Advances in Computational Mathematics最新文献_第9页

Error estimates for POD-DL-ROMs: a deep learning framework for reduced order modeling of nonlinear parametrized PDEs enhanced by proper orthogonal decomposition POD-DL-ROM 的误差估计：通过适当正交分解增强的非线性参数化 PDE 减阶建模深度学习框架

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-04-24 DOI: 10.1007/s10444-024-10110-1

Simone Brivio, Stefania Fresca, Nicola Rares Franco, Andrea Manzoni

{"title":"Error estimates for POD-DL-ROMs: a deep learning framework for reduced order modeling of nonlinear parametrized PDEs enhanced by proper orthogonal decomposition","authors":"Simone Brivio, Stefania Fresca, Nicola Rares Franco, Andrea Manzoni","doi":"10.1007/s10444-024-10110-1","DOIUrl":"10.1007/s10444-024-10110-1","url":null,"abstract":"<div>POD-DL-ROMs have been recently proposed as an extremely versatile strategy to build accurate and reliable reduced order models (ROMs) for nonlinear parametrized partial differential equations, combining (i) a preliminary dimensionality reduction obtained through proper orthogonal decomposition (POD) for the sake of efficiency, (ii) an autoencoder architecture that further reduces the dimensionality of the POD space to a handful of latent coordinates, and (iii) a dense neural network to learn the map that describes the dynamics of the latent coordinates as a function of the input parameters and the time variable. Within this work, we aim at justifying the outstanding approximation capabilities of POD-DL-ROMs by means of a thorough error analysis, showing how the sampling required to generate training data, the dimension of the POD space, and the complexity of the underlying neural networks, impact on the solutions us to formulate practical criteria to control the relative error in the approximation of the solution field of interest, and derive general error estimates. Furthermore, we show that, from a theoretical point of view, POD-DL-ROMs outperform several deep learning-based techniques in terms of model complexity. Finally, we validate our findings by means of suitable numerical experiments, ranging from parameter-dependent operators analytically defined to several parametrized PDEs.</div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10110-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140642700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Artificial neural networks with uniform norm-based loss functions 基于统一规范损失函数的人工神经网络

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-04-23 DOI: 10.1007/s10444-024-10124-9

Vinesha Peiris, Vera Roshchina, Nadezda Sukhorukova

引用次数: 0

The improvement of the truncated Euler-Maruyama method for non-Lipschitz stochastic differential equations 非 Lipschitz 随机微分方程的截断 Euler-Maruyama 方法的改进

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-04-22 DOI: 10.1007/s10444-024-10131-w

Weijun Zhan, Yuyuan Li

引用次数: 0

The adjoint double layer potential on smooth surfaces in (mathbb {R}^3) and the Neumann problem $$mathbb{R}^3$$中光滑表面上的邻接双层势和诺伊曼问题

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-04-19 DOI: 10.1007/s10444-024-10111-0

J. Thomas Beale, Michael Storm, Svetlana Tlupova

{"title":"The adjoint double layer potential on smooth surfaces in (mathbb {R}^3) and the Neumann problem","authors":"J. Thomas Beale, Michael Storm, Svetlana Tlupova","doi":"10.1007/s10444-024-10111-0","DOIUrl":"10.1007/s10444-024-10111-0","url":null,"abstract":"<div>We present a simple yet accurate method to compute the adjoint double layer potential, which is used to solve the Neumann boundary value problem for Laplace’s equation in three dimensions. An expansion in curvilinear coordinates leads us to modify the expression for the adjoint double layer so that the singularity is reduced when evaluating the integral on the surface. Then, to regularize the integral, we multiply the Green’s function by a radial function with length parameter (delta ) chosen so that the product is smooth. We show that a natural regularization has error (O(delta ^3)), and a simple modification improves the error to (O(delta ^5)). The integral is evaluated numerically without the need of special coordinates. We use this treatment of the adjoint double layer to solve the classical integral equation for the interior Neumann problem, altered to account for the solvability condition, and evaluate the solution on the boundary. Choosing (delta = ch^{4/5}), we find about (O(h^4)) convergence in our examples, where h is the spacing in a background grid.</div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10111-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140620315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

(mathcal {H}_2) optimal rational approximation on general domains 一般域上的 $$mathcal {H}_2$$ 最佳理性逼近

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-04-18 DOI: 10.1007/s10444-024-10125-8

Alessandro Borghi, Tobias Breiten

{"title":"(mathcal {H}_2) optimal rational approximation on general domains","authors":"Alessandro Borghi, Tobias Breiten","doi":"10.1007/s10444-024-10125-8","DOIUrl":"10.1007/s10444-024-10125-8","url":null,"abstract":"<div>Optimal model reduction for large-scale linear dynamical systems is studied. In contrast to most existing works, the systems under consideration are not required to be stable, neither in discrete nor in continuous time. As a consequence, the underlying rational transfer functions are allowed to have poles in general domains in the complex plane. In particular, this covers the case of specific conservative partial differential equations such as the linear Schrödinger and the undamped linear wave equation with spectra on the imaginary axis. By an appropriate modification of the classical continuous time Hardy space (varvec{mathcal {H}}_{varvec{2}}), a new (varvec{mathcal {H}}_{varvec{2}})-like optimal model reduction problem is introduced and first-order optimality conditions are derived. As in the classical (varvec{mathcal {H}}_{varvec{2}}) case, these conditions exhibit a rational Hermite interpolation structure for which an iterative model reduction algorithm is proposed. Numerical examples demonstrate the effectiveness of the new method.</div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10125-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140607911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Structured barycentric forms for interpolation-based data-driven reduced modeling of second-order systems 基于插值的数据驱动二阶系统还原建模的结构化巴里心形式

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-04-11 DOI: 10.1007/s10444-024-10118-7

Ion Victor Gosea, Serkan Gugercin, Steffen W. R. Werner

引用次数: 0

Computing equivariant matrices on homogeneous spaces for geometric deep learning and automorphic Lie algebras 计算同质空间上的等变矩阵，用于几何深度学习和非定态李代数

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-04-11 DOI: 10.1007/s10444-024-10126-7

Vincent Knibbeler

引用次数: 0

Numerical simulation of resistance furnaces by using distributed and lumped models 利用分布式和集合式模型对电阻炉进行数值模拟

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-04-10 DOI: 10.1007/s10444-024-10120-z

A. Bermúdez, D. Gómez, D. González

引用次数: 0

Robust space-time finite element methods for parabolic distributed optimal control problems with energy regularization 带能量正则化的抛物分布式最优控制问题的鲁棒时空有限元方法

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-04-10 DOI: 10.1007/s10444-024-10123-w

Ulrich Langer, Olaf Steinbach, Huidong Yang

{"title":"Robust space-time finite element methods for parabolic distributed optimal control problems with energy regularization","authors":"Ulrich Langer, Olaf Steinbach, Huidong Yang","doi":"10.1007/s10444-024-10123-w","DOIUrl":"10.1007/s10444-024-10123-w","url":null,"abstract":"<div>As in our previous work (SINUM 59(2):660–674, 2021) we consider space-time tracking optimal control problems for linear parabolic initial boundary value problems that are given in the space-time cylinder (Q = Omega times (0,T)), and that are controlled by the right-hand side (z_varrho ) from the Bochner space (L^2(0,T;H^{-1}(Omega ))). So it is natural to replace the usual (L^2(Q)) norm regularization by the energy regularization in the (L^2(0,T;H^{-1}(Omega ))) norm. We derive new a priori estimates for the error (Vert widetilde{u}_{varrho h} - overline{u}Vert _{L^2(Q)}) between the computed state (widetilde{u}_{varrho h}) and the desired state (overline{u}) in terms of the regularization parameter (varrho ) and the space-time finite element mesh size h, and depending on the regularity of the desired state (overline{u}). These new estimates lead to the optimal choice (varrho = h^2). The approximate state (widetilde{u}_{varrho h}) is computed by means of a space-time finite element method using piecewise linear and continuous basis functions on completely unstructured simplicial meshes for Q. The theoretical results are quantitatively illustrated by a series of numerical examples in two and three space dimensions. We also provide performance studies for different solvers.</div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 2","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10123-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140541246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Robust low tubal rank tensor recovery using discrete empirical interpolation method with optimized slice/feature selection 利用优化切片/特征选择的离散经验插值法进行稳健的低管阶张量恢复

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-04-06 DOI: 10.1007/s10444-024-10117-8

Salman Ahmadi-Asl, Anh-Huy Phan, Cesar F. Caiafa, Andrzej Cichocki

引用次数: 0