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The convergence properties of infeasible inexact proximal alternating linearized minimization

Science China Mathematics 2023年第10期66卷 2385-2410页

作者： Yukuan Hu Xin Liu State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

The proximal alternating linearized minimization(PALM)method suits well for solving blockstructured optimization problems,which are ubiquitous in real *** the cases where subproblems do not have closed-form solutions,e.g.,due to complex constraints,infeasible subsolvers are indispensable,giving rise to an infeasible inexact PALM(PALM-I).Numerous efforts have been devoted to analyzing the feasible PALM,while little attention has been paid to the *** usage of the PALM-I thus lacks a theoretical *** essential difficulty of analysis consists in the objective value nonmonotonicity induced by the *** study in the present work the convergence properties of the *** particular,we construct a surrogate sequence to surmount the nonmonotonicity issue and devise an implementable inexact *** upon these,we manage to establish the stationarity of any accumulation point,and moreover,show the iterate convergence and the asymptotic convergence rates under the assumption of the Lojasiewicz *** prominent advantages of the PALM-I on CPU time are illustrated via numerical experiments on problems arising from quantum physics and 3-dimensional anisotropic frictional contact.

关键词： proximal alternating linearized minimization infeasibility nonmonotonicity surrogate sequence inexact criterion iterate convergence asymptotic convergence rate

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A Domain Decomposition Method for Nonconforming Finite Element Approximations of Eigenvalue Problems

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Communications on Applied Mathematics and Computation 2025年第2期7卷 606-636页

作者： Qigang Liang Wei Wang Xuejun Xu School of Mathematical Science Tongji UniversityShanghai200092China Key Laboratory of Intelligent Computing and Applications(Tongji University) Ministry of EducationShanghai200092China LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing100190China

Since the nonconforming finite elements(NFEs)play a significant role in approximating PDE eigenvalues from below,this paper develops a new and parallel two-level preconditioned Jacobi-Davidson(PJD)method for solving the large scale discrete eigenvalue problems resulting from NFE discretization of 2mth(m=1.2)order elliptic eigenvalue *** a spectral projection on the coarse space and an overlapping domain decomposition(DD),a parallel preconditioned system can be solved in each iteration.A rigorous analysis reveals that the convergence rate of our two-level PJD method is optimal and *** results supporting our theory are given.

关键词： PDE eigenvalue problems Nonconforming finite elements(NFEs) Preconditioned Jacobi-Davidson(PJD)method Overlapping domain decomposition(DD)

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A New Sixth-Order WENO Scheme for Solving Hyperbolic Conservation Laws

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Communications on Applied Mathematics and Computation 2023年第1期5卷 3-30页

作者： Kunlei Zhao Yulong Du Li Yuan State Key Laboratory of Scientific and Engineering Computing(LSEC)and Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100190China School of Mathematical Sciences Beihang UniversityBeijing 100191China

In this paper,we develop a new sixth-order WENO scheme by adopting a convex combina-tion of a sixth-order global reconstruction and four low-order local *** the classical WENO schemes,the associated linear weights of the new scheme can be any positive numbers with the only requirement that their sum equals ***,a very simple smoothness indicator for the global stencil is *** new scheme can achieve sixth-order accuracy in smooth *** tests in some one-and two-dimensional bench-mark problems show that the new scheme has a little bit higher resolution compared with the recently developed sixth-order WENO-Z6 scheme,and it is more efficient than the classical fifth-order WENO-JS5 scheme and the recently developed sixth-order WENO6-S scheme.

关键词： Global smoothness indicator Linear weights Sixth-order accuracy WENO

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The spectrum and stability of travelling pulses in a coupled FitzHugh-Nagumo equation

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Science China Mathematics 2024年第5期67卷 975-1010页

作者： Qi Qiao Xiang Zhang School of Mathematical Sciences Shanghai Jiao Tong UniversityShanghai 200240China Key Laboratory of Scientific and Engineering Computing(Ministry of Education) and Shanghai Frontier Science Center of Modern Analysis(CMA-Shanghai)Shanghai Jiao Tong UniversityShanghai 200240China

For a coupled slow-fast FitzHugh-Nagumo(FHN)equation derived from a reaction-diffusionmechanics(RDM)model,Holzer et al.(2013)studied the existence and stability of the travelling pulse,which consists of two fast orbit arcs and two slow ones,where one fast segment passes the unique fold point with algebraic decreasing and two slow ones follow normally hyperbolic critical curve *** and Zhang(2021)obtained the existence of the travelling pulse,whose two fast orbit arcs both exponentially decrease,and one of the slow orbit arcs could be normally hyperbolic or not at the ***,we characterize both the nonlinear and spectral stability of this travelling pulse.

关键词： coupled FitzHugh-Nagumo equation singular perturbation travelling pulse spectrum nonlinear stability spectral stability

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SHARP POINTWISE-IN-TIME ERROR ESTIMATE OF L1 SCHEME FOR NONLINEAR SUBDIFFUSION EQUATIONS

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Journal of Computational Mathematics 2024年第3期42卷 662-678页

作者： Dongfang Li Hongyu Qin Jiwei Zhang School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China School of Mathematics and Statistics Wuhan UniversityWuhan 430072China

An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial *** weak regularity of the solution is usually characterized by a regularity parameterσ∈(0,1)∪(1,2).Under this general regularity assumption,we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results,i.e.,a refined discrete fractional-type Grönwall inequality(DFGI).After that,we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion *** present results fill the gap on some interesting convergence results of L1 scheme onσ∈(0,α)∪(α,1)∪(1,2].Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.

关键词： Sharp pointwise-in-time error estimate Ll scheme Nonlinear subdiffusion equations Non-smooth solutions

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MC-Nonlocal-PINNs:Handling Nonlocal Operators in PINNs Via Monte Carlo Sampling

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Numerical Mathematics(Theory,Methods and Applications) 2023年第3期16卷 769-791页

作者： Xiaodong Feng Yue Qian Wanfang Shen Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina Shandong Key Laboratory of Blockchain Finance Shandong University of Finance and EconomicsJinan 250014China

We propose Monte Carlo Nonlocal physics-informed neural networks(MC-Nonlocal-PINNs),which are a generalization of MC-fPINNs in *** et al.(*** ***.400(2022),115523)for solving general nonlocal models such as integral equations and nonlocal *** to MC-fPINNs,our MC-Nonlocal-PINNs handle nonlocal operators in a Monte Carlo way,resulting in a very stable approach for high dimensional *** present a variety of test problems,including high dimensional Volterra type integral equations,hypersingular integral equations and nonlocal PDEs,to demonstrate the effectiveness of our approach.

关键词： Nonlocal models PINNs Monte Carlo sampling deep neural networks

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TexCIL: Text-Guided Continual Learning of Disease with Vision-Language Model

TexCIL: Text-Guided Continual Learning of Disease with Visio...

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2024 IEEE International Conference on Bioinformatics and Biomedicine, BIBM 2024

作者： Zhang, Wentao Zhao, Defeng Zheng, Wei-Shi Wang, Ruixuan Sun Yat-sen University School of Computer Science and Engineering Guangzhou China Peng Cheng Laboratory Shenzhen China Moe Key Laboratory of Machine Intelligence and Advanced Computing Guangzhou China

ISBN: (纸本)9798350386226

Current intelligent diagnostic systems often catastrophically forget old knowledge when learning new diseases only from the training dataset of the new diseases. Inspired by human learning of visual classes with the effective help of language, we propose a continual learning framework based on a pre-trained visual-language model (VLM) without storing any image of previously learned diseases. In this framework, textual prior knowledge of each new disease can be obtained by utilizing the frozen VLM's text encoder, and then used to guide the visual learning of the new disease. This framework innovatively utilizes the textual prior knowledge of all previously learned diseases as out-of-distribution (OOD) information to help differentiate currently being-learned diseases from others. Extensive empirical evaluations on both medical and natural image datasets confirm the superiority of the proposed method over existing state-of-the-art methods in continual learning of new visual classes. The source code is available at https://***/OpenMedIA/TexCIL. © 2024 IEEE.

关键词： Contrastive Learning

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A NEW PROOF OF GAFFNEY’S INEQUALITY FOR DIFFERENTIAL FORMS ON MANIFOLDS-WITH-BOUNDARY:THE VARIATIONAL APPROACH à LA KOZONO-YANAGISAWA

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Acta Mathematica Scientia 2022年第4期42卷 1427-1452页

作者： Siran LI School of Mathematical Sciences IMA-Shanghai&Key Laboratory of Scientific and Engineering Computing(Ministry of Education)Shanghai Jiao Tong UniversityShanghai 200240China

Let(M,g_(0))be a compact Riemannian *** present a new proof of the classical Gaffney inequality for differential forms in boundary value spaces over M,via a variational approach a la Kozono-Yanagisawa[Lr-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains,Indiana ***.J.58(2009),1853-1920],combined with global computations based on the Bochner technique.

关键词： Gaffney’s inequality differential form Sobolev spaces on manifolds Bochner technique variational approach

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A Riemannian Exponential Augmented Lagrangian Method for computing the Projection Robust Wasserstein Distance 37

A Riemannian Exponential Augmented Lagrangian Method for Com...

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37th Conference on Neural Information Processing Systems, NeurIPS 2023

作者： Jiang, Bo Liu, Ya-Feng Ministry of Education Key Laboratory of NSLSCS School of Mathematical Sciences Nanjing Normal University Nanjing210023 China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

ISBN: (纸本)9781713899921

Projection robust Wasserstein (PRW) distance is recently proposed to efficiently mitigate the curse of dimensionality in the classical Wasserstein distance. In this paper, by equivalently reformulating the computation of the PRW distance as an optimization problem over the Cartesian product of the Stiefel manifold and the Euclidean space with additional nonlinear inequality constraints, we propose a Riemannian exponential augmented Lagrangian method (REALM) for solving this problem. Compared with the existing Riemannian exponential penalty-based approaches, REALM can potentially avoid too small penalty parameters and exhibit more stable numerical performance. To solve the subproblems in REALM efficiently, we design an inexact Riemannian Barzilai-Borwein method with Sinkhorn iteration (iRBBS), which selects the stepsizes adaptively rather than tuning the stepsizes in efforts as done in the existing methods. We show that iRBBS can return an Ε-stationary point of the original PRW distance problem within O(Ε-3) iterations, which matches the best known iteration complexity result. Extensive numerical results demonstrate that our proposed methods outperform the state-of-the-art solvers for computing the PRW distance. © 2023 Neural information processing systems foundation. All rights reserved.

关键词： Constrained optimization

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A HIGH ORDER SCHEME FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH THE CAPUTO-HADAMARD DERIVATIVE

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Journal of Computational Mathematics 2025年第3期43卷 615-640页

作者： Xingyang Ye Junying Cao Chuanju Xu School of Science Jimei UniversityXiamen 361021China School of Data Science and Information Engineering Guizhou Minzu UniversityGuiyang 550025China School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper,we consider numerical solutions of the fractional diffusion equation with theαorder time fractional derivative defined in the Caputo-Hadamard sense.A high order time-stepping scheme is constructed,analyzed,and numerically *** contribution of the paper is twofold:1)regularity of the solution to the underlying equation is investigated,2)a rigorous stability and convergence analysis for the proposed scheme is performed,which shows that the proposed scheme is 3+αorder *** numerical examples are provided to verify the theoretical statement.

关键词： Caputo-Hadamard derivative Fractional differential equations High order scheme Stability and convergence analysis

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