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Convergence of proximal algorithms with stepsize controls for non-linear inverse problems and application to sparse non-negative matrix factorization

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NUMERICAL ALGORITHMS 2020年第4期85卷 1255-1279页

作者： Pham Quy Muoi Lachmund, Delf Hao Dinh Nho Univ Danang Univ Sci & Educ Dept Math 459 Ton Duc Thang Danang Vietnam Univ Bremen Ctr Ind Math Bibliothekstr 1 D-28359 Bremen Germany VAST Hanoi Inst Math 18 Hoang Quoc Viet Rd Hanoi Vietnam

We consider a general ill-posed inverse problem in a Hilbert space setting by minimizing a misfit functional coupling with a multi-penalty regularization for stabilization. For solving this minimization problem, we investigate two proximal algorithms with stepsize controls: a proximal fixed point algorithm and an alternating proximal algorithm. We prove the decrease of the objective functional and the convergence of both update schemes to a stationary point under mild conditions on the stepsizes. These algorithms are then applied to the sparse and non-negative matrix factorization problems. Based on a priori information of non-negativity and sparsity of the exact solution, the problem is regularized by corresponding terms. In both cases, the implementation of our proposed algorithms is straight-forward since the evaluation of the proximal operators in these problems can be done explicitly. Finally, we test the proposed algorithms for the non-negative sparse matrix factorization problem with both simulated and real-world data and discuss reconstruction performance, convergence, as well as achieved sparsity.

关键词： Sparse matrix factorization Sparse non-negative matrix factorization Proximal fixed point iteration Alternating proximal iteration non-smooth Optimization non-linear inverse problems Sparsity regularization non-negative sparsity regularization 49Mxx 65Kxx

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On log-concave approximations of high-dimensional posterior measures and stability properties in non-linear inverse problems

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ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES 2024年第4期60卷 2619-2667页

作者： Bohr, Jan Nickl, Richard Univ Bonn Math Inst Bonn Germany Univ Cambridge Dept Pure Math & Math Stat Cambridge England

The problem of efficiently generating random samples from high-dimensional and non-log-concave posterior measures arising in nonlinear regression problems is considered. Extending investigations from (J. Eur. Math. Soc. (JEMS) 26 (2024) 1031- 1112), local and global stability properties of the model are identified under which such posterior distributions can be approximated in Wasserstein distance by suitable log-concave measures. This allows the use of fast gradient based sampling algorithms, for which convergence guarantees are established that scale polynomially in all relevant quantities (assuming 'warm' initialisation). The scope of the general theory is illustrated in a non-linear inverse problem from integral geometry for which new stability results are derived.

关键词： non-linear inverse problems Bayesian non-parametric statistics non-Abelian X-ray transform Polynomial time sampling

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A Unified 2D/3D Large-Scale Software Environment for nonlinear inverse problems

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ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 2019年第1期45卷 7-7页

作者： Da Silva, Curt Herrmann, Felix Univ British Columbia Dept Math Room 1211984 Math Rd Vancouver BC V6T 1Z2 Canada Georgia Inst Technol Dept Earth & Atmospher Sci Room 203310 Ferst Dr NW Atlanta GA 30332 USA

Large-scale parameter estimation problems are among some of the most computationally demanding problems in numerical analysis. An academic researcher's domain-specific knowledge often precludes that of software design, which results in inversion frameworks that are technically correct but not scalable to realistically sized problems. On the other hand, the computational demands for realistic problems result in industrial codebases that are geared solely for high performance, rather than comprehensibility or flexibility. We propose a new software design for inverse problems constrained by partial differential equations that bridges the gap between these two seemingly disparate worlds. A hierarchical and modular design reduces the cognitive burden on the user while exploiting high-performance primitives at the lower levels. Our code has the added benefit of actually reflecting the underlying mathematics of the problem, which lowers the cognitive load on the user using it and reduces the initial startup period before a researcher can be fully productive. We also introduce a new preconditioner for the 3D Helmholtz equation that is suitable for fault-tolerant distributed systems. Numerical experiments on a variety of 2D and 3D test problems demonstrate the effectiveness of this approach on scaling algorithms from small-to large-scale problems with minimal code changes.

关键词： non-linear inverse problems acoustic wave equation Laplace's equation sparsity-promoting optimization full waveform inversion

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Landweber-type method with uniformly convex constraints under conditional stability assumptions

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APPLIED MATHEMATICS LETTERS 2023年第1期144卷

作者： Zhong, Min Qiu, Lingyun Wang, Wei Southeast Univ Sch Math Nanjing 210096 Jiangsu Peoples R China Nanjing Ctr Appl Math Nanjing 211135 Jiangsu Peoples R China Tsinghua Univ Yau Math Sci Ctr Beijing 100084 Peoples R China Yanqi Lake Beijing Inst Math Sci & Applicat Beijing 101408 Peoples R China Jiaxing Univ Coll Data Sci Jiaxing 314001 Peoples R China Key Lab Med Elect & Digital Hlth Zhejiang Prov Jiaxing 314001 Peoples R China

The Landweber-type iteration method is one of the most prominent regularization methods for solving ill-posed problems when the data is corrupted by noise. By extending the previous results proposed by de Hoop et al. (2012), the current work systematically investigates the convergence rate of Landweber-type iteration in Banach spaces for nonlinear ill-posed problems. Two types of stopping rules including the discrepancy principle and the heuristic Hanke-Raus rule are provided which are novel in view of existing research work.(c) 2023 Elsevier Ltd. All rights reserved.

关键词： non-linear inverse problems Convex constraints Convergence rate Conditional stability assumptions

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Quantitative imaging from diffracted fields intensities: an inversion method and its experimental validation

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JOURNAL OF MODERN OPTICS 2010年第9期57卷 777-782页

作者： Crocco, Lorenzo D'Urso, Michele Isernia, Tommaso CNR IREA I-80124 Naples Italy Giugliano Res Ctr Integrated Syst Anal Dept Selex Sistemi Integrati I-80014 Naples Italy Univ Mediterranea Reggio Calabria DIMET I-89060 Reggio Di Calabria Italy

In many diagnostics and imaging applications intensity only measurements of the diffracted field are available. Hence, it is of interest to develop procedures able to solve the underlying inverse scattering problem under these conditions. In this paper, we introduce an approach in which the geometrical and electromagnetic properties of unknown complex targets are reconstructed from the squared amplitudes of the diffracted fields. The approach relies on some preliminary considerations on the spatial bandwidth of the diffracted intensities which allow one to fix rules to effectively measure the data and on an alternative formulation of the electromagnetic scattering that lowers the 'degree of non-linearity' of the relationship among the electric features of the targets and the fields. The proposed procedure is tested against benchmark experimental data in the microwave frequency range and the obtained results confirm the possibility of achieving faithful quantitative images of unknown targets from phaseless data without making approximations in the inverse scattering problem.

关键词： inverse scattering non-linear inverse problems image processing phase retrieval

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WAVEFORM INVERSION OF TIME-CORRECTED REFRACTION DATA

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GEOPHYSICAL JOURNAL-OXFORD 1988年第1期93卷 75-90页

作者： SHAW, PR Woods Hole Oceanographic Institution Woods Hole MA 02543 USA

Seismic refraction data collected in land or marine surveys are often complicated by small, near-random time shifts of the individual seismograms relative to one another. For the case of fixed-receiver geometry, these shifts arise primarily from basement topography, often masked by sediments, beneath each source. Although a topographic timing correction is normally applied to the seismograms, the effects of poorly-known topography produce residual shifts in the seismograms. The application of additional time corrections (statics) can bias solutions computed using iterative waveform inversion, because a suitable set of statics can make a poor starting velocity model appear to match the data. Without any correction, however, seismograms computed from even the best-fitting laterally homogeneous model may not agree with the data to within one-half period. In this paper I present a formalism for iterative waveform inversion of refraction data containing residual time shifts. In this method the statics are directly modelled in the synthetic seismograms, and new values estimated in the inversion together with new velocity model parameters. The method is similar to studies of statics in seismic reflection data, but in this case values for the statics are estimated simultaneously with the velocity model; with the use of suitable norms the new velocity model is required to be smooth and the statics required to be small. Convergence is achieved when the velocity profile's roughness, the RMS size of the statics, and the waveform misfit are in equilibrium. Because the starting synthetics need not match the data to within one-half period, a much larger set of velocity models used for an initial guess will converge on the family of solutions than for waveform inversions in which time shifts are not used. The original, arbitrary starting model loses its importance after the first iteration, and solutions meeting the convergence criterion appear to be independent of the particular c

关键词： WKBJ waveform inversion non-linear inverse problems statics

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FDEMtools: a MATLAB package for FDEM data inversion

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NUMERICAL ALGORITHMS 2020年第4期84卷 1313-1327页

作者： Deidda, G. P. Diaz de Alba, P. Fenu, C. Lovicu, G. Rodriguez, G. Univ Cagliari Dept Civil & Environm Engn & Architecture Via Marengo 2 I-09123 Cagliari Italy Univ Cagliari Dept Math & Comp Sci Viale Merello 92 I-09123 Cagliari Italy Univ Cagliari Dept Elect & Elect Engn Via Marengo 2 I-09123 Cagliari Italy

Electromagnetic induction surveys are among the most popular techniques for non-destructive investigation of soil properties, in order to detect the presence of both ground inhomogeneities and particular substances. This paper introduces a MATLAB package, called FDEMtools, for the inversion of frequency domain electromagnetic data collected by a ground conductivity meter, which includes a graphical user interface to interactively modify the parameters of the computation and visualize the results. Based on a non-linear forward model used to describe the interaction between an electromagnetic field and the soil, the software reconstructs the distribution of either the electrical conductivity or the magnetic permeability with respect to depth, by a regularized damped Gauss-Newton method. The regularization part of the algorithm is based on a low-rank approximation of the Jacobian of the non-linear model. The package allows the user to experiment with synthetic and experimental data sets, and different regularization strategies, in order to compare them and draw conclusions.

关键词： FDEM induction non-linear inverse problems Gauss-Newton method TGSVD MATLAB toolbox

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On polynomial-time computation of high-dimensional posterior measures by Langevin-type algorithms

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JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY 2024年第3期26卷 1031-1112页

作者： Nickl, Richard Wang, Sven Univ Cambridge Ctr Math Sci Cambridge CB3 0WB England MIT IDSS Cambridge MA 02139 USA

The problem of generating random samples of high-dimensional posterior distributions is considered. The main results consist of non-asymptotic computational guarantees for Langevintype MCMC algorithms which scale polynomially in key quantities such as the dimension of the model, the desired precision level, and the number of available statistical measurements. As a direct consequence, it is shown that posterior mean vectors as well as optimisation based maximum a posteriori (MAP) estimates are computable in polynomial time, with high probability under the distribution of the data. These results are complemented by statistical guarantees for recovery of the ground truth parameter generating the data. Our results are derived in a general high-dimensional non-linear regression setting (with Gaussian process priors) where posterior measures are not necessarily log-concave, employing a set of local 'geometric' assumptions on the parameter space, and assuming that a good initialiser of the algorithm is available. The theory is applied to a representative non-linear example from PDEs involving a steady-state Schr & ouml;dinger equation.

关键词： . Bayesian inference Markov Chain Monte Carlo non-linear inverse problems Bayesian computation Markov chain mixing times

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2-D inverse scattering: degree of non-linearity, solution strategies and polarization effects

2-D Inverse scattering: degree of non-linearity, solution st...

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Conference on Image Reconstruction from Incomplete Data

作者： Bucci, OM Cardace, N Crocco, L Isernia, T Univ Naples Federico II Dipartimento Ingn Elettr & Telecommun I-80125 Naples Italy

ISBN: (纸本)0819437689

Quantifying the "degree of non linearity" with respect to the permittivity profile represents an interesting possibility for quantifying the difficulty of both forward and inverse scattering problems. With reference to the 2-D case, in this paper new tools are proposed that allow to determine the "degree of non linearity" while changing maximum value, dimensions and spatial frequency content of the unknown permittivity profile, as well as polarization of the incident field. As such, the proposed tools allows to identify new solution strategy for the inverse problem and the simpler nature of the TE (vectorial) problem. Numerical examples support the conclusions.

关键词： inverse scattering non-linear inverse problems scattering non-linearity polarization

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On statistical Calderón problems

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Mathematical Statistics and Learning 2019年第2期2卷 165-216页

作者： Abraham, Kweku Nickl, Richard Université Paris-Saclay CNRS Laboratoire de mathématiques d'Orsay Orsay 91405 France Department of Pure Mathematics and Mathematical Statistics University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom

For D a bounded domain in Rd, d ≥ 2, with smooth boundary ∂D, the non-linear inverse problem of recovering the unknown conductivity γ determining solutions u = uγf of the partial differential equation from noisy ob... 详细信息

For D a bounded domain in R^d, d ≥ 2, with smooth boundary ∂D, the non-linear inverse problem of recovering the unknown conductivity γ determining solutions u = u_γf of the partial differential equation from noisy observations Y of the Dirichlet-to-Neumann map with ∂/∂v denoting the outward normal derivative, is considered. The data Y consists of ∧_ycorrupted by additive Gaussian noise at noise level ε > 0, and a statistical algorithm y(Y) is constructed which is shown to recover γ in supremum-norm loss at a statistical convergence rate of the order log(1/ɛ)^-δ as ε → 0. It is further shown that this convergence rate is optimal, up to the precise value of the exponent δ > 0, in an information theoretic sense. The estimator y(Y) has a Bayesian interpretation in terms of the posterior mean of a suitable Gaussian process prior and can be computed by MCMC methods. © European Mathematical Society

关键词： asymptotics of nonparametric Bayes procedures electrical impedance tomography elliptic partial differential equations non-linear inverse problems

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