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Topic

Inverse problem

About: Inverse problem is a(n) research topic. Over the lifetime, 29795 publication(s) have been published within this topic receiving 604246 citation(s). The topic is also known as: F(x,y)=0.
Papers
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Journal ArticleDOI
TL;DR: A new fast iterative shrinkage-thresholding algorithm (FISTA) which preserves the computational simplicity of ISTA but with a global rate of convergence which is proven to be significantly better, both theoretically and practically.
Abstract: We consider the class of iterative shrinkage-thresholding algorithms (ISTA) for solving linear inverse problems arising in signal/image processing. This class of methods, which can be viewed as an extension of the classical gradient algorithm, is attractive due to its simplicity and thus is adequate for solving large-scale problems even with dense matrix data. However, such methods are also known to converge quite slowly. In this paper we present a new fast iterative shrinkage-thresholding algorithm (FISTA) which preserves the computational simplicity of ISTA but with a global rate of convergence which is proven to be significantly better, both theoretically and practically. Initial promising numerical results for wavelet-based image deblurring demonstrate the capabilities of FISTA which is shown to be faster than ISTA by several orders of magnitude.

9,684 citations


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

4,634 citations


Journal ArticleDOI
TL;DR: It is proved that replacing the usual quadratic regularizing penalties by weighted 𝓁p‐penalized penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem.
Abstract: We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted p-penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem. Use of such p-penalized problems with p < 2 is often advocated when one expects the underlying ideal noiseless solution to have a sparse expansion with respect to the basis under consideration. To compute the corresponding regularized solutions, we analyze an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. © 2004 Wiley Periodicals, Inc.

3,809 citations


Book
17 Jul 2000-
TL;DR: This book treats the inverse problem of remote sounding comprehensively, and discusses a wide range of retrieval methods for extracting atmospheric parameters of interest from the quantities such as thermal emission that can be measured remotely.
Abstract: Remote sounding of the atmosphere has proved to be a fruitful method of obtaining global information about the atmospheres of the earth and planets. This book treats the inverse problem of remote sounding comprehensively, and discusses a wide range of retrieval methods for extracting atmospheric parameters of interest from the quantities such as thermal emission that can be measured remotely. Inverse theory is treated in depth from an estimation-theory point of view, but practical questions are also emphasized, for example designing observing systems to obtain the maximum quantity of information, efficient numerical implementation of algorithms for processing of large quantities of data, error analysis and approaches to the validation of the resulting retrievals, The book is targeted at both graduate students and working scientists.

3,806 citations


Posted Content
Abstract: We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary pre-assigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted l^p-penalties on the coefficients of such expansions, with 1 < or = p < or =2, still regularizes the problem. If p < 2, regularized solutions of such l^p-penalized problems will have sparser expansions, with respect to the basis under consideration. To compute the corresponding regularized solutions we propose an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. We also review some potential applications of this method.

3,640 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202233
20211,464
20201,574
20191,466
20181,357
20171,378

Top Attributes

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Topic's top 5 most impactful authors

Michael V. Klibanov

130 papers, 3.6K citations

Gunther Uhlmann

104 papers, 3.9K citations

Daniel Lesnic

104 papers, 2.2K citations

Vjacheslav Yurko

74 papers, 1K citations

Simon R. Arridge

69 papers, 4.9K citations