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Sparse Recovery Beyond Compressed Sensing: Separable nonlinear inverse problems

IEEE TRANSACTIONS ON INFORMATION THEORY 2020年第9期66卷 5904-5926页

作者： Bernstein, Brett Liu, Sheng Papadaniil, Chrysa Fernandez-Granda, Carlos NYU Courant Inst Math Sci New York NY 10011 USA NYU Ctr Data Sci New York NY 10011 USA NYU Ctr Brain Imaging New York NY 10011 USA

Extracting information from nonlinear measurements is a fundamental challenge in data analysis. In this work, we consider separable inverse problems, where the data are modeled as a linear combination of functions that depend nonlinearly on certain parameters of interest. These parameters may represent neuronal activity in a human brain, frequencies of electromagnetic waves, fluorescent probes in a cell, or magnetic relaxation times of biological tissues. Separable nonlinear inverse problems can be reformulated as underdetermined sparse-recovery problems, and solved using convex programming. This approach has had empirical success in a variety of domains, from geophysics to medical imaging, but lacks a theoretical justification. In particular, compressed-sensing theory does not apply, because the measurement operators are deterministic and violate incoherence conditions such as the restricted-isometry property. Our main contribution is a theory for sparse recovery adapted to deterministic settings. We show that convex programming succeeds in recovering the parameters of interest, as long as their values are sufficiently distinct with respect to the correlation structure of the measurement operator. The theoretical results are illustrated through numerical experiments for two applications: heat-source localization and estimation of brain activity from electroencephalography data.

关键词： Geophysical measurements inverse problems Extraterrestrial measurements Brain modeling Heating systems Mathematical model Atomic measurements Sparse recovery convex programming incoherence correlated measurements dual certificates nonlinear inverse problems source localization

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An iteration regularization method with general convex penalty for nonlinear inverse problems in Banach spaces

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2019年 361卷 472-486页

作者： Wang, Jing Wang, Wei Han, Bo Heilongjiang Univ Sch Math Sci Harbin 150080 Heilongjiang Peoples R China Jiaxing Univ Coll Math Phys & Informat Engn Jiaxing 314001 Zhejiang Peoples R China Harbin Inst Technol Dept Math Harbin 150001 Heilongjiang Peoples R China

This paper investigates the construction, analysis and implementation of a novel iterative regularization scheme with general convex penalty term for nonlinear inverse problems in Banach spaces. The method is formulated by combining the homotopy perturbation technique. The penalty term is allowed to be nonsmooth to include L-1-norm or total variation-like penalty functionals, especially suitable for detecting the special features of the sought solutions such as sparsity or piecewise constant. By using tools from convex analysis in Banach spaces, we establish the detailed convergence and stability results. Numerical simulations for one-dimensional and two-dimensional parameter identification problems are performed to validate that our approach is competitive in terms of reducing the overall computational time in comparison with the existing Landweber iteration with general convex penalty. Moreover, our approach is used to the non-Gaussian noisy data to illustrate the iterative regularization in Banach spaces. (C) 2019 Elsevier B.V. All rights reserved.

关键词： nonlinear inverse problems Iteration regularization method Banach spaces General convex penalty

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Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems

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JOURNAL OF inverse AND ILL-POSED problems 2016年第4期24卷 435-447页

作者： Kaltenbacher, Barbara Rendl, Franz Resmerita, Elena Alpen Adria Univ Klagenfurt Univ Str 65-67 A-9020 Klagenfurt Austria

In this paper we present a method for the regularized solution of nonlinear inverse problems, based on Ivanov regularization (also called method of quasi solutions or constrained least squares regularization). This leads to the minimization of a nonconvex cost function under a norm constraint, where nonconvexity is caused by nonlinearity of the inverse problem. Minimization is done by iterative approximation, using (nonconvex) quadratic Taylor expansions of the cost function. This leads to repeated solution of quadratic trust region subproblems with possibly indefinite Hessian. Thus, the key step of the method consists in application of an efficient method for solving such quadratic subproblems, developed by Rendl and Wolkowicz [10]. We here present a convergence analysis of the overall method as well as numerical experiments.

关键词： nonlinear inverse problems regularization quasi solutions nonconvex trust region method

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A FAST AND SCALABLE METHOD FOR A-OPTIMAL DESIGN OF EXPERIMENTS FOR INFINITE-DIMENSIONAL BAYESIAN nonlinear inverse problems

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2016年第1期38卷 A243-A272页

作者： Alexanderian, Alen Petra, Noemi Stadler, Georg Ghattas, Omar N Carolina State Univ Dept Math Box 8205 Raleigh NC 27695 USA Univ Calif Sch Nat Sci Appl Math Merced CA 95343 USA NYU Courant Inst Math Sci 251 Mercer St New York NY 10012 USA Univ Texas Austin Dept Mech Engn Inst Computat Engn & Sci Dept Geol Sci Austin TX 78712 USA

We address the problem of optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs). The inverse problem seeks to infer an infinite-dimensional parameter from experimental data observed at a set of sensor locations and from the governing PDEs. The goal of the OED problem is to find an optimal placement of sensors so as to minimize the uncertainty in the inferred parameter field. Specifically, we seek an optimal subset of sensors from among a fixed set of candidate sensor locations. We formulate the OED objective function by generalizing the classical A-optimal experimental design criterion using the expected value of the trace of the posterior covariance. This expected value is computed through sample averaging over the set of likely experimental data. To cope with the infinite-dimensional character of the parameter field, we construct a Gaussian approximation to the posterior at the maximum a posteriori probability (MAP) point, and use the resulting covariance operator to define the OED objective function. We use randomized trace estimation to compute the trace of this covariance operator, which is defined only implicitly. The resulting OED problem includes as constraints the system of PDEs characterizing the MAP point, and the PDEs describing the action of the covariance (of the Gaussian approximation to the posterior) to vectors. We control the sparsity of the sensor configurations using sparsifying penalty functions. Variational adjoint methods are used to efficiently compute the gradient of the PDE-constrained OED objective function. We elaborate our OED method for the problem of determining the optimal sensor configuration to best infer the coefficient of an elliptic PDE. Furthermore, we provide numerical results for inference of the log permeability field in a porous medium flow problem. Numerical results show that the number of PDE solves required for the evaluation of the OED objective functi

关键词： optimal experimental design A-optimal design Bayesian inference sensor placement nonlinear inverse problems randomized trace estimator sparsified designs

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REGINN-IT method with general convex penalty terms for nonlinear inverse problems

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APPLICABLE ANALYSIS 2022年第17期101卷 5949-5973页

作者： Fu, Zhenwu Chen, Yong Han, Bo Harbin Inst Technol Dept Math Harbin Peoples R China

In the spirit of the method in Margotti and Rieder (An inexact Newton regularization in Banach spaces based on the non-stationary iterated Tikhonov method. J inverse Ill-Posed Probl. 2015;23:373-392), we propose an inexact Newton regularization which is based on the non-stationary iterated Tikhonov method for non-smooth solutions of nonlinear inverse problems in Banach spaces. The method consists of an inner iteration and an outer iteration. The outer iteration is terminated by the discrepancy principle and consists of an inexact Newton regularization method. The inner iteration is performed by the non-stationary iterated Tikhonov method. The remarkable point is that the penalty terms can be general convex functions. Under certain assumptions on nonlinear operators, the convergence analysis of our method is given by making use of tools from convex analysis. Furthermore, numerical simulations are provided to support the theoretical results and test the performance of the method.

关键词： Inexact Newton regularization non-stationary iterated Tikhonov method nonlinear inverse problems general convex penalty terms

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An Efficient Quasi-Newton Method for nonlinear inverse problems via Learned Singular Values

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IEEE SIGNAL PROCESSING LETTERS 2021年 28卷 748-752页

作者： Smyl, Danny Tallman, Tyler N. Liu, Dong Hauptmann, Andreas Univ Sheffield Dept Civil & Struct Engn Sheffield S Yorkshire England Purdue Univ Sch Aeronaut & Astronaut W Lafayette IN 47907 USA Univ Sci & Technol China Dept Modern Phys Hefei 230026 Peoples R China Univ Oulu Res Unit Math Sci Oulu 90570 Finland UCL Dept Comp Sci London WC1E 6BT England

Solving complex optimization problems in engineering and the physical sciences requires repetitive computation of multi-dimensional function derivatives, which commonly require computationally-demanding numerical differentiation such as perturbation techniques. In particular, Gauss-Newton methods are used for nonlinear inverse problems that require iterative updates to be computed from the Jacobian and allow for flexible incorporation of prior knowledge. Computationally more efficient alternatives are Quasi-Newton methods, where the repeated computation of the Jacobian is replaced by an approximate update, but unfortunately are often too restrictive for highly ill-posed problems. To overcome this limitation, we present a highly efficient data-driven Quasi-Newton method applicable to nonlinear inverse problems, by using the singular value decomposition and learning a mapping from model outputs to the singular values to compute the updated Jacobian. Enabling time-critical applications and allowing for flexible incorporation of prior knowledge necessary to solve ill-posed problems. We present results for the highly non-linear inverse problem of electrical impedance tomography with experimental data.

关键词： Jacobian matrices Artificial neural networks inverse problems Training Computational modeling Electrical impedance tomography Conductivity Quasi-Newton method nonlinear inverse problems neural networks electrical impedance tomography

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ON A CLASS OF FROZEN REGULARIZED GAUSS-NEWTON METHODS FOR nonlinear inverse problems

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MATHEMATICS OF COMPUTATION 2010年第272期79卷 2191-2211页

作者： Jin, Qinian Univ Texas Austin Dept Math Austin TX 78712 USA

In this paper we consider a class of regularized Gauss-Newton methods for solving nonlinear inverse problems for which an a posteriori stopping rule is proposed to terminate the iteration. Such methods have the frozen feature that they require only the computation of the Frechet derivative at the initial approximation. Thus the computational work is considerably reduced. Under certain mild conditions, we give the convergence analysis and derive various estimates, including the order optimality, on these methods.

关键词： nonlinear inverse problems frozen regularized Gauss-Newton method a posteriori stopping rule convergence order optimality

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Proximal algorithm for minimization problems in l0-regularization for nonlinear inverse problems

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NUMERICAL ALGORITHMS 2022年第1期91卷 367-388页

作者： Muoi, Pham Quy Hiep, Duong Xuan Univ Sci & Educ Univ Danang Dept Math Ton Duc Thang 459 Danang Vietnam

In this paper, we study a proximal method for the minimization problem arising from l(0)-regularization for nonlinear inverse problems. First of all, we prove the existence of solutions and give an optimality condition for solutions to the minimization problem. Then, we propose and prove the convergence of the proximal method for this minimization problem, which is controlled by step size conditions. Finally, we illustrate performance of the proximal method by applying it to a numerical example.

关键词： Proximal method l(0)-regularization nonlinear inverse problems Nonsmooth optimization

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Solving nonlinear inverse problems by evolution equations based on GaussNewton methods

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APPLICABLE ANALYSIS 2013年第3期92卷 527-548页

作者： Jin, Qinian Australian Natl Univ Inst Math Sci Canberra ACT 0200 Australia

In this article, we solve nonlinear inverse problems by an evolution equation method which can be viewed as the continuous analogue of the GaussNewton method. Under certain conditions we prove the convergence and deri... 详细信息

关键词： nonlinear inverse problems evolution equation based on Gauss-Newton method the discrepancy principle convergence rates of convergence

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RETRACTED: A fast adaptive algorithm for nonlinear inverse problems with convex penalty

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COMPUTATIONAL & APPLIED MATHEMATICS 2023年第4期42卷 1-19页

作者： Ren, Gang Gao, Guanglan Henan Inst Technol Coll Comp Sci & Technol 699 Pingyuan Rd Xinxiang Peoples R China

In this paper, we propose a fast adaptive algorithm for solving nonlinear inverse problems in Hilbert spaces. The iterative process of the proposed method combines classical two point gradient method and adaptive accelerate strategy. In practice, it is often encountered that the reconstruction solution has special feature, such as sparsity and slicing smoothness. To capture the special feature of solution, convex functions are utilized to be penalty terms in iterative format. Meanwhile, a complete convergence analysis is given to show the theoretical rationality of the algorithm. The numerical simulations are provided to demonstrate the effectiveness and acceleration effect of the proposed method.

关键词： nonlinear inverse problems Two point gradient method Adaptive Convex penalty terms

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