arXiv - MATH - Numerical Analysis最新文献_第8页

Anisotropic Error Analysis of Weak Galerkin finite element method for Singularly Perturbed Biharmonic Problems 弱 Galerkin 有限元方法对奇异扰动双谐波问题的各向异性误差分析

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-11 DOI: arxiv-2409.07217

Aayushman Raina, Srinivasan Natesan, Şuayip Toprakseven

引用次数: 0

Dynamic Error-Bounded Hierarchical Matrices in Neural Network Compression 神经网络压缩中的动态误差约束层次矩阵

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-11 DOI: arxiv-2409.07028

John Mango, Ronald Katende

{"title":"Dynamic Error-Bounded Hierarchical Matrices in Neural Network Compression","authors":"John Mango, Ronald Katende","doi":"arxiv-2409.07028","DOIUrl":"https://doi.org/arxiv-2409.07028","url":null,"abstract":"This paper presents an innovative framework that integrates hierarchical\u0000matrix (H-matrix) compression techniques into the structure and training of\u0000Physics-Informed Neural Networks (PINNs). By leveraging the low-rank properties\u0000of matrix sub-blocks, the proposed dynamic, error-bounded H-matrix compression\u0000method significantly reduces computational complexity and storage requirements\u0000without compromising accuracy. This approach is rigorously compared to\u0000traditional compression techniques, such as Singular Value Decomposition (SVD),\u0000pruning, and quantization, demonstrating superior performance, particularly in\u0000maintaining the Neural Tangent Kernel (NTK) properties critical for the\u0000stability and convergence of neural networks. The findings reveal that H-matrix\u0000compression not only enhances training efficiency but also ensures the\u0000scalability and robustness of PINNs for complex, large-scale applications in\u0000physics-based modeling. This work offers a substantial contribution to the\u0000optimization of deep learning models, paving the way for more efficient and\u0000practical implementations of PINNs in real-world scenarios.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181964","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

Finite element approximation of stationary Fokker--Planck--Kolmogorov equations with application to periodic numerical homogenization 静态福克--普朗克--科尔莫戈罗夫方程的有限元近似与周期数值同质化的应用

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-11 DOI: arxiv-2409.07371

Timo Sprekeler, Endre Süli, Zhiwen Zhang

引用次数: 0

Dual scale Residual-Network for turbulent flow sub grid scale resolving: A prior analysis 用于湍流子网格尺度解析的双尺度残差网络：先验分析

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-11 DOI: arxiv-2409.07605

Omar Sallam, Mirjam Fürth

{"title":"Dual scale Residual-Network for turbulent flow sub grid scale resolving: A prior analysis","authors":"Omar Sallam, Mirjam Fürth","doi":"arxiv-2409.07605","DOIUrl":"https://doi.org/arxiv-2409.07605","url":null,"abstract":"This paper introduces generative Residual Networks (ResNet) as a surrogate\u0000Machine Learning (ML) tool for Large Eddy Simulation (LES) Sub Grid Scale (SGS)\u0000resolving. The study investigates the impact of incorporating Dual Scale\u0000Residual Blocks (DS-RB) within the ResNet architecture. Two LES SGS resolving\u0000models are proposed and tested for prior analysis test cases: a\u0000super-resolution model (SR-ResNet) and a SGS stress tensor inference model\u0000(SGS-ResNet). The SR-ResNet model task is to upscale LES solutions from coarse\u0000to finer grids by inferring unresolved SGS velocity fluctuations, exhibiting\u0000success in preserving high-frequency velocity fluctuation information, and\u0000aligning with higher-resolution LES solutions' energy spectrum. Furthermore,\u0000employing DS-RB enhances prediction accuracy and precision of high-frequency\u0000velocity fields compared to Single Scale Residual Blocks (SS-RB), evident in\u0000both spatial and spectral domains. The SR-ResNet model is tested and trained on\u0000filtered/downsampled 2-D LES planar jet injection problems at two Reynolds\u0000numbers, two jet configurations, and two upscale ratios. In the case of SGS\u0000stress tensor inference, both SS-RB and DS-RB exhibit higher prediction\u0000accuracy over the Smagorinsky model with reference to the true DNS SGS stress\u0000tensor, with DS-RB-based SGS-ResNet showing stronger statistical alignment with\u0000DNS data. The SGS-ResNet model is tested on a filtered/downsampled 2-D DNS\u0000isotropic homogenous decay turbulence problem. The adoption of DS-RB incurs\u0000notable increases in network size, training time, and forward inference time,\u0000with the network size expanding by over tenfold, and training and forward\u0000inference times increasing by approximately 0.5 and 3 times, respectively.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181925","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 construction of canonical nonconforming finite element spaces for elliptic equations of any order in any dimension 构建任意维度任意阶椭圆方程的典型非符合有限元空间

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-10 DOI: arxiv-2409.06134

Jia Li, Shuonan Wu

引用次数: 0

Distributed Cooperative AI for Large-Scale Eigenvalue Computations Using Neural Networks 利用神经网络进行大规模特征值计算的分布式合作人工智能

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-10 DOI: arxiv-2409.06746

Ronald Katende

引用次数: 0

A tutorial on automatic differentiation with complex numbers 复数自动微分教程

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-10 DOI: arxiv-2409.06752

Nicholas Krämer

{"title":"A tutorial on automatic differentiation with complex numbers","authors":"Nicholas Krämer","doi":"arxiv-2409.06752","DOIUrl":"https://doi.org/arxiv-2409.06752","url":null,"abstract":"Automatic differentiation is everywhere, but there exists only minimal\u0000documentation of how it works in complex arithmetic beyond stating \"derivatives\u0000in $mathbb{C}^d$\" $cong$ \"derivatives in $mathbb{R}^{2d}$\" and, at best,\u0000shallow references to Wirtinger calculus. Unfortunately, the equivalence\u0000$mathbb{C}^d cong mathbb{R}^{2d}$ becomes insufficient as soon as we need to\u0000derive custom gradient rules, e.g., to avoid differentiating \"through\"\u0000expensive linear algebra functions or differential equation simulators. To\u0000combat such a lack of documentation, this article surveys forward- and\u0000reverse-mode automatic differentiation with complex numbers, covering topics\u0000such as Wirtinger derivatives, a modified chain rule, and different gradient\u0000conventions while explicitly avoiding holomorphicity and the Cauchy--Riemann\u0000equations (which would be far too restrictive). To be precise, we will derive,\u0000explain, and implement a complex version of Jacobian-vector and vector-Jacobian\u0000products almost entirely with linear algebra without relying on complex\u0000analysis or differential geometry. This tutorial is a call to action, for users\u0000and developers alike, to take complex values seriously when implementing custom\u0000gradient propagation rules -- the manuscript explains how.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"171 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181961","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 low-rank Runge-Kutta methods 随机低阶 Runge-Kutta 方法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-10 DOI: arxiv-2409.06384

Hei Yin Lam, Gianluca Ceruti, Daniel Kressner

{"title":"Randomized low-rank Runge-Kutta methods","authors":"Hei Yin Lam, Gianluca Ceruti, Daniel Kressner","doi":"arxiv-2409.06384","DOIUrl":"https://doi.org/arxiv-2409.06384","url":null,"abstract":"This work proposes and analyzes a new class of numerical integrators for\u0000computing low-rank approximations to solutions of matrix differential equation.\u0000We combine an explicit Runge-Kutta method with repeated randomized low-rank\u0000approximation to keep the rank of the stages limited. The so-called generalized\u0000Nystr\"om method is particularly well suited for this purpose; it builds\u0000low-rank approximations from random sketches of the discretized dynamics. In\u0000contrast, all existing dynamical low-rank approximation methods are\u0000deterministic and usually perform tangent space projections to limit rank\u0000growth. Using such tangential projections can result in larger error compared\u0000to approximating the dynamics directly. Moreover, sketching allows for\u0000increased flexibility and efficiency by choosing structured random matrices\u0000adapted to the structure of the matrix differential equation. Under suitable\u0000assumptions, we establish moment and tail bounds on the error of our randomized\u0000low-rank Runge-Kutta methods. When combining the classical Runge-Kutta method\u0000with generalized Nystr\"om, we obtain a method called Rand RK4, which exhibits\u0000fourth-order convergence numerically -- up to the low-rank approximation error.\u0000For a modified variant of Rand RK4, we also establish fourth-order convergence\u0000theoretically. Numerical experiments for a range of examples from the\u0000literature demonstrate that randomized low-rank Runge-Kutta methods compare\u0000favorably with two popular dynamical low-rank approximation methods, in terms\u0000of robustness and speed of convergence.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181966","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

An Eulerian Vortex Method on Flow Maps 流图上的欧拉旋涡法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-10 DOI: arxiv-2409.06201

Sinan Wang, Yitong Deng, Molin Deng, Hong-Xing Yu, Junwei Zhou, Duowen Chen, Taku Komura, Jiajun Wu, Bo Zhu

{"title":"An Eulerian Vortex Method on Flow Maps","authors":"Sinan Wang, Yitong Deng, Molin Deng, Hong-Xing Yu, Junwei Zhou, Duowen Chen, Taku Komura, Jiajun Wu, Bo Zhu","doi":"arxiv-2409.06201","DOIUrl":"https://doi.org/arxiv-2409.06201","url":null,"abstract":"We present an Eulerian vortex method based on the theory of flow maps to\u0000simulate the complex vortical motions of incompressible fluids. Central to our\u0000method is the novel incorporation of the flow-map transport equations for line\u0000elements, which, in combination with a bi-directional marching scheme for flow\u0000maps, enables the high-fidelity Eulerian advection of vorticity variables. The\u0000fundamental motivation is that, compared to impulse $mathbf{m}$, which has\u0000been recently bridged with flow maps to encouraging results, vorticity\u0000$boldsymbol{omega}$ promises to be preferable for its numerically stability\u0000and physical interpretability. To realize the full potential of this novel\u0000formulation, we develop a new Poisson solving scheme for vorticity-to-velocity\u0000reconstruction that is both efficient and able to accurately handle the\u0000coupling near solid boundaries. We demonstrate the efficacy of our approach\u0000with a range of vortex simulation examples, including leapfrog vortices, vortex\u0000collisions, cavity flow, and the formation of complex vortical structures due\u0000to solid-fluid interactions.","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":"142181972","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 Liang-Kleeman Causality Analysis based on Linear Inverse Modeling 基于线性反演模型的梁-克莱曼因果关系分析

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-10 DOI: arxiv-2409.06797

Justin Lien

{"title":"A Liang-Kleeman Causality Analysis based on Linear Inverse Modeling","authors":"Justin Lien","doi":"arxiv-2409.06797","DOIUrl":"https://doi.org/arxiv-2409.06797","url":null,"abstract":"Causality analysis is a powerful tool for determining cause-and-effect\u0000relationships between variables in a system by quantifying the influence of one\u0000variable on another. Despite significant advancements in the field, many\u0000existing studies are constrained by their focus on unidirectional causality or\u0000Gaussian external forcing, limiting their applicability to complex real-world\u0000problems. This study proposes a novel data-driven approach to causality\u0000analysis for complex stochastic differential systems, integrating the concepts\u0000of Liang-Kleeman information flow and linear inverse modeling. Our method\u0000models environmental noise as either memoryless Gaussian white noise or\u0000memory-retaining Ornstein-Uhlenbeck colored noise, and allows for self and\u0000mutual causality, providing a more realistic representation and interpretation\u0000of the underlying system. Moreover, this LIM-based approach can identify the\u0000individual contribution of dynamics and correlation to causality. We apply this\u0000approach to re-examine the causal relationships between the El\u0000Ni~{n}o-Southern Oscillation (ENSO) and the Indian Ocean Dipole (IOD), two\u0000major climate phenomena that significantly influence global climate patterns.\u0000In general, regardless of the type of noise used, the causality between ENSO\u0000and IOD is mutual but asymmetric, with the causality map reflecting an\u0000ENSO-like pattern consistent with previous studies. Notably, in the case of\u0000colored noise, the noise memory map reveals a hotspot in the Ni~no 3 region,\u0000which is further related to the information flow. This suggests that our\u0000approach offers a more comprehensive framework and provides deeper insights\u0000into the causal inference of global climate systems.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181960","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