error in measurement to a band in pressure prediction and pressure derivative of the impulse response.This study developed a more robust method for deconvolution of pressure measurements. In addition, the mathematical insights obtained in the study allow for a more general understanding of how the deconvolution problem can be deconstructed into its relevant parts – likely allowing for expanded capability of deconvolution algorithms in industrial practice.