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Probabilistic constraints for nonlinear inverse problems

CONSTRAINTS 2013年第3期18卷 344-376页

作者： Carvalho, Elsa Cruz, Jorge Barahona, Pedro Univ Nova Lisboa CENTRIA Dept Informat Fac Ciencias & Tecnol P-2829516 Caparica Portugal Univ Madeira Ctr Ciencias Exactas & Engn Funchal Portugal

The probabilistic continuous constraint framework complements the representation of uncertainty by means of intervals with a probabilistic distribution of values within such intervals. This paper describes how nonlinear inverse problems can be cast into this framework, highlighting its ability to deal with all the uncertainty aspects of such problems. In previous work we have formalized the framework, relying on simplified integration methods to characterize the uncertainty distributions. In this paper we (1) provide validated constraint-based algorithms to compute these distributions, (2) discuss approximations obtained by their hybridization with Monte-Carlo methods, and (3) obtain a better uncertainty characterization, by including methods to compute expected values and standard deviations. The paper illustrates this new methodology in Ocean Color (OC), a research area which is widely used in climate change studies and has potential applications in water quality monitoring. OC semi-analytical approaches rely on forward models that relate optically active seawater compounds (OC products) to remote sensing measurements of the sea-surface reflectance. OC products are derived by inverting the forward model on a spectral-reflectance basis. Based on a set of preliminary experiments we show that the probabilistic constraint framework is able to provide a valuable characterization of the uncertainty of all scenarios consistent with the model and the measurements. Moreover, the framework can be used to derive how measurements accuracy affects the uncertainty distribution of the retrieved OC products, which may constitute an important contribution to the OC community.

关键词： Probabilistic continuous constraints nonlinear inverse problems Uncertainty representation and reasoning Ocean color remote sensing

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Sequential subspace optimization for nonlinear inverse problems

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JOURNAL OF inverse AND ILL-POSED problems 2017年第1期25卷 99-117页

作者： Wald, Anne Schuster, Thomas Univ Saarland Dept Math POB 15 11 50 D-66041 Saarbrucken Germany

In this work we discuss a method to adapt sequential subspace optimization (SESOP), which has so far been developed for linear inverse problems in Hilbert and Banach spaces, to the case of nonlinear inverse problems. We start by revising the technique for linear problems. In a next step, we introduce a method using multiple search directions that are especially designed to fit the nonlinearity of the forward operator. To this end, we iteratively project the initial value onto stripes whose width is determined by the search direction, the nonlinearity of the operator and the noise level. We additionally propose a fast algorithm that uses two search directions. Finally, we will show convergence and regularization properties for the presented method.

关键词： nonlinear inverse problems iterative methods sequential subspace optimization multiple search directions regularization metric projection

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A regularizing multilevel approach for nonlinear inverse problems

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APPLIED NUMERICAL MATHEMATICS 2019年 135卷 297-315页

作者： Zhong, Min Wang, Wei Southeast Univ Sch Math Nanjing 210096 Jiangsu Peoples R China Jiaxing Univ Coll Math Phys & Informat Engn Jiaxing 314001 Zhejiang Peoples R China

In this paper, we propose a multilevel method for solving nonlinear inverse problems F(x) = y in Banach spaces. By minimizing the discretized version of the regularized functionals at different levels, we define a sequence of regularized approximations to the sought solution, which is shown to be stable and globally convergent. The penalty term Theta in regularized functionals is allowed to be non-smooth to include L-p - L-1 or L-p - TV (Total Variation) reconstructions, which are significant in reconstructing special features of solutions such as sparsity and discontinuities. Two parameter identification examples are presented to validate the theoretical analysis and to verify the effectiveness of the method. (C) 2018 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： nonlinear inverse problems Multilevel approach Global minimization Non-smooth penalties

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A GENERAL CONVERGENCE ANALYSIS OF SOME NEWTON-TYPE METHODS FOR nonlinear inverse problems

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2011年第2期49卷 549-573页

作者： Jin, Qinian Virginia Tech Dept Math Blacksburg VA 24061 USA

We consider the methods x(n+1)(delta) - x(n)(delta) - g(alpha n) (F'(x(n)(delta))* F'(x(n)(delta)))F'(x(n)(delta))*(F(x(n)(delta)) - y(delta)) for solving nonlinear ill-posed inverse problems F(x) = y using the only available noise data y(delta) satisfying parallel to y(delta) - y parallel to <= delta with a given small noise level delta > 0. We terminate the iteration by the discrepancy principle parallel to F(x(n delta)(delta))-y(delta)parallel to <= tau delta < parallel to F(x(n)(delta))-y(delta)parallel to, 0 <= n < n(delta), with a given number tau > 1. Under certain conditions on {alpha(n)} and F, we prove for a large class of spectral filter functions {g(alpha)} the convergence of x(n delta)(delta) to a true solution as delta -> 0. Moreover, we derive the order optimal rates of convergence when certain Holder source conditions hold. Numerical examples are given to test the theoretical results.

关键词： nonlinear inverse problems Newton-type methods discrepancy principle convergence order optimal convergence rates

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RANDOMIZE-THEN-OPTIMIZE: A METHOD FOR SAMPLING FROM POSTERIOR DISTRIBUTIONS IN nonlinear inverse problems

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2014年第4期36卷 A1895-A1910页

作者： Bardsley, Johnathan M. Solonen, Antti Haario, Heikki Laine, Marko Univ Montana Dept Math Sci Missoula MT 59812 USA MIT Dept Aeronaut & Astronaut Cambridge MA 02139 USA Lappeenranta Univ Technol Dept Math & Phys Lappeenranta Finland Finnish Meteorol Inst FIN-00101 Helsinki Finland

High-dimensional inverse problems present a challenge for Markov chain Monte Carlo (MCMC)-type sampling schemes. Typically, they rely on finding an efficient proposal distribution, which can be difficult for large-scale problems, even with adaptive approaches. Moreover, the autocorrelations of the samples typically increase with dimension, which leads to the need for long sample chains. We present an alternative method for sampling from posterior distributions in nonlinear inverse problems, when the measurement error and prior are both Gaussian. The approach computes a candidate sample by solving a stochastic optimization problem. In the linear case, these samples are directly from the posterior density, but this is not so in the nonlinear case. We derive the form of the sample density in the nonlinear case, and then show how to use it within both a Metropolis-Hastings and importance sampling framework to obtain samples from the posterior distribution of the parameters. We demonstrate, with various small- and medium-scale problems, that randomize-then-optimize can be efficient compared to standard adaptive MCMC algorithms.

关键词： nonlinear inverse problems Bayesian methods uncertainty quantification computational statistics sampling methods

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A projected gradient method for nonlinear inverse problems with αl1 - βl2 sparsity regularization

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JOURNAL OF inverse AND ILL-POSED problems 2024年第3期32卷 513-528页

作者： Zhao, Zhuguang Ding, Liang Northeast Forestry Univ Dept Math Harbin 150040 Peoples R China

The non-convex alpha|| center dot ||(l1) - beta|| center dot|| l(2) (alpha >= beta >= 0) regularization is a new approach for sparse recovery. A minimizer of the alpha|| center dot ||(l1) - beta|| center dot ||(l2) regularized function can be computed by applying the ST-(alpha l1 - beta l2) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). Unfortunately, It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. Nevertheless, the current applicability of the PG method is limited to linear inverse problems. In this paper, we extend the PG method based on a surrogate function approach to nonlinear inverse problems with the alpha|| center dot ||(l1) - beta|| center dot|| l(2) (alpha >= beta >= 0) regularization in the finite-dimensional space Rn. It is shown that the presented algorithm converges subsequentially to a stationary point of a constrained Tikhonov-type functional for sparsity regularization. Numerical experiments are given in the context of a nonlinear compressive sensing problem to illustrate the efficiency of the proposed approach.

关键词： Projected gradient method nonlinear inverse problems surrogate function approach alpha l(1)-beta l(2) sparsity regularization

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Tikhonov-type regularization for nonlinear inverse problems with multiple repeated measurement data

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PHYSICA SCRIPTA 2024年第12期99卷 125222-125222页

作者： Hou, Bingxue Wang, Wei Zhong, Min Jiaxing Univ Coll Data Sci Jiaxing 314000 Zhejiang Peoples R China Zhejiang Normal Univ Sch Math Sci Jinhua 321004 Zhejiang Peoples R China Southeast Univ Sch Math Nanjing 210096 Jiangsu Peoples R China Nanjing Ctr Appl Math Nanjing 211135 Jiangsu Peoples R China

Tikhonov-type regularization for nonlinear inverse problems with multiple repeated measurement data

关键词： nonlinear inverse problems Tikhonov-type regularization multiple repeated measurement data convergence rates globally convergent algorithms

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A homotopy method for nonlinear inverse problems

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APPLIED MATHEMATICS AND COMPUTATION 2006年第2期183卷 1270-1279页

作者： Fu, H. S. Han, B. Harbin Inst Technol Dept Math Harbin 150001 Peoples R China Dalian Maritime Univ Dept Math Dalian 116026 Peoples R China

We develop a homotopy method for nonlinear inverse problems, where the forward problems are governed by some forms of differential equations. A Tikhonov-style regularization approach yields an optimization problem. Ordinary iterative methods may fail to solve this problem, due to their locally convergent properties. Then the fixed-point homotopy method is introduced to solving the normal equation of the optimization problem, and a new and globally convergent algorithm is constructed, which is highly effective in the aspects of speed of computation, ability of noise suppression and wide region of convergence. As a practical application, the method is used to solve the inverse problem of 2-D acoustic wave equation. We demonstrate the merits and effectiveness of our algorithm on two realistic model problems. (c) 2006 Elsevier Inc. All rights reserved.

关键词： homotopy method nonlinear inverse problems Tikhonov regularization acoustic wave equation geophysical prospecting

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A multigrid-homotopy method for nonlinear inverse problems

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2020年第6期79卷 1706-1717页

作者： Liu, Tao Northeastern Univ Qinhuangdao Sch Math & Stat Qinhuangdao 066004 Hebei Peoples R China

In the present contribution, we develop a novel method combining the multigrid idea and the homotopy technique for nonlinear inverse problems, in which the forward problems are modeled by some forms of partial differential equations. The method first attempts to use the multigrid method to decompose the original inverse problem into a sequence of sub-inverse problems which depend on the grid variables and are solved in proper order according to the grid size from the coarsest to the finest, and then carries out the inversion on the coarsest grid by the homotopy method. The strategy may give a rapidly and globally convergent method. As a practical application, this method is used to solve the nonlinear inverse problem of a nonlinear convection-diffusion equation, which is the saturation equation within the two-phase porous media flow. We demonstrate the effectiveness and merits of the multigrid-homotopy method on two actual model problems. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Multigrid Homotopy Regularization nonlinear inverse problems Convection-diffusion equation

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A projected homotopy perturbation method for nonlinear inverse problems in Banach spaces

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JOURNAL OF inverse AND ILL-POSED problems 2023年第6期31卷 849-872页

作者： Xia, Yuxin Han, Bo Wang, Wei Harbin Inst Technol Dept Math Harbin 150001 Heilongjiang Peoples R China Jiaxing Univ Coll Data Sci Jiaxing 314001 Zhejiang Peoples R China

In this paper, we propose and analyze a projected homotopy perturbation method based on sequential Bregman projections for nonlinear inverse problems in Banach spaces. To expedite convergence, the approach uses two search directions given by homotopy perturbation iteration, and the new iteration is calculated as the projection of the current iteration onto the intersection of stripes decided by above directions. The method allows to use L-1 -like penalty terms, which is significant to reconstruct sparsity solutions. Under reasonable conditions, we establish the convergence and regularization properties of the method. Finally, two parameter identification problems are presented to indicate the effectiveness of capturing the property of the sparsity solutions and the acceleration effect of the proposed method.

关键词： nonlinear inverse problems homotopy perturbation iteration sequential Bregman projections parameter identification problems

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