检索结果-内蒙古大学图书馆

arXiv 2024年

作者： Xu, Shu Cao, Liqun School of Mathematical Sciences Peking University Beijing100871 China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China State Key Laboratory of Scientific and Engineering Computing Beijing100190 China National Center for Mathematics and Interdisciplinary Sciences Chinese Academy of Sciences Beijing100190 China

We investigate the convergence of a backward Euler finite element discretization applied to a multi-domain and multi-scale elliptic-parabolic system, derived from the Doyle-Fuller-Newman model for lithium-ion batteries. We establish optimal-order error estimates for the solution in the norms l2(H1) and l2(L2(Hrq)), q = 0,1. To improve computational efficiency, we propose a novel solver that accelerates the solution process and controls memory usage. Numerical experiments with realistic battery parameters validate the theoretical error rates and demonstrate the significantly superior performance of the proposed solver over existing solvers. Copyright © 2024, The Authors. All rights reserved.

关键词： computational efficiency

Generalization Error Estimates of Machine Learning Methods for Solving High Dimensional Schrödinger Eigenvalue Problems

学校读者我要写书评

暂无评论

arXiv 2024年

作者： Yu, Hao Guo, Yixiao Ming, Pingbing 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

We propose a machine learning method for computing eigenvalues and eigenfunctions of the Schrödinger operator on a d-dimensional hypercube with Dirichlet boundary conditions. The cut-off function technique is employed to construct trial functions that precisely satisfy the homogeneous boundary conditions. This approach eliminates the error caused by the standard boundary penalty method, improves the overall accuracy of the method, as demonstrated by the typical numerical examples. Under the assumption that the eigenfunctions belong to a spectral Barron space, we derive an explicit convergence rate of the generalization error of the proposed method, which does not suffer from the curse of dimensionality. We verify the assumption by proving a new regularity shift result for the eigenfunctions when the potential function belongs to an appropriate spectral Barron space. Moreover, we extend the generalization error bound to the normalized penalty method, which is widely used in practice. Copyright © 2024, The Authors. All rights reserved.

关键词： Boundary conditions

Adaptive neural network basis methods for partial differential equations with low-regular solutions

学校读者我要写书评

暂无评论

arXiv 2024年

作者： Huang, Jianguo Wu, Haohao Zhou, Tao School of Mathematical Sciences MOE-LSC Shanghai Jiao Tong University Shanghai200240 China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

This paper aims to devise an adaptive neural network basis method for numerically solving a second-order semilinear partial differential equation (PDE) with low-regular solutions in two/three dimensions. The method is obtained by combining basis functions from a class of shallow neural networks and the resulting multi-scale analogues, a residual strategy in adaptive methods and the non-overlapping domain decomposition method. At the beginning, in view of the solution residual, we partition the total domain Ω into K + 1 non-overlapping subdomains, denoted respectively as {Ωk}Kk=0, where the exact solution is smooth on subdomain Ω0 and low-regular on subdomain Ωk (1 ≤ k ≤ K). Secondly, the low-regular solutions on different subdomains Ωk (1 ≤ k ≤ K) are approximated by neural networks with different scales, while the smooth solution on subdomain Ω0 is approximated by the initialized neural network. Thirdly, we determine the undetermined coefficients by solving the linear least squares problems directly or the nonlinear least squares problem via the Gauss-Newton method. The proposed method can be extended to multi-level case naturally. Finally, we use this adaptive method for several peak problems in two/three dimensions to show its high-efficient computational performance. Copyright © 2024, The Authors. All rights reserved.

关键词： Partial differential equations

CPAFT: A Consistent Parallel Advancing Front Technique for Unstructured Triangular/Tetrahedral Mesh Generation

学校读者我要写书评

暂无评论

arXiv 2024年

作者： Ma, Chengdi Huang, Jizu Luo, Hao Yang, Chao School of Mathematical Sciences Peking University Beijing100871 China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China PKU-Changsha Institute for Computing and Digital Economy Peking University Hunan410006 China

Compared with the remarkable progress made in parallel numerical solvers of partial differential equations, the development of algorithms for generating unstructured triangular/tetrahedral meshes has been relatively sluggish. In this paper, we propose a novel, consistent parallel advancing front technique (CPAFT) by combining the advancing front technique, the domain decomposition method based on space-filling curves, the distributed forest-of-overlapping-trees approach, and the consistent parallel maximal independent set algorithm. The newly proposed CPAFT algorithm can mathematically ensure that the generated unstructured triangular/tetrahedral meshes are independent of the number of processors and the implementation of domain decomposition. Several numerical tests are conducted to validate the parallel consistency and outstanding parallel efficiency of the proposed algorithm, which scales effectively up to two thousand processors. This is, as far as we know, the first parallel unstructured triangular/tetrahedral mesh generator with scalability to O(1,000) CPU *** Codes 65M50, 65M55, 68W10 Copyright © 2024, The Authors. All rights reserved.

关键词： Domain decomposition methods

A riemannian exponential augmented lagrangian method for computing the projection robust Wasserstein distance 23

学校读者我要写书评

暂无评论

A riemannian exponential augmented lagrangian method for com...

Proceedings of the 37th International Conference on Neural Information Processing Systems

作者： Bo Jiang Ya-Feng Liu Ministry of Education Key Laboratory of NSLSCS School of Mathematical Sciences Nanjing Normal University Nanjing 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 Beijing China

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 step-sizes 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.

关键词：

Quasi-Monte Carlo integration for feedback control under uncertainty

学校读者我要写书评

暂无评论

arXiv 2024年

作者： Guth, Philipp A. Kritzer, Peter Kunisch, Karl Johann Radon Institute for Computational and Applied Mathematics ÖAW Altenbergerstrasse 69 Linz4040 Austria Institute of Mathematics and Scientific Computing Karl-Franzens University Graz Heinrichstrasse 36 Graz8010 Austria

A control in feedback form is derived for linear quadratic, time-invariant optimal control problems subject to parabolic partial differential equations with coefficients depending on a countably infinite number of uncertain parameters. It is shown that the Riccati-based feedback operator depends analytically on the parameters provided that the system operator depends analytically on the parameters, as is the case, for instance, in diffusion problems when the diffusion coefficient is parameterized by a Karhunen–Loève expansion. These novel parametric regularity results allow the application of quasi-Monte Carlo (QMC) methods to efficiently compute an a-priori chosen feedback law based on the expected value. Moreover, under moderate assumptions on the input random field, QMC methods achieve superior error rates compared to ordinary Monte Carlo methods, independently of the stochastic dimension of the problem. Indeed, our paper for the first time studies Banach-space-valued integration by higher-order QMC *** Codes 35R60, 49N10, 65D30, 65D32, 93B52 © 2024, CC BY.

关键词： Stochastic systems

An Effective Two-Stage Column Generation Strategy for Solving Unit Commitment

学校读者我要写书评

暂无评论

An Effective Two-Stage Column Generation Strategy for Solvin...

IEEE Conference on Energy Internet and Energy System Integration (EI2)

作者： Zhao-Gang Su Sheng-Jie Chen Yu-Fei Li Liang Chen Jia-Yi Deng Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China Institute of Applied Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China College of Sciences Beijing University of Posts and Telecommunications Beijing China

ISBN: (数字)9798331523527

ISBN: (纸本)9798331523534

Unit commitment (UC) has been extensively studied in power systems literature and is typically formulated as a mixed-integer programming (MIP) problem. However, existing methods and commercial MIP solvers often struggle to efficiently solve large-scale UC instances. This paper proposes a novel MIP model for addressing UC by employing an innovative column generation-based two-stage approach to efficiently compute high-quality solutions. Our method leverages parallel computing to solve pricing problems and generate columns effectively. Following column generation, we solve the master problem as a MIP model to obtain high-quality feasible solutions for the UC problem. Comprehensive numerical experiments demonstrate that our proposed method can effectively tackle large-scale UC problems involving over 1,000 units, significantly outperforming state-of-the-art MIP solvers in both solution quality and computational efficiency. The results underscore the potential of our approach for practical application in large-scale power system scheduling.

关键词： Processor scheduling computational modeling Refining Pricing System integration Parallel processing Programming computational efficiency Power system reliability Reliability

Asymptotic Estimates for Spectral Estimators of Rotationally Invariant Matrices

学校读者我要写书评

暂无评论

Asymptotic Estimates for Spectral Estimators of Rotationally...

IEEE International Symposium on Information Theory

作者： Zhuohang He Xiaojun Yuan Junjie Ma National Key Laboratory of Wireless Communications University of Electronic Science and Technology of China Chengdu China Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences Beijing China

ISBN: (数字)9798350382846

ISBN: (纸本)9798350382853

In this paper, we consider the recovery of low-rank matrices from noisy observations using spectral denoisers, where the singular values are denoised through an identical scalar smoothing function. We explore the asymptotic mean squared error (AMSE) of these denoisers within a framework where the rank of the matrix to be recovered grows linearly with the matrix size. We demonstrate that, under arbitrary i.i.d. noise and some mild regularity assumptions, the AMSE converges in probability to a deterministic function of the noise power. Our results are applicable to commonly used denoisers, including the best-rank-r denoiser, the singular-value soft-threshold denoiser, and the singular-value hard-threshold denoiser. To the best of our knowledge, this is the first study to establish an analytical expression for the asymptotic MSE under arbitrary i.i.d. noise. The derived analytical expression depends solely on the empirical distribution of the singular values of the low-rank matrix and the specific form of the spectral denoiser employed.

关键词： Smoothing methods Limiting Noise Noise reduction Noise measurement Information theory

A Scalable Real-Time Data Assimilation Framework for Predicting Turbulent Atmosphere Dynamics

学校读者我要写书评

暂无评论

A Scalable Real-Time Data Assimilation Framework for Predict...

2024 Workshops of the International Conference for High Performance computing, Networking, Storage and Analysis, SC Workshops 2024

作者： Yin, Junqi Liang, Siming Liu, Siyan Bao, Feng Chipilski, Hristo G. Lu, Dan Zhang, Guannan Oak Ridge National Laboratory National Center for Computational Science Oak RidgeTN United States Florida State University Department of Mathematics TallahasseeFL United States Oak Ridge National Laboratory Computational Science and Engineering Division Oak RidgeTN United States Florida State University Department of Scientific Computing TallahasseeFL United States Oak Ridge National Laboratory Computer Science and Mathematics Division Oak RidgeTN United States

ISBN: (纸本)9798350355543

AI-based foundation models like FourCastNet, GraphCast, ClimaX, and Pangu-Weather are revolutionizing weather and climate predictions but are not yet ready for operational use. Their limitation lies in the absence of a data assimilation system to incorporate real-time Earth system observations, crucial for accurately forecasting events like tropical cyclones and atmospheric rivers. To overcome these obstacles, we introduce a generic real-time data assimilation framework and demonstrate its end-to-end performance on the Frontier supercomputer. This framework comprises two primary modules: an ensemble score filter (EnSF), which significantly outperforms the state-of-the-art data assimilation method, e.g., the Local Ensemble Transform Kalman Filter (LETKF);and a vision transformer-based surrogate capable of real-time adaptation through the integration of observational data. We demonstrate both the strong and weak scaling of our framework up to 1024 GPUs on the Exascale supercomputer, Frontier. Our results not only illustrate the framework's exceptional scalability on high-performance computing systems, but also demonstrate the importance of supercomputers in real-time data assimilation for weather and climate predictions. Even though the proposed framework is tested only on a benchmark surface quasi-geostrophic (SQG) turbulence system, it has the potential to be combined with existing AI-based foundation models, making it suitable for future operational implementations. © 2024 IEEE.

关键词： data assimilation Diffusion models foundation models generative AI high-performance computing uncertainty quantification

Using a research-based assessment instrument to explore undergraduate students’ proficiencies around measurement uncertainty in physics lab contexts

学校读者我要写书评

暂无评论

Physical Review Physics Education Research 2024年第2期20卷 020105-020105页

作者： Gayle Geschwind Michael Vignal Marcos D. Caballero H. J. Lewandowski Department of Physics Department of Physics and Astronomy and CREATE for STEM Institute Department of Computational Mathematics Science and Engineering Department of Physics and Center for Computing in Science Education

Concepts and practices surrounding measurement uncertainty are vital knowledge for physicists and are often emphasized in undergraduate physics laboratory courses. We have previously developed a research-based assessment instrument—the Survey of Physics Reasoning on Uncertainty Concepts in Experiments (SPRUCE)—to examine student proficiency with measurement uncertainty along a variety of axes, including sources of uncertainty, handling of uncertainty, and distributions and repeated measurements. We present here initial results from the assessment representing over 1500 students from 20 institutions. We analyze students’ performance pre- and postinstruction in lab courses and examine how instruction impacts students with different majors and gender. We find that students typically excel in certain areas, such as reporting the mean of a distribution as their result, while they struggle in other areas, such as comparing measurements with uncertainty and correctly propagating errors using formulas. Additionally, we find that the importance that an instructor places in certain areas of measurement uncertainty is uncorrelated with student performance in those areas.

关键词： Assessment Instructional strategies Student preparation