A Computationally Efficient Tool for Assessing the Depth Resolution in Potential-Field Inversion

In potential-field inversion problems, it can be difficult to obtain reliable information about the source distribution with respect to depth. Moreover, spatial resolution of the solution decreases with depth, and in fact the more ill-posed the problem – and the more noisy the data – the less reliable the depth information. Based on early work in depth resolution, defined in terms of the singular value decomposition, we introduce a tool APPROXDRP which uses an approximation of the singular vectors obtained by the iterative Lanczos bidiagonalization algorithm, making it well suited for large-scale problems. This tool allows a computational/visual analysis of how much the depth resolution in a computational potential-field inversion problem can be obtained from the given data.We show that when used in combination with a plot of the approximate SVD quantities, APPROXDRP may successfully show the limitations of depth resolution resulting from noise in the data. This allows a reliable analysis of the retrievable depth information and effectively guides the user in choosing the optimal number of iterations, for a given problem

A rounding error analysis of the joint bidiagonalization process with applications to the GSVD computation

The joint bidiagonalization(JBD) process is a useful algorithm for approximating some extreme generalized singular values and vectors of a large sparse or structured matrix pair {A,L\}. We present a rounding error analysis of the JBD process, which establishes connections between the JBD process and the two joint Lanczos bidiagonalizations. We investigate the loss of orthogonality of the computed Lanczos vectors. Based on the results of rounding error analysis, we investigate the convergence and accuracy of the approximate generalized singular values and vectors of {A,L\}. The results show that semiorthogonality of the Lanczos vectors is enough to guarantee the accuracy and convergence of the approximate generalized singular values, which is a guidance for designing an efficient semiorthogonalization strategy for the JBD process. We also investigate the residual norm appeared in the computation of the generalized singular value decomposition (GSVD), and show that its upper bound can be used as a stopping criterion.Comment: 28 pages, 9 figure

Computing Singular Values of Large Matrices with an Inverse-Free Preconditioned Krylov Subspace Method

We present an efficient algorithm for computing a few extreme singular values of a large sparse m×n matrix C. Our algorithm is based on reformulating the singular value problem as an eigenvalue problem for CTC. To address the clustering of the singular values, we develop an inverse-free preconditioned Krylov subspace method to accelerate convergence. We consider preconditioning that is based on robust incomplete factorizations, and we discuss various implementation issues. Extensive numerical tests are presented to demonstrate efficiency and robustness of the new algorithm

Numerické metody pro řešení diskrétních inverzních úloh

Název práce: Numerické metody pro řešení diskrétních inverzních úloh Autor: Marie Kubínová Katedra: Katedra numerické matematiky Vedoucí disertační práce: RNDr. Iveta Hnětynková, Ph.D., Katedra numerické matematiky Abstrakt: Inverzní úlohy představují širokou skupinu problémů rekonstrukce neznámých veličin z naměřených dat, přičemž společným rysem těchto problémů je vysoká citlivost řešení na změny v datech. Úkolem numerických metod je zkonstruovat výpočetně nenáročným způsobem aproximaci řešení a zároveň pot- lačit vliv nepřesností v datech, tzv. šumu, který je vždy přítomen. Vlastnosti šumu a jeho chování v regularizačních metodách hrají klíčovou roli při konstruk- ci a analýze těchto metod. Tato práce se zaměřuje na některé aspekty řešení diskrétních inverzních úloh, a to konkrétně: na propagaci šumu v iteračních metodách a jeho reprezentaci v příslušných residuích, včetně studia vlivu arit- metiky s konečnou přesností, na odhad hladiny šumu a na řešení problémů s daty zatíženými šumem z různých zdrojů. Klíčová slova: diskrétní inverzní úlohy, iterační metody, odhadování šumu, smíšený šum, aritmetika s konečnou přesností - v -Title: Numerical Methods in Discrete Inverse Problems Author: Marie Kubínová Department: Department of Numerical Mathematics Supervisor: RNDr. Iveta Hnětynková, Ph.D., Department of Numerical Mathe- matics Abstract: Inverse problems represent a broad class of problems of reconstruct- ing unknown quantities from measured data. A common characteristic of these problems is high sensitivity of the solution to perturbations in the data. The aim of numerical methods is to approximate the solution in a computationally efficient way while suppressing the influence of inaccuracies in the data, referred to as noise, that are always present. Properties of noise and its behavior in reg- ularization methods play crucial role in the design and analysis of the methods. The thesis focuses on several aspects of solution of discrete inverse problems, in particular: on propagation of noise in iterative methods and its representation in the corresponding residuals, including the study of influence of finite-precision computation, on estimating the noise level, and on solving problems with data polluted with noise coming from various sources. Keywords: discrete inverse problems, iterative solvers, noise estimation, mixed noise, finite-precision arithmetic - iii -Katedra numerické matematikyDepartment of Numerical MathematicsMatematicko-fyzikální fakultaFaculty of Mathematics and Physic