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Computational and applied mathematics 2023年第3期42卷

作者： Wang, Mengjie Dai, Xinjie Yu, Yanyan Xiao, Aiguo Hunan Key Laboratory for Computation and Simulation in Science and Engineering National Center for Applied Mathematics in Hunan Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Yunnan University Yunnan Kunming650504 China

In this paper, a fast θ-Maruyama method is proposed for solving stochastic Volterra integral equations of convolution type with singular and Hölder continuous kernels based on the sum-of-exponentials approximation. Furthermore, the average storage O(N) and the calculation cost O(N2) of θ-Maruyama scheme are reduced to O(log N) and O(Nlog N) for T 1 or O(log 2N) and O(Nlog 2N) for T≈ 1 , respectively, which implies that the fast θ-Maruyama scheme is confirmed to improve the computational efficiency of the θ-Maruyama method. Under the local Lipschitz and linear growth conditions, strong convergence of the given numerical scheme are obtained. Then, for the linear test equation, we show the asymptotic behavior of solutions in mean square sense. Further, we obtain the explicit structure of the stability matrices and some numerical results of the mean-square stability for the fast θ-Maruyama method applied to the linear test equation. Finally, some numerical experiments are also given to illustrate the effectiveness of the method. © 2023, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.

关键词： Computational efficiency

A posteriori error estimation for an interior penalty virtual element method of Kirchhoff plates

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arXiv 2024年

作者： Feng, Fang Hu, Yuming Yu, Yue Zhao, Jikun School of Mathematics and Statistics Nanjing University of Science and Technology Nanjing China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education National Center for Applied Mathematics in Hunan School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Zhengzhou University Zhengzhou450001 China

In this paper, we develop a residual-type a posteriori error estimation for an interior penalty virtual element method (IPVEM) for the Kirchhoff plate bending problem. Building on the work in [18], we adopt a modified discrete variational formulation that incorporates the H1-elliptic projector in the jump and average terms. This allows us to simplify the numerical implementation by including the H1-elliptic projector in the computable error estimators. We derive the reliability and efficiency of the a posteriori error bound by constructing an enriching operator and establishing some related error estimates that align with C0-continuous interior penalty finite element methods. As observed in the a priori analysis, the interior penalty virtual elements exhibit similar behaviors to C0continuous elements despite its discontinuity. This observation extends to the a posteriori estimate since we do not need to account for the jumps of the function itself in the discrete scheme and the error estimators. As an outcome of the error estimator, an adaptive VEM is introduced by means of the mesh refinement strategy with the one-hanging-node rule. Numerical results from several benchmark tests confirm the robustness of the proposed error estimators and show the efficiency of the resulting adaptive VEM. Copyright © 2024, The Authors. All rights reserved.

关键词： Discrete element methods

A DISCONTINUOUS GALERKIN METHOD FOR THE EXTRACELLULAR MEMBRANE INTRACELLULAR MODEL

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arXiv 2024年

作者： Masri, Rami Kirk, Keegan L.A. Hauge, Eirill Kuchta, Miroslav Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway Division of Applied Mathematics Brown University ProvidenceRI02906 United States Department of Mathematical Sciences Center for Mathematics and Artificial Intelligence George Mason University FairfaxVA22030 United States

We formulate and analyze interior penalty discontinuous Galerkin methods for coupled elliptic PDEs modeling excitable tissue, represented by intracellular and extracellular domains sharing a common interface. The PDEs are coupled through a dynamic boundary condition, posed on the interface, that relates the normal gradients of the solutions to the time derivative of their jump. This system is referred to as the Extracellular Membrane Intracellular model or the cell-by-cell model. Due to the dynamic nature of the interface condition and to the presence of corner singularities, the analysis of discontinuous Galerkin methods is non-standard. We prove the existence and uniqueness of solutions by a reformulation of the problem to one posed on the membrane. Convergence is shown by utilizing face-to-element lifting operators and notions of weak consistency suitable for solutions with low spatial regularity. Further, we present parameter-robust preconditioned iterative solvers. Numerical examples in idealized geometries demonstrate our theoretical findings, and simulations in multiple cells portray the robustness of the *** Codes 65N30, 65M60 Copyright © 2024, The Authors. All rights reserved.

关键词： Galerkin methods

Memristive feature and mechanism induced by laser-doping in defect-free 2D semiconductor materials

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Journal of Semiconductors 2024年第7期45卷 85-91页

作者： Xiaoshan Du Shu Wang Qiaoxuan Zhang Shengyao Chen Fengyou Yang Zhenzhou Liu Zhengwei Fan Lijun Ma Lei Wang Lena Du Zhongchang Wang Cong Wang Bing Chen Qian Liu College of Electronic and Information Engineering Shandong University of Science and TechnologyQingdao 266590China National Center for Nanoscience and Technology&University of Chinese Academy of Sciences Beijing 100190China Hebei University of Water Resources and Electric Engineering Electrical Automation Department Cangzhou 061001China The MOE Key Laboratory of Weak-Light Nonlinear Photonics and International Sino-Slovenian Join Re-search Center on Liquid Crystal Pho-tonics TEDA Institute of Applied Physics and School of PhysicsNankai UniversityTianjin 300457China College of Mathematics and Physics Shandong Advanced Optoelectronic Materials and Technologies Engineering LaboratoryQingdao Uni-versity of Science and TechnologyQingdao 266061China Department of Physics Capital Normal UniversityBeijing 100048China Department of Advanced Materials and Computing International Iberian Nanotechnology Laboratory(INL)4715-330 BragaPortugal College of Mathematics and Physics Beijing University of Chemical TechnologyBeijing 100029China

Memristors as non-volatile memory devices have gained numerous attentions owing to their advantages in storage,in-memory computing, synaptic applications, etc. In recent years, two-dimensional(2D) materials with moderate defects have been discovered to exist memristive feature. However, it is very difficult to obtain moderate defect degree in 2D materials, and studied on modulation means and mechanism becomes urgent and essential. In this work, we realized memristive feature with a bipolar switching and a configurable on/off ratio in a two-terminal MoS_(2) device(on/off ratio ~100), for the first time, from absent to present using laser-modulation to few-layer defect-free MoS_(2)(about 10 layers), and its retention time in both high resistance state and low resistance state can reach 2×10^(4) s. The mechanism of the laser-induced memristive feature has been cleared by dynamic Monte Carlo simulations and first-principles calculations. Furthermore, we verified the universality of the laser-modulation by investigating other 2D materials of TMDs. Our work will open a route to modulate and optimize the performance of 2D semiconductor memristive devices.

关键词： 2D-material memristor laser doping laser direct writing memristive mechanism

An adaptive consensus based method for multi-objective optimization with uniform Pareto front approximation

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arXiv 2022年

作者： Borghi, Giacomo Herty, Michael Pareschi, Lorenzo RWTH Aachen University Institute for Geometry and Applied Mathematics Aachen Germany University of Ferrara Department of Mathematics and Computer Science Center for Modelling Computing and Statistics Ferrara Italy

In this work we are interested in stochastic particle methods for multi-objective optimization. The problem is formulated using parametrized, single-objective sub-problems which are solved simultaneously. To this end a consensus based multi-objective optimization method on the search space combined with an additional heuristic strategy to adapt parameters during the computations is proposed. The adaptive strategy aims to distribute the particles uniformly over the image space by using energy-based measures to quantify the diversity of the system. The resulting metaheuristic algorithm is mathematically analyzed using a mean-field approximation and convergence guarantees towards optimal points is rigorously proven. In addition, a gradient flow structure in the parameter space for the adaptive method is revealed and analyzed. Several numerical experiments shows the validity of the proposed stochastic particle dynamics and illustrate the theoretical *** Codes 35Q70, 35Q84, 35Q93, 90C29, 90C56 Copyright © 2022, The Authors. All rights reserved.

关键词： Multiobjective optimization

A Fast and Stable Marginal-Likelihood Calibration Method with Application to Quantum Characterization

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arXiv 2023年

作者： Motamed, Mohammad Petersson, N. Anders Department of Mathematics and Statistics University of New Mexico AlbuquerqueNM87131 United States Center for Applied Scientific Computing Lawrence Livermore National Laboratory LivermoreCA94550 United States

We present a marginal likelihood strategy integrated into the Kennedy-O’Hagan (KOH) Bayesian framework, where a Gaussian process serves as a prior for model discrepancy. The proposed method is both computationally efficient and numerically stable, even in large dataset regimes where the likelihood function approaches degeneracy. applied to the characterization of a super-conducting quantum device at Lawrence Livermore National Laboratory, the approach enhances the predictive accuracy of the Lindblad master equations for modeling Ramsey measurement data by effectively quantifying uncertainties consistent with the quantum data. © 2023, CC BY-NC-SA.

关键词： Potassium hydroxide

Balancing act: multivariate rational reconstruction for IBP

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arXiv 2023年

作者： Belitsky, A.V. Smirnov, A.V. Yakovlev, R.V. Department of Physics Arizona State University TempeAZ85287-1504 United States Research Computing Center Moscow State University Moscow119992 Russia Moscow Center for Fundamental and Applied Mathematics Moscow119992 Russia

We address the problem of unambiguous reconstruction of rational functions of many variables. This is particularly relevant for recovery of exact expansion coefficients in integration-by-parts identites (IBPs) based on modular arithmetic. These IBPs are indispensable in modern approaches to evaluation of multiloop Feynman integrals by means of differential equations. Modular arithmetic is far more superior to algebraic implementations when one deals with high-multiplicity situations involving a large number of Lorentz invariants. We introduce a new method based on balanced relations which allows one to achieve the goal of a robust functional restoration with minimal data input. The technique is implemented as a Mathematica package Reconstruction.m in the FIRE6 environment and thus successfully demonstrates a proof of concept. Copyright © 2023, The Authors. All rights reserved.

关键词： Rational functions

On error-based step size control for discontinuous Galerkin methods for compressible fluid dynamics

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arXiv 2022年

作者： Ranocha, Hendrik Winters, Andrew R. Castro, Hugo Guillermo Dalcin, Lisandro Schlottke-Lakemper, Michael Gassner, Gregor J. Parsani, Matteo Applied Mathematics University of Hamburg Germany Department of Mathematics Applied Mathematics Linköping University Sweden Thuwal23955-6900 Saudi Arabia Applied and Computational Mathematics RWTH Aachen University Germany High-Performance Computing Center Stuttgart University of Stuttgart Germany Department of Mathematics and Computer Science University of Cologne Germany Center for Data and Simulation Science University of Cologne Germany

We study temporal step size control of explicit Runge-Kutta methods for compressible computational fluid dynamics (CFD), including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations. We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy (CFL) number. Our numerical examples show that error-based step size control is easy to use, robust, and efficient, e.g., for (initial) transient periods, complex geometries, nonlinear shock capturing approaches, and schemes that use nonlinear entropy projections. We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases, the open source Julia packages *** with *** and the C/Fortran code SSDC based on *** Codes 65L06, 65M20, 65M70, 76M10, 76M22, 76N99, 35L50 Copyright © 2022, The Authors. All rights reserved.

关键词： Errors

Subspace method based on neural networks for eigenvalue problems

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arXiv 2024年

作者： Dai, Xiaoying Fan, Yunying Sheng, Zhiqiang LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China National Key Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing100088 China HEDPS Center for Applied Physics and Technology College of Engineering Peking University Beijing100871 China

With the rapid development of machine learning, numerical discretization methods based on deep neural networks have been widely used in many fields, especially in solving high-dimensional problems where traditional methods face bottlenecks. However, for low-dimensional problems, existing machine learning methods are not as accurate and efficient as traditional methods. In this paper, we propose a subspace method based on neural networks for eigenvalue problems with high accuracy and low cost. Our basic idea is to use neural network based basis functions to span a subspace, then calculate the parameters of the neural network based basis functions through appropriate training, and finally calculate the Galerkin projection of the eigenvalue problem onto the subspace and obtain an approximate solution. In addition, we reduce the dimension of the subspace by applying some dimensionality reduction technique, which can improve the accuracy of the approximate solution in further. Numerical experiments show that we can obtain approximate eigenvalues with accuracy of 10−11 but with less then 400 epochs, which is significantly superior to other existing neural network based *** Codes 65N25, 65L15, 68T07 Copyright © 2024, The Authors. All rights reserved.

关键词：

Graph Coarsening via Supervised Granular-Ball for Scalable Graph Neural Network Training

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arXiv 2024年

作者： Xia, Shuyin Ma, Xinjun Liu, Zhiyuan Liu, Cheng Zhao, Sen Wang, Guoyin Key Laboratory of Big Data Intelligent Computing China Chongqing Key Laboratory of Computational Intelligence China Chongqing University of Posts and Telecommunications Chongqing China National Center for Applied Mathematics in Chongqing Chongqing Normal University Chongqing401331 China

Graph Neural Networks (GNNs) have demonstrated significant achievements in processing graph data, yet scalability remains a substantial challenge. To address this, numerous graph coarsening methods have been developed. However, most existing coarsening methods are training-dependent, leading to lower efficiency, and they all require a predefined coarsening rate, lacking an adaptive approach. In this paper, we employ granular-ball computing to effectively compress graph data. We construct a coarsened graph network by iteratively splitting the graph into granular-balls based on a purity threshold and using these granular-balls as super vertices. This granulation process significantly reduces the size of the original graph, thereby greatly enhancing the training efficiency and scalability of GNNs. Additionally, our algorithm can adaptively perform splitting without requiring a predefined coarsening rate. Experimental results demonstrate that our method achieves accuracy comparable to training on the original graph. Noise injection experiments further indicate that our method exhibits robust performance. Moreover, our approach can reduce the graph size by up to 20 times without compromising test accuracy, substantially enhancing the scalability of GNNs. Copyright © 2024, The Authors. All rights reserved.

关键词： Graph neural networks