arXiv - MATH - Numerical Analysis最新文献_第7页

Deep learning methods for stochastic Galerkin approximations of elliptic random PDEs 椭圆随机 PDE 随机 Galerkin 近似的深度学习方法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-12 DOI: arxiv-2409.08063

Fabio Musco, Andrea Barth

{"title":"Deep learning methods for stochastic Galerkin approximations of elliptic random PDEs","authors":"Fabio Musco, Andrea Barth","doi":"arxiv-2409.08063","DOIUrl":"https://doi.org/arxiv-2409.08063","url":null,"abstract":"This work considers stochastic Galerkin approximations of linear elliptic\u0000partial differential equations with stochastic forcing terms and stochastic\u0000diffusion coefficients, that cannot be bounded uniformly away from zero and\u0000infinity. A traditional numerical method for solving the resulting\u0000high-dimensional coupled system of partial differential equations (PDEs) is\u0000replaced by deep learning techniques. In order to achieve this,\u0000physics-informed neural networks (PINNs), which typically operate on the strong\u0000residual of the PDE and can therefore be applied in a wide range of settings,\u0000are considered. As a second approach, the Deep Ritz method, which is a neural\u0000network that minimizes the Ritz energy functional to find the weak solution, is\u0000employed. While the second approach only works in special cases, it overcomes\u0000the necessity of testing in variational problems while maintaining mathematical\u0000rigor and ensuring the existence of a unique solution. Furthermore, the\u0000residual is of a lower differentiation order, reducing the training cost\u0000considerably. The efficiency of the method is demonstrated on several model\u0000problems.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227624","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 of the Hilbert Transform on the unit circle 单位圆上希尔伯特变换的近似值

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-12 DOI: arxiv-2409.07810

Luisa Fermo, Valerio Loi

引用次数: 0

Transformed Physics-Informed Neural Networks for The Convection-Diffusion Equation 用于对流扩散方程的变换物理信息神经网络

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-12 DOI: arxiv-2409.07671

Jiajing Guan, Howard Elman

引用次数: 0

Edge-Wise Graph-Instructed Neural Networks 边缘智图引导神经网络

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-12 DOI: arxiv-2409.08023

Francesco Della Santa, Antonio Mastropietro, Sandra Pieraccini, Francesco Vaccarino

引用次数: 0

Coordinate complexification for the Helmholtz equation with Dirichlet boundary conditions in a perturbed half-space 扰动半空间中带有狄里赫特边界条件的亥姆霍兹方程的坐标复合化

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-11 DOI: arxiv-2409.06988

Charles L. Epstein, Leslie Greengard, Jeremy Hoskins, Shidong Jiang, Manas Rachh

{"title":"Coordinate complexification for the Helmholtz equation with Dirichlet boundary conditions in a perturbed half-space","authors":"Charles L. Epstein, Leslie Greengard, Jeremy Hoskins, Shidong Jiang, Manas Rachh","doi":"arxiv-2409.06988","DOIUrl":"https://doi.org/arxiv-2409.06988","url":null,"abstract":"We present a new complexification scheme based on the classical double layer\u0000potential for the solution of the Helmholtz equation with Dirichlet boundary\u0000conditions in compactly perturbed half-spaces in two and three dimensions. The\u0000kernel for the double layer potential is the normal derivative of the\u0000free-space Green's function, which has a well-known analytic continuation into\u0000the complex plane as a function of both target and source locations. Here, we\u0000prove that - when the incident data are analytic and satisfy a precise\u0000asymptotic estimate - the solution to the boundary integral equation itself\u0000admits an analytic continuation into specific regions of the complex plane, and\u0000satisfies a related asymptotic estimate (this class of data includes both plane\u0000waves and the field induced by point sources). We then show that, with a\u0000carefully chosen contour deformation, the oscillatory integrals are converted\u0000to exponentially decaying integrals, effectively reducing the infinite domain\u0000to a domain of finite size. Our scheme is different from existing methods that\u0000use complex coordinate transformations, such as perfectly matched layers, or\u0000absorbing regions, such as the gradual complexification of the governing\u0000wavenumber. More precisely, in our method, we are still solving a boundary\u0000integral equation, albeit on a truncated, complexified version of the original\u0000boundary. In other words, no volumetric/domain modifications are introduced.\u0000The scheme can be extended to other boundary conditions, to open wave guides\u0000and to layered media. We illustrate the performance of the scheme with two and\u0000three dimensional examples.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"118 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181958","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 novel second order scheme with one step for forward backward stochastic differential equations 用于前向后向随机微分方程的新型一步二阶方案

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-11 DOI: arxiv-2409.07118

Qiang Han, Shihao Lan, Quanxin Zhu

引用次数: 0

$M$-QR decomposition and hyperpower iterative methods for computing outer inverses of tensors 计算张量外反的 $M$-QR 分解和超幂迭代法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-11 DOI: arxiv-2409.07007

Ratikanta Behera, Krushnachandra Panigrahy, Jajati Keshari Sahoo, Yimin Wei

引用次数: 0

Homogenisation for Maxwell and Friends 麦克斯韦和朋友们的同质化

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-11 DOI: arxiv-2409.07084

Andreas Buchinger, Sebastian Franz, Nathanael Skrepek, Marcus Waurick

{"title":"Homogenisation for Maxwell and Friends","authors":"Andreas Buchinger, Sebastian Franz, Nathanael Skrepek, Marcus Waurick","doi":"arxiv-2409.07084","DOIUrl":"https://doi.org/arxiv-2409.07084","url":null,"abstract":"We refine the understanding of continuous dependence on coefficients of\u0000solution operators under the nonlocal $H$-topology viz Schur topology in the\u0000setting of evolutionary equations in the sense of Picard. We show that certain\u0000components of the solution operators converge strongly. The weak convergence\u0000behaviour known from homogenisation problems for ordinary differential\u0000equations is recovered on the other solution operator components. The results\u0000are underpinned by a rich class of examples that, in turn, are also treated\u0000numerically, suggesting a certain sharpness of the theoretical findings.\u0000Analytic treatment of an example that proves this sharpness is provided too.\u0000Even though all the considered examples contain local coefficients, the main\u0000theorems and structural insights are of operator-theoretic nature and, thus,\u0000also applicable to nonlocal coefficients. The main advantage of the problem\u0000class considered is that they contain mixtures of type, potentially highly\u0000oscillating between different types of PDEs; a prototype can be found in\u0000Maxwell's equations highly oscillating between the classical equations and\u0000corresponding eddy current approximations.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181959","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 compatible finite element discretisation for moist shallow water equations 潮湿浅水方程的兼容有限元离散化

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-11 DOI: arxiv-2409.07182

Nell Hartney, Thomas M. Bendall, Jemma Shipton

{"title":"A compatible finite element discretisation for moist shallow water equations","authors":"Nell Hartney, Thomas M. Bendall, Jemma Shipton","doi":"arxiv-2409.07182","DOIUrl":"https://doi.org/arxiv-2409.07182","url":null,"abstract":"The moist shallow water equations offer a promising route for advancing\u0000understanding of the coupling of physical parametrisations and dynamics in\u0000numerical atmospheric models, an issue known as 'physics-dynamics coupling'.\u0000Without moist physics, the traditional shallow water equations are a simplified\u0000form of the atmospheric equations of motion and so are computationally cheap,\u0000but retain many relevant dynamical features of the atmosphere. Introducing\u0000physics into the shallow water model in the form of moisture provides a tool to\u0000experiment with numerical techniques for physics-dynamics coupling in a simple\u0000dynamical model. In this paper, we compare some of the different moist shallow\u0000water models by writing them in a general formulation. The general formulation\u0000encompasses three existing forms of the moist shallow water equations and also\u0000a fourth, previously unexplored formulation. The equations are coupled to a\u0000three-state moist physics scheme that interacts with the resolved flow through\u0000source terms and produces two-way physics-dynamics feedback. We present a new\u0000compatible finite element discretisation of the equations and apply it to the\u0000different formulations of the moist shallow water equations in three test\u0000cases. The results show that the models capture generation of cloud and rain\u0000and physics-dynamics interactions, and demonstrate some differences between\u0000moist shallow water formulations and the implications of these different\u0000modelling choices.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181930","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 Reynolds-semi-robust and pressure robust Hybrid High-Order method for the time dependent incompressible Navier--Stokes equations on general meshes 用于一般网格上时变不可压缩纳维-斯托克斯方程的雷诺稳态和压力稳健混合高阶方法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-11 DOI: arxiv-2409.07037

Daniel Castanon Quiroz, Daniele A. Di Pietro

引用次数: 0