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Dynamic thresholding algorithm with memory for linear inverse problems

inverse problems 2025年第1期41卷

作者： Sun, Zhong-Feng Zhao, Yun-Bin Zhou, Jin-Chuan Huang, Zheng-Hai Shandong Univ Technol Sch Math & Stat Zibo Shandong Peoples R China Chinese Univ Hong Kong Shenzhen Int Ctr Ind & Appl Math SRIBD Shenzhen Peoples R China Tianjin Univ Sch Math Tianjin Peoples R China

The relaxed optimal k-thresholding pursuit (ROTP) is a recent algorithm for linear inverse problems. This algorithm is based on the optimal k-thresholding technique which performs vector thresholding and error metric reduction simultaneously. Although ROTP can be used to solve small to medium-sized linear inverse problems, the computational cost of this algorithm is high when solving large-scale problems. By merging the optimal k-thresholding technique and iterative method with memory as well as optimization with sparse search directions, we propose the so-called dynamic thresholding algorithm with memory (DTAM), which iteratively and dynamically selects vector bases to construct the problem solution. At every step, the algorithm uses more than one or all iterates generated so far to construct a new search direction, and solves only the small-sized quadratic subproblems at every iteration. Thus the computational complexity of DTAM is remarkably lower than that of ROTP-type methods. It turns out that DTAM can locate the solution of linear inverse problems if the matrix involved satisfies the restricted isometry property. Experiments on synthetic data, audio signal reconstruction and image denoising demonstrate that the proposed algorithm performs comparably to several mainstream thresholding and greedy algorithms, and it works faster than the ROTP-type algorithms especially when the sparsity level of signal is relatively low.

关键词： linear inverse problems thresholding algorithm restricted isometry property signal reconstruction image denoising

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Increasing resolution and instability for linear inverse scattering problems

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JOURNAL OF FUNCTIONAL ANALYSIS 2025年第1期289卷

作者： Kow, Pu-Zhao Salo, Mikko Zou, Sen Natl Chengchi Univ Dept Math Sci 64Sec 2ZhiNan RdWenshan Dist Taipei 116302 Taiwan Univ Jyvaskyla Dept Math & Stat POB MaD 35 FI-40014 Jyvaskyla Finland Fudan Univ Sch Math Sci 220 Rd Handan Shanghai 200433 Peoples R China

In this work we study the increasing resolution of linear inverse scattering problems at a large fixed frequency. We consider the problem of recovering the density of a Herglotz wave function, and the linearized inverse scattering problem for a potential. It is shown that the number of features that can be stably recovered (stable region) becomes larger as the frequency increases, whereas one has strong instability for the rest of the features (unstable region). To show this rigorously, we prove that the singular values of the forward operator stay roughly constant in the stable region and decay exponentially in the unstable region. The arguments are based on structural properties of the problems and they involve the Courant min-max principle for singular values, quantitative AgmonH & ouml;rmander estimates, and a Schwartz kernel computation based on the coarea formula. (c) 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// ***/licenses/by/4.0/).

关键词： linear inverse problems Instability mechanisms Increasing stability/resolution Singular value

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linear inverse problems for the heat equation and non-local boundary value problems with generalized Samarskii-Ionkin condition

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BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA 2023年第3期29卷 1-16页

作者： Kozhanov, A. I. Shipina, T. N. Sobolev Inst Math Acad Koptyug Ave 4 Novosibirsk 630090 Russia Siberian Fed Univ Svobodny Ave 79 Krasnoyarsk 660049 Russia

The paper is devoted to the study of the solvability of linear inverse problems for a one-dimensional heat equation with an unknown right-hand side. The aim of the work is to obtain theorems of the existence and uniqueness of regular solutions (i.e., solutions having all weak derivatives in the sense of Sobolev occurring in the equation) The proofs will essentially use new results on the solvability of nonlocal problems with a generalized Samarskii-Ionkin boundary condition.

关键词： Heat equation linear inverse problems Generalized Samarskii-Ionkin condition Regular solutions Existence and uniqueness.

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linear inverse problems and Neural Networks

Linear Inverse Problems and Neural Networks

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作者： Dhaliwal, Jasjeet San Jose State University

学位级别：M.A., Master of Arts

We investigate two ideas in this thesis. First, we analyze the results of adapting recovery algorithms from linear inverse problems to defend neural networks against adversarial attacks. Second, we analyze the results of substituting sparsity priors with neural network priors in linear inverse problems. For the former, we are able to extend an existing compressive sensing framework to defend neural networks against ℓ0, ℓ2,and ℓ∞ norm attacks,and for the latter, we find that our method yields an improvement over reconstruction results of existing neural network based priors.

关键词： linear inverse problems Neural networks Recovery algorithms

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The use of linear programming in the construction of extremal solutions to linear inverse problems

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SIAM REVIEW 1996年第3期38卷 496-506页

作者： Huestis, SP UNIV NEW MEXICO DEPT EARTH & PLANETARY SCI ALBUQUERQUE NM 87131 USA

While a finite collection of data does not specify a unique solution to a linear inverse problem, it can allow bounds to be placed on certain nonlinear solution functionals. Using the Dirichlet problem for the unit disc as an example, this note demonstrates the use of linear programming in constructing extremal solutions associated with a variety of such bounds.

关键词： linear inverse problems extremal solutions linear programming

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ALTERNATING DIRECTION METHOD OF MULTIPLIERS FOR linear inverse problems

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2016年第4期54卷 2114-2137页

作者： Jiao, Yuling Jin, Qinian Lu, Xiliang Wang, Weijie Zhongnan Univ Econ & Law Sch Math & Stat Wuhan 430063 Peoples R China Australian Natl Univ Inst Math Sci Canberra ACT 2601 Australia Wuhan Univ Sch Math & Stat Wuhan 430072 Peoples R China Wuhan Univ Hubei Key Lab Computat Sci Wuhan 430072 Peoples R China

In this paper we propose an iterative method using alternating direction method of multipliers (ADMM) strategy to solve linear inverse problems in Hilbert spaces with a general convex penalty term. When the data is given exactly, we give a convergence analysis of our ADMM algorithm without assuming the existence of a Lagrange multiplier. In case the data contains noise, we show that our method is a regularization method as long as it is terminated by a suitable stopping rule. Various numerical simulations are performed to test the efficiency of the method.

关键词： linear inverse problems alternating direction method of multipliers Bregman distance convergence regularization property

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An Optimal Multi-Vector Iterative Algorithm in a Krylov Subspace for Solving the Ill-Posed linear inverse problems

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CMC-COMPUTERS MATERIALS & CONTINUA 2013年第2期33卷 175-198页

作者： Liu, Chein-Shan Natl Taiwan Univ Dept Civil Engn Taipei 10764 Taiwan

An optimal m-vector descent iterative algorithm in a Krylov subspace is developed, of which the m weighting parameters are optimized from a properly defined objective function to accelerate the convergence rate in solving an ill-posed linear problem. The optimal multi-vector iterative algorithm (OMVIA) is convergent fast and accurate, which is verified by numerical tests of several linear inverse problems, including the backward heat conduction problem, the heat source identification problem, the inverse Cauchy problem, and the external force recovery problem. Because the OMVIA has a good filtering effect, the numerical results recovered are quite smooth with small error, even under a large noise up to 10%.

关键词： linear inverse problems Ill-posed linear equations system Optimal multi-vector iterative algorithm (OMVIA) Future cone Invariant-manifold Krylov subspace method

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On general convergence behaviours of finite-dimensional approximants for abstract linear inverse problems

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ASYMPTOTIC ANALYSIS 2022年第1-2期127卷 167-189页

作者： Caruso, Noe Angelo Michelangeli, Alessandro Novati, Paolo Scuola Int Super Studi Avanzati Via Bonomea 265 I-34136 Trieste Italy Gran Sasso Sci Inst Viale F Crispi 7 I-67100 Laquila Italy Univ Bonn Inst Appl Math Endenicher Allee 60 D-53115 Bonn Germany Univ Bonn Hausdorff Ctr Math Endenicher Allee 60 D-53115 Bonn Germany Univ Trieste Piazzale Europa 1 I-34127 Trieste Italy

In the framework of abstract linear inverse problems in infinite-dimensional Hilbert space we discuss generic convergence behaviours of approximate solutions determined by means of general projection methods, namely outside the standard assumptions of Petrov-Galerkin truncation schemes. This includes a discussion of the mechanisms why the error or the residual generically fail to vanish in norm, and the identification of practically plausible sufficient conditions for such indicators to be small in some weaker sense. The presentation is based on theoretical results together with a series of model examples and numerical tests.

关键词： linear inverse problems infinite-dimensional Hilbert space ill-posed problems orthonormal basis discretisation bounded linear operators Krylov subspaces Krylov solution GMRES conjugate gradient LSQR

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On the Convergence of Stochastic Gradient Descent for linear inverse problems in Banach Spaces

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SIAM JOURNAL ON IMAGING SCIENCES 2023年第2期16卷 671-705页

作者： Jin, Bangti Kereta, Zeljko Chinese Univ Hong Kong Dept Math Shatin Hong Kong Peoples R China UCL Dept Comp Sci Gower St London WC1E 6BT England

In this work we consider stochastic gradient descent (SGD) for solving linear inverse problems in Banach spaces. SGD and its variants have been established as one of the most successful optimization methods in machine learning, imaging, and signal processing, to name a few. At each iteration SGD uses a single datum, or a small subset of data, resulting in highly scalable methods that are very attractive for large-scale inverse problems. Nonetheless, the theoretical analysis of SGD-based approaches for inverse problems has thus far been largely limited to Euclidean and Hilbert spaces. In this work we present a novel convergence analysis of SGD for linear inverse problems in general Banach spaces: we show the almost sure convergence of the iterates to the minimum norm solution and establish the regularizing property for suitable a priori stopping criteria. Numerical results are also presented to illustrate features of the approach.

关键词： stochastic gradient descent Banach spaces linear inverse problems convergence rate regularizing property almost sure convergence

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An optimal tri-vector iterative algorithm for solving ill-posed linear inverse problems

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inverse problems IN SCIENCE AND ENGINEERING 2013年第4期21卷 650-681页

作者： Liu, Chein-Shan Natl Taiwan Univ Dept Civil Engn Taipei 10764 Taiwan

A future cone in the Minkowski space, defined in terms of the square-norm of the residual vector for an ill-posed linear system to be solved, is used to derive an optimal tri-vector descent system of nonlinear ordinary differential equations (ODEs). Then, a simple Euler scheme is used to generate an iterative algorithm from these ODEs, of which the two parameters appeared are optimized from a properly defined merit function to accelerate the convergence speed in solving the ill-posed linear systems. The optimal tri-vector iterative algorithm (OTVIA) is fast convergent and accurate, which is proven by numerical tests of inverse problems, including the backward heat conduction problem, the Calderon inverse problem and the inverse Cauchy problems. By defining a suitable convergence rate, we assess the convergence speeds of OTVIA and the conjugate gradient method (CGM), which reveal that the performance of OTVIA is better than the CGM. Also by comparing the OTVIA with the generalized minimal residual method (GMRES), we observe that the OTVIA is better than the GMRES.

关键词： linear inverse problems ill-posed linear equations system conjugate gradient method generalized minimal residual method optimal tri-vector iterative algorithm invariant-manifold

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