arXiv - MATH - Numerical Analysis最新文献_第9页

Efficient Matrix Decomposition for High-Dimensional Structured Systems: Theory and Applications 高维结构系统的高效矩阵分解：理论与应用

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-10 DOI: arxiv-2409.06321

Ronald Katende

{"title":"Efficient Matrix Decomposition for High-Dimensional Structured Systems: Theory and Applications","authors":"Ronald Katende","doi":"arxiv-2409.06321","DOIUrl":"https://doi.org/arxiv-2409.06321","url":null,"abstract":"In this paper, we introduce a novel matrix decomposition method, referred to\u0000as the ( D )-decomposition, designed to improve computational efficiency and\u0000stability for solving high-dimensional linear systems. The decomposition\u0000factorizes a matrix ( A in mathbb{R}^{n times n} ) into three matrices (\u0000A = PDQ ), where ( P ), ( D ), and ( Q ) are structured to exploit\u0000sparsity, low rank, and other matrix properties. We provide rigorous proofs for\u0000the existence, uniqueness, and stability of the decomposition under various\u0000conditions, including noise perturbations and rank constraints. The ( D\u0000)-decomposition offers significant computational advantages, particularly for\u0000sparse or low-rank matrices, reducing the complexity from ( O(n^3) ) for\u0000traditional decompositions to ( O(n^2 k) ) or better, depending on the\u0000structure of the matrix. This method is particularly suited for large-scale\u0000applications in machine learning, signal processing, and data science.\u0000Numerical examples demonstrate the method's superior performance over\u0000traditional LU and QR decompositions, particularly in the context of\u0000dimensionality reduction and large-scale matrix factorization.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"118 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Block structured matrix-sequences and their spectral and singular value canonical distributions: a general theory 块结构矩阵序列及其频谱和奇异值规范分布：一般理论

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-10 DOI: arxiv-2409.06465

Isabella Furci, Andrea Adriani, Stefano Serra-Capizzano

引用次数: 0

Two-level Restricted Additive Schwarz preconditioner based on Multiscale Spectral Generalized FEM for Heterogeneous Helmholtz Problems 基于多尺度谱广义有限元的两级限制性加法施瓦茨预处理器，用于解决异质亥姆霍兹问题

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-10 DOI: arxiv-2409.06533

Chupeng Ma, Christian Alber, Robert Scheichl

{"title":"Two-level Restricted Additive Schwarz preconditioner based on Multiscale Spectral Generalized FEM for Heterogeneous Helmholtz Problems","authors":"Chupeng Ma, Christian Alber, Robert Scheichl","doi":"arxiv-2409.06533","DOIUrl":"https://doi.org/arxiv-2409.06533","url":null,"abstract":"We present and analyze a two-level restricted additive Schwarz (RAS)\u0000preconditioner for heterogeneous Helmholtz problems, based on a multiscale\u0000spectral generalized finite element method (MS-GFEM) proposed in [C. Ma, C.\u0000Alber, and R. Scheichl, SIAM. J. Numer. Anal., 61 (2023), pp. 1546--1584]. The\u0000preconditioner uses local solves with impedance boundary conditions, and a\u0000global coarse solve based on the MS-GFEM approximation space constructed from\u0000local eigenproblems. It is derived by first formulating MS-GFEM as a Richardson\u0000iterative method, and without using an oversampling technique, reduces to the\u0000preconditioner recently proposed and analyzed in [Q. Hu and Z.Li, arXiv\u00002402.06905]. We prove that both the Richardson iterative method and the preconditioner\u0000used within GMRES converge at a rate of $Lambda$ under some reasonable\u0000conditions, where $Lambda$ denotes the error of the underlying MS-GFEM\u0000rs{approximation}. Notably, the convergence proof of GMRES does not rely on\u0000the `Elman theory'. An exponential convergence property of MS-GFEM, resulting\u0000from oversampling, ensures that only a few iterations are needed for\u0000convergence with a small coarse space. Moreover, the convergence rate $Lambda$\u0000is not only independent of the fine-mesh size $h$ and the number of subdomains,\u0000but decays with increasing wavenumber $k$. In particular, in the\u0000constant-coefficient case, with $hsim k^{-1-gamma}$ for some $gammain\u0000(0,1]$, it holds that $Lambda sim k^{-1+frac{gamma}{2}}$.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"118 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A Second Moment Method for k-Eigenvalue Acceleration with Continuous Diffusion and Discontinuous Transport Discretizations 连续扩散和非连续传输离散的 k 特征值加速度第二矩方法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-10 DOI: arxiv-2409.06162

Zachary K. Hardy, Jim E. Morel, Jan I. C. Vermaak

{"title":"A Second Moment Method for k-Eigenvalue Acceleration with Continuous Diffusion and Discontinuous Transport Discretizations","authors":"Zachary K. Hardy, Jim E. Morel, Jan I. C. Vermaak","doi":"arxiv-2409.06162","DOIUrl":"https://doi.org/arxiv-2409.06162","url":null,"abstract":"The second moment method is a linear acceleration technique which couples the\u0000transport equation to a diffusion equation with transport-dependent additive\u0000closures. The resulting low-order diffusion equation can be discretized\u0000independent of the transport discretization, unlike diffusion synthetic\u0000acceleration, and is symmetric positive definite, unlike quasi-diffusion. While\u0000this method has been shown to be comparable to quasi-diffusion in iterative\u0000performance for fixed source and time-dependent problems, it is largely\u0000unexplored as an eigenvalue problem acceleration scheme due to thought that the\u0000resulting inhomogeneous source makes the problem ill-posed. Recently, a\u0000preliminary feasibility study was performed on the second moment method for\u0000eigenvalue problems. The results suggested comparable performance to\u0000quasi-diffusion and more robust performance than diffusion synthetic\u0000acceleration. This work extends the initial study to more realistic reactor\u0000problems using state-of-the-art discretization techniques. Results in this\u0000paper show that the second moment method is more computationally efficient than\u0000its alternatives on complex reactor problems with unstructured meshes.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A Canonical Gauge for Computing of Eigenpairs of the Magnetic Schrödinger Operator 计算磁薛定谔算子特征对的典型量纲

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-09 DOI: arxiv-2409.06023

Jeffrey S. Ovall, Li Zhu

引用次数: 0

Communication in Multiplex Transportation Networks 多路运输网络中的通信

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-09 DOI: arxiv-2409.05575

Silvia Noschese, Lothar Reichel

引用次数: 0

A divergence-free projection method for quasiperiodic photonic crystals in three dimensions 三维准周期光子晶体的无发散投影法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-09 DOI: arxiv-2409.05528

Zixuan Gao, Zhenli Xu, Zhiguo Yang

{"title":"A divergence-free projection method for quasiperiodic photonic crystals in three dimensions","authors":"Zixuan Gao, Zhenli Xu, Zhiguo Yang","doi":"arxiv-2409.05528","DOIUrl":"https://doi.org/arxiv-2409.05528","url":null,"abstract":"This paper presents a point-wise divergence-free projection method for\u0000numerical approximations of photonic quasicrystals problems. The original\u0000three-dimensional quasiperiodic Maxwell's system is transformed into a periodic\u0000one in higher dimensions through a variable substitution involving the\u0000projection matrix, such that periodic boundary condition can be readily\u0000applied. To deal with the intrinsic divergence-free constraint of the Maxwell's\u0000equations, we present a quasiperiodic de Rham complex and its associated\u0000commuting diagram, based on which a point-wise divergence-free quasiperiodic\u0000Fourier spectral basis is proposed. With the help of this basis, we then\u0000propose an efficient solution algorithm for the quasiperiodic source problem\u0000and conduct its rigorous error estimate. Moreover, by analyzing the decay rate\u0000of the Fourier coefficients of the eigenfunctions, we further propose a\u0000divergence-free reduced projection method for the quasiperiodic Maxwell\u0000eigenvalue problem, which significantly alleviates the computational cost.\u0000Several numerical experiments are presented to validate the efficiency and\u0000accuracy of the proposed method.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A semi-Lagrangian method for the direct numerical simulation of crystallization and precipitation at the pore scale 孔隙尺度结晶和沉淀直接数值模拟的半拉格朗日方法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-09 DOI: arxiv-2409.05449

Sarah Perez, Jean-Matthieu Etancelin, Philippe Poncet

引用次数: 0

On a shrink-and-expand technique for block eigensolvers 关于分块求解器的收缩与扩展技术

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-09 DOI: arxiv-2409.05572

Yuqi Liu, Yuxin Ma, Meiyue Shao

引用次数: 0

Asymptotic Preserving Linearly Implicit Additive IMEX-RK Finite Volume Schemes for Low Mach Number Isentropic Euler Equations 低马赫数等熵欧拉方程的渐近保线性隐含加法 IMEX-RK 有限体积方案

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-09 DOI: arxiv-2409.05854

Saurav Samantaray

{"title":"Asymptotic Preserving Linearly Implicit Additive IMEX-RK Finite Volume Schemes for Low Mach Number Isentropic Euler Equations","authors":"Saurav Samantaray","doi":"arxiv-2409.05854","DOIUrl":"https://doi.org/arxiv-2409.05854","url":null,"abstract":"We consider the compressible Euler equations of gas dynamics with isentropic\u0000equation of state. In the low Mach number regime i.e. when the fluid velocity\u0000is very very small in comparison to the sound speed in the medium, the solution\u0000of the compressible system converges to the solution of its incompressible\u0000counter part. Standard numerical schemes fail to respect this transition\u0000property and hence are plagued with inaccuracies as well as instabilities. In\u0000this paper we introduce an extra flux term to the momentum flux. This extra\u0000term is brought to fore by looking at the incompressibility constraints of the\u0000asymptotic limit system. This extra flux term enables us to get a suitable flux\u0000splitting, so that an additive IMEX-RK scheme could be applied. Using an\u0000elliptic reformulation the scheme boils down to just solving a linear elliptic\u0000problem for the density and then explicit updates for the momentum. The IMEX\u0000schemes developed are shown to be formally asymptotically consistent with the\u0000low Mach number limit of the Euler equations. A second order space time fully\u0000discrete scheme is obtained in the finite volume framework using a combination\u0000of Rusanov flux for the explicit part and simple central differences for the\u0000implicit part. Numerical results are reported which elucidate the theoretical\u0000assertions regarding the scheme and its robustness.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"389 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0