arXiv - CS - Numerical Analysis最新文献

An adaptive finite element multigrid solver using GPU acceleration 利用 GPU 加速的自适应有限元多网格求解器

arXiv - CS - Numerical Analysis Pub Date : 2024-05-08 DOI: arxiv-2405.05047

Manuel Liebchen, Utku Kaya, Christian Lessig, Thomas Richter

{"title":"An adaptive finite element multigrid solver using GPU acceleration","authors":"Manuel Liebchen, Utku Kaya, Christian Lessig, Thomas Richter","doi":"arxiv-2405.05047","DOIUrl":"https://doi.org/arxiv-2405.05047","url":null,"abstract":"Adaptive finite elements combined with geometric multigrid solvers are one of\u0000the most efficient numerical methods for problems such as the instationary\u0000Navier-Stokes equations. Yet despite their efficiency, computations remain\u0000expensive and the simulation of, for example, complex flow problems can take\u0000many hours or days. GPUs provide an interesting avenue to speed up the\u0000calculations due to their very large theoretical peak performance. However, the\u0000large degree of parallelism and non-standard API make the use of GPUs in\u0000scientific computing challenging. In this work, we develop a GPU acceleration\u0000for the adaptive finite element library Gascoigne and study its effectiveness\u0000for different systems of partial differential equations. Through the systematic\u0000formulation of all computations as linear algebra operations, we can employ\u0000GPU-accelerated linear algebra libraries, which simplifies the implementation\u0000and ensures the maintainability of the code while achieving very efficient GPU\u0000utilizations. Our results for a transport-diffusion equation, linear\u0000elasticity, and the instationary Navier-Stokes equations show substantial\u0000speedups of up to 20X compared to multi-core CPU implementations.","PeriodicalId":501061,"journal":{"name":"arXiv - CS - Numerical Analysis","volume":"112 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939097","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 posteriori error analysis of hybrid higher order methods for the elliptic obstacle problem 椭圆障碍问题混合高阶方法的后验误差分析

arXiv - CS - Numerical Analysis Pub Date : 2024-05-08 DOI: arxiv-2405.04961

Kamana Porwal, Ritesh Singla

引用次数: 0

A general error analysis for randomized low-rank approximation with application to data assimilation 应用于数据同化的随机低阶近似的一般误差分析

arXiv - CS - Numerical Analysis Pub Date : 2024-05-08 DOI: arxiv-2405.04811

Alexandre Scotto Di Perrotolo, Youssef Diouane, Selime Gürol, Xavier Vasseur

{"title":"A general error analysis for randomized low-rank approximation with application to data assimilation","authors":"Alexandre Scotto Di Perrotolo, Youssef Diouane, Selime Gürol, Xavier Vasseur","doi":"arxiv-2405.04811","DOIUrl":"https://doi.org/arxiv-2405.04811","url":null,"abstract":"Randomized algorithms have proven to perform well on a large class of\u0000numerical linear algebra problems. Their theoretical analysis is critical to\u0000provide guarantees on their behaviour, and in this sense, the stochastic\u0000analysis of the randomized low-rank approximation error plays a central role.\u0000Indeed, several randomized methods for the approximation of dominant eigen- or\u0000singular modes can be rewritten as low-rank approximation methods. However,\u0000despite the large variety of algorithms, the existing theoretical frameworks\u0000for their analysis rely on a specific structure for the covariance matrix that\u0000is not adapted to all the algorithms. We propose a general framework for the\u0000stochastic analysis of the low-rank approximation error in Frobenius norm for\u0000centered and non-standard Gaussian matrices. Under minimal assumptions on the\u0000covariance matrix, we derive accurate bounds both in expectation and\u0000probability. Our bounds have clear interpretations that enable us to derive\u0000properties and motivate practical choices for the covariance matrix resulting\u0000in efficient low-rank approximation algorithms. The most commonly used bounds\u0000in the literature have been demonstrated as a specific instance of the bounds\u0000proposed here, with the additional contribution of being tighter. Numerical\u0000experiments related to data assimilation further illustrate that exploiting the\u0000problem structure to select the covariance matrix improves the performance as\u0000suggested by our bounds.","PeriodicalId":501061,"journal":{"name":"arXiv - CS - Numerical Analysis","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939273","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

Analysis of the SQP Method for Hyperbolic PDE-Constrained Optimization in Acoustic Full Waveform Inversion 声学全波形反演中双曲 PDE 受限优化的 SQP 方法分析

arXiv - CS - Numerical Analysis Pub Date : 2024-05-08 DOI: arxiv-2405.05158

Luis Ammann, Irwin Yousept

引用次数: 0

Detection of a piecewise linear crack with one incident wave 用一个入射波探测片状线性裂纹

arXiv - CS - Numerical Analysis Pub Date : 2024-05-08 DOI: arxiv-2405.05179

Xiaoxu Xu, Guanqiu Ma, Guanghui Hu

引用次数: 0

A score-based particle method for homogeneous Landau equation 基于分数的同质朗道方程粒子法

arXiv - CS - Numerical Analysis Pub Date : 2024-05-08 DOI: arxiv-2405.05187

Yan Huang, Li Wang

{"title":"A score-based particle method for homogeneous Landau equation","authors":"Yan Huang, Li Wang","doi":"arxiv-2405.05187","DOIUrl":"https://doi.org/arxiv-2405.05187","url":null,"abstract":"We propose a novel score-based particle method for solving the Landau\u0000equation in plasmas, that seamlessly integrates learning with\u0000structure-preserving particle methods [arXiv:1910.03080]. Building upon the\u0000Lagrangian viewpoint of the Landau equation, a central challenge stems from the\u0000nonlinear dependence of the velocity field on the density. Our primary\u0000innovation lies in recognizing that this nonlinearity is in the form of the\u0000score function, which can be approximated dynamically via techniques from\u0000score-matching. The resulting method inherits the conservation properties of\u0000the deterministic particle method while sidestepping the necessity for kernel\u0000density estimation in [arXiv:1910.03080]. This streamlines computation and\u0000enhances scalability with dimensionality. Furthermore, we provide a theoretical\u0000estimate by demonstrating that the KL divergence between our approximation and\u0000the true solution can be effectively controlled by the score-matching loss.\u0000Additionally, by adopting the flow map viewpoint, we derive an update formula\u0000for exact density computation. Extensive examples have been provided to show\u0000the efficiency of the method, including a physically relevant case of Coulomb\u0000interaction.","PeriodicalId":501061,"journal":{"name":"arXiv - CS - Numerical Analysis","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926824","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

Randomized quasi-Monte Carlo and Owen's boundary growth condition: A spectral analysis 随机准蒙特卡罗和欧文边界增长条件：光谱分析

arXiv - CS - Numerical Analysis Pub Date : 2024-05-08 DOI: arxiv-2405.05181

Yang Liu

引用次数: 0

Exponential time propagators for elastodynamics 弹性力学的指数时间传播者

arXiv - CS - Numerical Analysis Pub Date : 2024-05-08 DOI: arxiv-2405.05213

Paavai Pari, Bikash Kanungo, Vikram Gavini

{"title":"Exponential time propagators for elastodynamics","authors":"Paavai Pari, Bikash Kanungo, Vikram Gavini","doi":"arxiv-2405.05213","DOIUrl":"https://doi.org/arxiv-2405.05213","url":null,"abstract":"We propose a computationally efficient and systematically convergent approach\u0000for elastodynamics simulations. We recast the second-order dynamical equation\u0000of elastodynamics into an equivalent first-order system of coupled equations,\u0000so as to express the solution in the form of a Magnus expansion. With any\u0000spatial discretization, it entails computing the exponential of a matrix acting\u0000upon a vector. We employ an adaptive Krylov subspace approach to inexpensively\u0000and and accurately evaluate the action of the exponential matrix on a vector.\u0000In particular, we use an apriori error estimate to predict the optimal Kyrlov\u0000subspace size required for each time-step size. We show that the Magnus\u0000expansion truncated after its first term provides quadratic and superquadratic\u0000convergence in the time-step for nonlinear and linear elastodynamics,\u0000respectively. We demonstrate the accuracy and efficiency of the proposed method\u0000for one linear (linear cantilever beam) and three nonlinear (nonlinear\u0000cantilever beam, soft tissue elastomer, and hyperelastic rubber) benchmark\u0000systems. For a desired accuracy in energy, displacement, and velocity, our\u0000method allows for $10-100times$ larger time-steps than conventional\u0000time-marching schemes such as Newmark-$beta$ method. Computationally, it\u0000translates to a $sim$$1000times$ and $sim$$10-100times$ speed-up over\u0000conventional time-marching schemes for linear and nonlinear elastodynamics,\u0000respectively.","PeriodicalId":501061,"journal":{"name":"arXiv - CS - Numerical Analysis","volume":"154 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926731","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

Approximation properties relative to continuous scale space for hybrid discretizations of Gaussian derivative operators 高斯导数算子混合离散化相对于连续尺度空间的逼近特性

arXiv - CS - Numerical Analysis Pub Date : 2024-05-08 DOI: arxiv-2405.05095

Tony Lindeberg

{"title":"Approximation properties relative to continuous scale space for hybrid discretizations of Gaussian derivative operators","authors":"Tony Lindeberg","doi":"arxiv-2405.05095","DOIUrl":"https://doi.org/arxiv-2405.05095","url":null,"abstract":"This paper presents an analysis of properties of two hybrid discretization\u0000methods for Gaussian derivatives, based on convolutions with either the\u0000normalized sampled Gaussian kernel or the integrated Gaussian kernel followed\u0000by central differences. The motivation for studying these discretization\u0000methods is that in situations when multiple spatial derivatives of different\u0000order are needed at the same scale level, they can be computed significantly\u0000more efficiently compared to more direct derivative approximations based on\u0000explicit convolutions with either sampled Gaussian kernels or integrated\u0000Gaussian kernels. While these computational benefits do also hold for the genuinely discrete\u0000approach for computing discrete analogues of Gaussian derivatives, based on\u0000convolution with the discrete analogue of the Gaussian kernel followed by\u0000central differences, the underlying mathematical primitives for the discrete\u0000analogue of the Gaussian kernel, in terms of modified Bessel functions of\u0000integer order, may not be available in certain frameworks for image processing,\u0000such as when performing deep learning based on scale-parameterized filters in\u0000terms of Gaussian derivatives, with learning of the scale levels. In this paper, we present a characterization of the properties of these\u0000hybrid discretization methods, in terms of quantitative performance measures\u0000concerning the amount of spatial smoothing that they imply, as well as the\u0000relative consistency of scale estimates obtained from scale-invariant feature\u0000detectors with automatic scale selection, with an emphasis on the behaviour for\u0000very small values of the scale parameter, which may differ significantly from\u0000corresponding results obtained from the fully continuous scale-space theory, as\u0000well as between different types of discretization methods.","PeriodicalId":501061,"journal":{"name":"arXiv - CS - Numerical Analysis","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939163","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

Energy stable gradient flow schemes for shape and topology optimization in Navier-Stokes flows 纳维-斯托克斯流中形状和拓扑优化的能量稳定梯度流方案

arXiv - CS - Numerical Analysis Pub Date : 2024-05-08 DOI: arxiv-2405.05098

Jiajie Li, Shengfeng Zhu

引用次数: 0