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

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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Extended Milstein Approximation to the Stochastic Allen-Cahn Equation with Random Diffusion Coefficient Field and Multiplicative Noise

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数学研究（英文） 2023年第4期56卷 366-391页

作者： Xiao Qi School of Mathematical Sciences Xiamen UniversityXiamen 361005China Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005China

This paper studies the stochastic Allen-Cahn equation driven by a random diffu-sion coefficient field and multiplicative force noise.A new time-stepping scheme based on a stabilized approach and Milstein scheme is proposed and *** proposed method is unconditionally stable in the sense that a discrete energy is dissipative when the multi-plicative noise is *** strong convergence rate of a spatio-temporal fully discrete scheme is *** experiments are finally reported to confirm the theoretical result and show that the new scheme is much more robust than the classical semi-implicit Euler-Maruyama scheme,especially when the interface width parameter is small.

关键词： extended Mil-stein scheme multiplicative noise stability Stochastic Allen-Cahn equation strong convergence

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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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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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An Enriched Petrov-Galerkin Method for Darcy Flow in Fractured Porous Media

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Annals of Applied Mathematics 2024年第4期40卷 346-392页

作者： Huangxin Chen Zixuan Wang School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamenFujian 361005China

We develop a locally mass-conservative enriched Petrov-Galerkin(EPG)method without any penalty term for the simulation of Darcy flow in fractured porous *** discrete fracture model is applied to model the fractures as the lower dimensional fracture *** new method enriches the approximation trial space of the conforming continuous Galerkin(CG)method with bubble functions and enriches the approximation test space of the CG method with piecewise constant functions in the fractures and the surrounding porous *** propose a framework for constructing the bubble functions and consider a decoupled algorithm for the EPG *** solution of the pressure can be decoupled into two steps with a standard CG method and a post-processing *** post-processing correction based on the bubble functions in the matrix and the fractures can be solved separately,which is useful for parallel *** derive a priori and a posteriori error estimates for the *** examples are presented to illustrate the performance of the proposed method.

关键词： Discrete fracture model enriched Petrov-Galerkin method local mass conservation post-processing error analysis

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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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Dynamic road crime risk prediction with urban open data

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Frontiers of Computer Science 2022年第1期16卷 113-125页

作者： Binbin ZHOU Longbiao CHEN Fangxun ZHOU Shijian LI Sha ZHAO Gang PAN College of Computer Science Zhejiang UniversityHangzhou310027China Fujian Key Laboratory of Sensing and Computing for Smart Cities School of Information Science and EngineeringXiamen UniversityXiamen361005China

Crime risk prediction is helpful for urban safety and citizens’life ***,existing crime studies focused on coarse-grained prediction,and usually failed to capture the dynamics of urban *** key challenge is data sparsity,since that 1)not all crimes have been recorded,and 2)crimes usually occur with low *** this paper,we propose an effective framework to predict fine-grained and dynamic crime risks in each road using heterogeneous urban ***,to address the issue of unreported crimes,we propose a cross-aggregation soft-impute(CASI)method to deal with possible unreported ***,we use a novel crime risk measurement to capture the crime dynamics from the perspective of influence propagation,taking into consideration of both time-varying and location-varying risk *** on the dynamically calculated crime risks,we design contextual features(i.e.,POI distributions,taxi mobility,demographic features)from various urban data sources,and propose a zero-inflated negative binomial regression(ZINBR)model to predict future crime risks in *** experiments using the real-world data from New York City show that our framework can accurately predict road crime risks,and outperform other baseline methods.

关键词： crime prediction road crime risk urban computing data sparsity zero-inflated negative binomial regression

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Polygonal finite element-based content-aware image warping

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Computational Visual Media 2023年第2期9卷 367-383页

作者： Juan Cao Xiaoyi Zhang Jiannan Huang Yongjie Jessica Zhang School of Mathematical Sciences Xiamen UniversityXiamen 361005China Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation Xiamen UniversityXiamen 361005China Department of Mechanical Engineering Carnegie Mellon UniversityPittsburghPA 15213USA

Mesh-based image warping techniques typically represent image deformation using linear functions on triangular meshes or bilinear functions on rectangular *** enables simple and efficient implementation,but in turn,restricts the representation capability of the deformation,often leading to unsatisfactory warping *** present a novel,flexible polygonal finite element(poly-FEM)method for content-aware image *** deformation is represented by high-order poly-FEMs on a content-aware polygonal mesh with a cell distribution adapted to saliency information in the source *** allows highly adaptive meshes and smoother warping with fewer degrees of freedom,thus significantly extending the flexibility and capability of the warping *** from the continuous formulation of image deformation,our polyFEM warping method is able to compute the optimal image deformation by minimizing existing or even newly designed warping energies consisting of penalty terms for specific *** demonstrate the versatility of the proposed poly-FEM warping method in representing different deformations and its superiority by comparing it to other existing state-ofthe-art methods.

关键词： image warping finite element method(FEM) polygonal element mesh generation

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A Transformer-Based Approach for Effective Visual Question Answering 10

A Transformer-Based Approach for Effective Visual Question A...

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10th IEEE Smart World Congress, SWC 2024

作者： Zakari, Rufai Yusuf Owusu, Jim Wilson Qin, Ke Sagir, Auwal Muhammad University of Electronic Science and Technology of China School of Computer Science and Engineering China Xiamen University Fujian Key Laboratory of Sensing and Computing for Smart Cities Xiamen China

ISBN: (纸本)9798331520861

Visual Question Answering (VQA) presents a multifaceted challenge, requiring models to process and integrate both textual and visual data. The Transformer architecture, with its self-attention mechanism, has proven highly effective in managing global dependencies, making it a popular choice for many VQA models. However, a key challenge lies in achieving an optimal balance between global and local dependency modeling within conventional Transformer frameworks. Addressing this issue, our research introduces an innovative Transformer-based approach that employs region-specific self-attention for VQA. In contrast to models that concentrate exclusively on global dependencies, our method concurrently executes intra and inter-window attention operations to effectively capture contextual information from images. By utilizing regional windows for visual feature processing, our model eliminates redundancy in global self-attention while still preserving rich contextual details. The efficacy of our proposed technique, which relies on the utilization of visual features, has been extensively evaluated through experiments. These assessments were conducted using widely recognized VQA benchmark datasets, specifically VQA 2.0 and CLEVR. Our findings demonstrate that the model outperforms existing benchmark alternatives across various performance metrics. Notably, it achieved superior test results of 7 1. 9 3 % and 9 8. 6 2 % on the VQA 2.0 and CLEVR datasets, respectively, underscoring its effectiveness in tackling complex VQA tasks. © 2024 IEEE.

关键词： Visual languages

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Boundary Control of Coupled Wave Systems with Spatially-Varying Coefficients

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Journal of Systems Science & Complexity 2022年第4期35卷 1310-1329页

作者： FENG Xiaodan ZHANG Zhifei School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan430074China

This paper considers the stabilization of the coupled wave systems with spatially-varying *** authors design a state feedback controller by backstepping *** contrast to the previous work in the literature,the kernel equations become more complicated and the main difficulty lies in proving the existence and uniqueness of the solution to the kernel ***,using the backstepping approach,the authors verify the kernel equations,which is a system of coupled hyperbolic equations with spatially-varying ***,the existence and uniqueness of the kernel matrices is ***,the authors use a Lyapunov function to get the exponential stabilization of the closed-loop system.A numerical example is presented to illustrate the effectiveness of the proposed controller.

关键词： Backstepping boundary control coupled wave systems exponential stabilization spatially-varying coefficients

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