arXiv - MATH - Numerical Analysis最新文献_第10页

Uniform Approximation of Eigenproblems of a Large-Scale Parameter-Dependent Hermitian Matrix 大规模参数相关赫米矩阵特征问题的统一逼近

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-09 DOI: arxiv-2409.05791

Mattia Manucci, Emre Mengi, Nicola Guglielmi

{"title":"Uniform Approximation of Eigenproblems of a Large-Scale Parameter-Dependent Hermitian Matrix","authors":"Mattia Manucci, Emre Mengi, Nicola Guglielmi","doi":"arxiv-2409.05791","DOIUrl":"https://doi.org/arxiv-2409.05791","url":null,"abstract":"We consider the approximation of the smallest eigenvalue of a large\u0000parameter-dependent Hermitian matrix over a continuum compact domain. Our\u0000approach is based on approximating the smallest eigenvalue by the one obtained\u0000by projecting the large matrix onto a suitable small subspace, a practice\u0000widely employed in the literature. The projection subspaces are constructed\u0000iteratively (to reduce the error of the approximation where it is large) with\u0000the addition of the eigenvectors of the parameter-dependent matrix at the\u0000parameter values where a surrogate error is maximal. The surrogate error is the\u0000gap between the approximation and a lower bound for the smallest eigenvalue\u0000proposed in [Sirkovic and Kressner, SIAM J. Matrix Anal. Appl., 37(2), 2016].\u0000Unlike the classical approaches, such as the successive constraint method, that\u0000maximize such surrogate errors over a discrete and finite set, we maximize the\u0000surrogate error over the continuum of all permissible parameter values\u0000globally. We put particular attention to the lower bound, which enables us to\u0000formally prove the global convergence of our framework both in\u0000finite-dimensional and infinite-dimensional settings. In the second part, we\u0000focus on the approximation of the smallest singular value of a large\u0000parameter-dependent matrix, in case it is non-Hermitian, and propose another\u0000subspace framework to construct a small parameter-dependent non-Hermitian\u0000matrix whose smallest singular value approximates the original large-scale\u0000smallest singular value. We perform numerical experiments on synthetic\u0000examples, as well as on real examples arising from parametric PDEs. The\u0000numerical experiments show that the proposed techniques are able to drastically\u0000reduce the size of the large parameter-dependent matrix, while ensuring an\u0000approximation error for the smallest eigenvalue/singular value below the\u0000prescribed tolerance.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181978","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

Two-level trust-region method with random subspaces 具有随机子空间的两级信任区域法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-09 DOI: arxiv-2409.05479

Andrea Angino, Alena Kopaničáková, Rolf Krause

引用次数: 0

Strong convergence of some Magnus-type schemes for the finite element discretization of non-autonomous parabolic SPDEs driven by additive fractional Brownian motion and Poisson random measure 某些马格努斯型方案在加性分数布朗运动和泊松随机量驱动的非自治抛物 SPDE 的有限元离散化中的强收敛性

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-09 DOI: arxiv-2409.06045

Aurelien Junior Noupelah, Jean Daniel Mukam, Antoine Tambue

{"title":"Strong convergence of some Magnus-type schemes for the finite element discretization of non-autonomous parabolic SPDEs driven by additive fractional Brownian motion and Poisson random measure","authors":"Aurelien Junior Noupelah, Jean Daniel Mukam, Antoine Tambue","doi":"arxiv-2409.06045","DOIUrl":"https://doi.org/arxiv-2409.06045","url":null,"abstract":"The aim of this work is to provide the strong convergence results of\u0000numerical approximations of a general second order non-autonomous semilinear\u0000stochastic partial differential equation (SPDE) driven simultaneously by an\u0000additive fractional Brownian motion (fBm) with Hurst parameter H in (1/2,1)\u0000and a Poisson random measure, more realistic in modelling real world phenomena. Approximations in space are performed by the standard finite element method\u0000and in time by the stochastic Magnus-type integrator or the linear\u0000semi-implicit Euler method. We investigate the mean-square errors estimates of\u0000our fully discrete schemes and the results show how the convergence orders\u0000depend on the regularity of the initial data and the driven processes. To the\u0000best of our knowledge, these two schemes are the first numerical methods to\u0000approximate the non-autonomous semilinear stochastic partial differential\u0000equation (SPDE) driven simultaneously by an additive fractional Brownian motion\u0000with Hurst parameter H and a Poisson random measure.","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":"142181970","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

Differential Inversion of the Implicit Euler Method: Symbolic Analysis 隐式欧拉法的微分反演：符号分析

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-09 DOI: arxiv-2409.05445

Uwe Naumann

引用次数: 0

A relaxation approach to the coupling of a two-phase fluid with a linear-elastic solid 两相流体与线弹性固体耦合的松弛方法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-09 DOI: arxiv-2409.05473

Niklas Kolbe, Siegfried Müller

引用次数: 0

Differentiable programming across the PDE and Machine Learning barrier 跨越 PDE 和机器学习障碍的可微编程

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-09 DOI: arxiv-2409.06085

Nacime Bouziani, David A. Ham, Ado Farsi

{"title":"Differentiable programming across the PDE and Machine Learning barrier","authors":"Nacime Bouziani, David A. Ham, Ado Farsi","doi":"arxiv-2409.06085","DOIUrl":"https://doi.org/arxiv-2409.06085","url":null,"abstract":"The combination of machine learning and physical laws has shown immense\u0000potential for solving scientific problems driven by partial differential\u0000equations (PDEs) with the promise of fast inference, zero-shot generalisation,\u0000and the ability to discover new physics. Examples include the use of\u0000fundamental physical laws as inductive bias to machine learning algorithms,\u0000also referred to as physics-driven machine learning, and the application of\u0000machine learning to represent features not represented in the differential\u0000equations such as closures for unresolved spatiotemporal scales. However, the\u0000simulation of complex physical systems by coupling advanced numerics for PDEs\u0000with state-of-the-art machine learning demands the composition of specialist\u0000PDE solving frameworks with industry-standard machine learning tools.\u0000Hand-rolling either the PDE solver or the neural net will not cut it. In this\u0000work, we introduce a generic differentiable programming abstraction that\u0000provides scientists and engineers with a highly productive way of specifying\u0000end-to-end differentiable models coupling machine learning and PDE-based\u0000components, while relying on code generation for high performance. Our\u0000interface automates the coupling of arbitrary PDE-based systems and machine\u0000learning models and unlocks new applications that could not hitherto be\u0000tackled, while only requiring trivial changes to existing code. Our framework\u0000has been adopted in the Firedrake finite-element library and supports the\u0000PyTorch and JAX ecosystems, as well as downstream libraries.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181973","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 robust fourth-order finite-difference discretization for the strongly anisotropic transport equation in magnetized plasmas 磁化等离子体中强各向异性输运方程的稳健四阶有限差分离散法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-09 DOI: arxiv-2409.06070

L. Chacon, Jason Hamilton, Natalia Krasheninnikova

{"title":"A robust fourth-order finite-difference discretization for the strongly anisotropic transport equation in magnetized plasmas","authors":"L. Chacon, Jason Hamilton, Natalia Krasheninnikova","doi":"arxiv-2409.06070","DOIUrl":"https://doi.org/arxiv-2409.06070","url":null,"abstract":"We propose a second-order temporally implicit, fourth-order-accurate spatial\u0000discretization scheme for the strongly anisotropic heat transport equation\u0000characteristic of hot, fusion-grade plasmas. Following [Du Toit et al., Comp.\u0000Phys. Comm., 228 (2018)], the scheme transforms mixed-derivative diffusion\u0000fluxes (which are responsible for the lack of a discrete maximum principle)\u0000into nonlinear advective fluxes, amenable to nonlinear-solver-friendly\u0000monotonicity-preserving limiters. The scheme enables accurate multi-dimensional\u0000heat transport simulations with up to seven orders of magnitude of\u0000heat-transport-coefficient anisotropies with low cross-field numerical error\u0000pollution and excellent algorithmic performance, with the number of linear\u0000iterations scaling very weakly with grid resolution and grid anisotropy, and\u0000scaling with the square-root of the implicit timestep. We propose a multigrid\u0000preconditioning strategy based on a second-order-accurate approximation that\u0000renders the scheme efficient and scalable under grid refinement. Several\u0000numerical tests are presented that display the expected spatial convergence\u0000rates and strong algorithmic performance, including fully nonlinear\u0000magnetohydrodynamics simulations of kink instabilities in a Bennett pinch in 2D\u0000helical geometry and of ITER in 3D toroidal geometry.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"98 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181974","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

DeepTV: A neural network approach for total variation minimization DeepTV：总变异最小化的神经网络方法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-09 DOI: arxiv-2409.05569

Andreas Langer, Sara Behnamian

{"title":"DeepTV: A neural network approach for total variation minimization","authors":"Andreas Langer, Sara Behnamian","doi":"arxiv-2409.05569","DOIUrl":"https://doi.org/arxiv-2409.05569","url":null,"abstract":"Neural network approaches have been demonstrated to work quite well to solve\u0000partial differential equations in practice. In this context approaches like\u0000physics-informed neural networks and the Deep Ritz method have become popular.\u0000In this paper, we propose a similar approach to solve an infinite-dimensional\u0000total variation minimization problem using neural networks. We illustrate that\u0000the resulting neural network problem does not have a solution in general. To\u0000circumvent this theoretic issue, we consider an auxiliary neural network\u0000problem, which indeed has a solution, and show that it converges in the sense\u0000of $Gamma$-convergence to the original problem. For computing a numerical\u0000solution we further propose a discrete version of the auxiliary neural network\u0000problem and again show its $Gamma$-convergence to the original\u0000infinite-dimensional problem. In particular, the $Gamma$-convergence proof\u0000suggests a particular discretization of the total variation. Moreover, we\u0000connect the discrete neural network problem to a finite difference\u0000discretization of the infinite-dimensional total variation minimization\u0000problem. Numerical experiments are presented supporting our theoretical\u0000findings.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181981","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

Some Results on Neural Network Stability, Consistency, and Convergence: Insights into Non-IID Data, High-Dimensional Settings, and Physics-Informed Neural Networks 关于神经网络稳定性、一致性和收敛性的一些结果：对非 IID 数据、高维设置和物理信息神经网络的启示

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-08 DOI: arxiv-2409.05030

Ronald Katende, Henry Kasumba, Godwin Kakuba, John M. Mango

引用次数: 0

Polynomial Methods for Ensuring Data Integrity in Financial Systems 确保金融系统数据完整性的多项式方法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-08 DOI: arxiv-2409.07490

Ignacio Brasca

引用次数: 0