Definitions and examples of inverse and ill-posed problems
01 Jan 2008-Journal of Inverse and Ill-posed Problems (Walter de Gruyter GmbH & Co. KG)-Vol. 16, Iss: 4, pp 317-357
TL;DR: In this article, the authors consider definitions and classification of inverse and ill-posed problems and describe some approaches which have been proposed by outstanding Russian mathematicians A. N. Tikhonov, V. K. Ivanov and M. M. Lavrentiev.
Abstract: Abstract The terms “inverse problems” and “ill-posed problems” have been steadily and surely gaining popularity in modern science since the middle of the 20th century. A little more than fifty years of studying problems of this kind have shown that a great number of problems from various branches of classical mathematics (computational algebra, differential and integral equations, partial differential equations, functional analysis) can be classified as inverse or ill-posed, and they are among the most complicated ones (since they are unstable and usually nonlinear). At the same time, inverse and ill-posed problems began to be studied and applied systematically in physics, geophysics, medicine, astronomy, and all other areas of knowledge where mathematical methods are used. The reason is that solutions to inverse problems describe important properties of media under study, such as density and velocity of wave propagation, elasticity parameters, conductivity, dielectric permittivity and magnetic permeability, and properties and location of inhomogeneities in inaccessible areas, etc. In this paper we consider definitions and classification of inverse and ill-posed problems and describe some approaches which have been proposed by outstanding Russian mathematicians A. N. Tikhonov, V. K. Ivanov and M. M. Lavrentiev.
Citations
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TL;DR: It is shown that approximate Metropolis-adjusted Langevin sampling allows an efficient Bayesian inversion of model parameters obtained from a prior represented by a deep generative model, obtaining a diverse set of realizations that reflect the observed seismic response.
Abstract: We present an application of deep generative models in the context of partial differential equation constrained inverse problems. We combine a generative adversarial network representing an a priori model that generates geological heterogeneities and their petrophysical properties, with the numerical solution of the partial-differential equation governing the propagation of acoustic waves within the earth’s interior. We perform Bayesian inversion using an approximate Metropolis-adjusted Langevin algorithm to sample from the posterior distribution of earth models given seismic observations. Gradients with respect to the model parameters governing the forward problem are obtained by solving the adjoint of the acoustic wave equation. Gradients of the mismatch with respect to the latent variables are obtained by leveraging the differentiable nature of the deep neural network used to represent the generative model. We show that approximate Metropolis-adjusted Langevin sampling allows an efficient Bayesian inversion of model parameters obtained from a prior represented by a deep generative model, obtaining a diverse set of realizations that reflect the observed seismic response.
140 citations
Cites background from "Definitions and examples of inverse..."
...The observed data or measurements are often noisy and/or sparse, and therefore lead to an ill-posed inverse problem where numerous realizations of the underlying model parameters may lead to a model response that matches observed data (Kabanikhin 2008)....
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TL;DR: An overview of the application of computational intelligence technologies in optical remote sensing image processing, including: 1) feature representation and selection; 2) classification and clustering; and 3) change detection are provided.
133 citations
TL;DR: Evidence is provided that tools are treated by the nervous system as sensory extensions of the body rather than as simple distal links between the hand and the environment, and that this sensory capability emerges from the functional coupling between the material, biomechanical and neural levels of information processing.
Abstract: The ability to extend sensory information processing beyond the nervous system1 has been observed throughout the animal kingdom; for example, when rodents palpate objects using whiskers2 and spiders localize prey using webs3 We investigated whether the ability to sense objects with tools4–9 represents an analogous information processing scheme in humans Here we provide evidence from behavioural psychophysics, structural mechanics and neuronal modelling, which shows that tools are treated by the nervous system as sensory extensions of the body rather than as simple distal links between the hand and the environment10,11 We first demonstrate that tool users can accurately sense where an object contacts a wooden rod, just as is possible on the skin We next demonstrate that the impact location is encoded by the modal response of the tool upon impact, reflecting a pre-neuronal stage of mechanical information processing akin to sensing with whiskers2 and webs3 Lastly, we use a computational model of tactile afferents12 to demonstrate that impact location can be rapidly re-encoded into a temporally precise spiking code This code predicts the behaviour of human participants, providing evidence that the information encoded in motifs shapes localization Thus, we show that this sensory capability emerges from the functional coupling between the material, biomechanical and neural levels of information processing13,14 Tools are embodied by the human somatosensory system, serving as sensory extensions of the human body
115 citations
01 Jan 2018
TL;DR: This chapter provides an entry point to recent developments in geological modeling methods, helps researchers in the field to better consider uncertainties, and supports the integration of geological observations and knowledge in geophysical interpretation, modeling and inverse approaches.
Abstract: The Earth below ground is the subject of interest for many geophysical as well as geological investigations. Even though most practitioners would agree that all available information should be used in such an investigation, it is common practice that only a part of geological and geophysical information is actually integrated in structural geological models. We believe that some reasons for this omission are (a) an incomplete picture of available geological modeling methods, and (b) the problem of the perceived static picture of an inflexible geological representation in an image or geological model. With this work, we aim to contribute to the problem of subsurface interface detection through (a) the review of state-of-the-art geological modeling methods that allow the consideration of multiple aspects of geological realism in the form of observations, information, and knowledge, cast in geometric representations of subsurface structures, and (b) concepts and methods to analyze, quantify, and communicate related uncertainties in these models. We introduce a formulation for geological model representation and interpolation and uncertainty analysis methods with the aim to clarify similarities and differences in the diverse set of approaches that developed in recent years. We hope that this chapter provides an entry point to recent developments in geological modeling methods, helps researchers in the field to better consider uncertainties, and supports the integration of geological observations and knowledge in geophysical interpretation, modeling and inverse approaches.
110 citations
Cites background from "Definitions and examples of inverse..."
...In fact, the ill-posed nature of geophysical problems lead to many of the mathematical developments in this field (e.g. Kabanikhin, 2008)....
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01 Dec 2018
TL;DR: In this article, the authors present results of accuracy evaluation of numerous numerical algorithms for the numerical approximation of the Inverse Laplace Transform, including Stehfest, Abate and Whitt, Vlach and Singhai.
Abstract: In the paper we present results of accuracy evaluation of numerous numerical algorithms for the numerical approximation of the Inverse Laplace Transform. The selected algorithms represent diverse lines of approach to this problem and include methods by Stehfest, Abate and Whitt, Vlach and Singhai, De Hoog, Talbot, Zakian and a one in which the FFT is applied for the Fourier series convergence acceleration. We use C++ and Python languages with arbitrary precision mathematical libraries to address some crucial issues of numerical implementation. The test set includes Laplace transforms considered as difficult to compute as well as some others commonly applied in fractional calculus. Evaluation results enable to conclude that the Talbot method which involves deformed Bromwich contour integration, the De Hoog and the Abate and Whitt methods using Fourier series expansion with accelerated convergence can be assumed as general purpose high-accuracy algorithms. They can be applied to a wide variety of inversion problems.
69 citations
References
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Book•
20 Dec 2004
TL;DR: This chapter discusses Monte Carol methods, the least-absolute values criterion and the minimax criterion, and their applications to functional inverse problems.
Abstract: 1 The general discrete inverse problem 2 Monte Carol methods 3 The least-squares criterion 4 Least-absolute values criterion and minimax criterion 5 Functional inverse problems 6 Appendices 7 Problems References Index
5,249 citations
"Definitions and examples of inverse..." refers background in this paper
...Many problems of mathematical statistics in a sense may be considered as inverse to some problems of probability theory (Bertero and Boccacci, 1998; Engl et al, 1999; Tarantola, 2005)....
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...In this case ‖A−1‖ = sup f 6=0 ‖A−1f‖ ‖f‖A = 1 and thereforeA−1 is continuous....
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...Example 5.4(A. Tarantola, 2005)....
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Book•
01 Jan 1992
TL;DR: Inverse Medium Problem (IMP) as discussed by the authors is a generalization of the Helmholtz Equation for direct acoustical obstacle scattering in an Inhomogeneous Medium (IMM).
Abstract: Introduction.- The Helmholtz Equation.- Direct Acoustic Obstacle Scattering.- III-Posed Problems.- Inverse Acoustic Obstacle Scattering.- The Maxwell Equations.- Inverse Electromagnetic Obstacle Scattering.- Acoustic Waves in an Inhomogeneous Medium.- Electromagnetic Waves in an Inhomogeneous Medium.- The Inverse Medium Problem.-References.- Index
5,126 citations
"Definitions and examples of inverse..." refers background in this paper
...A lot of theoretical results and applications can be found in (Chadan and Sabatier, 1989; Colton and Kress, 1992; Kress, 2007)....
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Book•
31 Jul 1996
TL;DR: Inverse problems have been studied in this article, where Tikhonov regularization of nonlinear problems has been applied to weighted polynomial minimization problems, and the Conjugate Gradient Method has been used for numerical realization.
Abstract: Preface. 1. Introduction: Examples of Inverse Problems. 2. Ill-Posed Linear Operator Equations. 3. Regularization Operators. 4. Continuous Regularization Methods. 5. Tikhonov Regularization. 6. Iterative Regularization Methods. 7. The Conjugate Gradient Method. 8. Regularization with Differential Operators. 9. Numerical Realization. 10. Tikhonov Regularization of Nonlinear Problems. 11. Iterative Methods for Nonlinear Problems. A. Appendix: A.1. Weighted Polynomial Minimization Problems. A.2. Orthogonal Polynomials. A.3. Christoffel Functions. Bibliography. Index.
4,690 citations
"Definitions and examples of inverse..." refers background in this paper
...The inverse problem (2.2)–(2.5) is said to beretrospective(Alifanov et al, 1995; Engl et al, 1996; Kabanikhin et al, 2006) if it is required to determine the initial conditions, i.e., the functionsϕ(x) andψ(x) in (2.3)....
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...The Picard criterion (Engl et al, 1996) f ∈ D(A†) ⇐⇒ ∑ σn 6=0 |〈f, un〉|2 σ2n ∞ says that the best approximate solutionqnp exists only if the (generalized) Fourier coefficients〈f, un〉 with respect to singular functionsun decay fast enough relative to the singular values....
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...Example 5.1(H. W. Engl et al, 1996)....
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Book•
01 Jan 1986TL;DR: In this paper, the Radon transform and related transforms have been studied for stability, sampling, resolution, and accuracy, and quite a bit of attention is given to the derivation, analysis, and practical examination of reconstruction algorithm, for both standard problems and problems with incomplete data.
Abstract: The Mathematics of Computerized Tomography covers the relevant mathematical theory of the Radon transform and related transforms and also studies more practical questions such as stability, sampling, resolution, and accuracy. Quite a bit of attention is given to the derivation, analysis, and practical examination of reconstruction algorithm, for both standard problems and problems with incomplete data.
3,600 citations
Book•
21 Dec 2021TL;DR: Part 2 Linear inverse problems: examples of linear inverse problems singular value decomposition (SVD) inversion methods revisited Fourier based methods for specific problems comments and concluding remarks.
Abstract: This is a graduate textbook on the principles of linear inverse problems, methods of their approximate solution, and practical application in imaging The level of mathematical treatment is kept as low as possible to make the book suitable for a wide range of readers from different backgrounds in science and engineering Mathematical prerequisites are first courses in analysis, geometry, linear algebra, probability theory, and Fourier analysis The authors concentrate on presenting easily implementable and fast solution algorithms With examples and exercises throughout, the book will provide the reader with the appropriate background for a clear understanding of the essence of inverse problems (ill-posedness and its cure) and, consequently, for an intelligent assessment of the rapidly growing literature on these problems
2,027 citations
"Definitions and examples of inverse..." refers background in this paper
...Many problems of mathematical statistics in a sense may be considered as inverse to some problems of probability theory (Bertero and Boccacci, 1998; Engl et al, 1999; Tarantola, 2005)....
[...]