arXiv - CS - Numerical Analysis最新文献_第5页

Gaussian process learning of nonlinear dynamics 非线性动力学的高斯过程学习

arXiv - CS - Numerical Analysis Pub Date : 2023-12-19 DOI: arxiv-2312.12193

Dongwei Ye, Mengwu Guo

{"title":"Gaussian process learning of nonlinear dynamics","authors":"Dongwei Ye, Mengwu Guo","doi":"arxiv-2312.12193","DOIUrl":"https://doi.org/arxiv-2312.12193","url":null,"abstract":"One of the pivotal tasks in scientific machine learning is to represent\u0000underlying dynamical systems from time series data. Many methods for such\u0000dynamics learning explicitly require the derivatives of state data, which are\u0000not directly available and can be approximated conventionally by finite\u0000differences. However, the discrete approximations of time derivatives may\u0000result in a poor estimation when state data are scarce and/or corrupted by\u0000noise, thus compromising the predictiveness of the learned dynamical models. To\u0000overcome this technical hurdle, we propose a new method that learns nonlinear\u0000dynamics through a Bayesian inference of characterizing model parameters. This\u0000method leverages a Gaussian process representation of states, and constructs a\u0000likelihood function using the correlation between state data and their\u0000derivatives, yet prevents explicit evaluations of time derivatives. Through a\u0000Bayesian scheme, a probabilistic estimate of the model parameters is given by\u0000the posterior distribution, and thus a quantification is facilitated for\u0000uncertainties from noisy state data and the learning process. Specifically, we\u0000will discuss the applicability of the proposed method to two typical scenarios\u0000for dynamical systems: parameter identification and estimation with an affine\u0000structure of the system, and nonlinear parametric approximation without prior\u0000knowledge.","PeriodicalId":501061,"journal":{"name":"arXiv - CS - Numerical Analysis","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138818017","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

Plane Wave Discontinuous Galerkin methods for scattering by periodic structures 周期性结构散射的平面波非连续伽勒金方法

arXiv - CS - Numerical Analysis Pub Date : 2023-12-19 DOI: arxiv-2312.12045

Armando Maria Monforte

{"title":"Plane Wave Discontinuous Galerkin methods for scattering by periodic structures","authors":"Armando Maria Monforte","doi":"arxiv-2312.12045","DOIUrl":"https://doi.org/arxiv-2312.12045","url":null,"abstract":"This thesis explores the application of Plane Wave Discontinuous Galerkin\u0000(PWDG) methods for the numerical simulation of electromagnetic scattering by\u0000periodic structures. Periodic structures play a pivotal role in various\u0000engineering and scientific applications, including antenna design, metamaterial\u0000characterization, and photonic crystal analysis. Understanding and accurately\u0000predicting the scattering behavior of electromagnetic waves from such\u0000structures is crucial in optimizing their performance and advancing\u0000technological advancements. The thesis commences with an overview of the theoretical foundations of\u0000electromagnetic scattering by periodic structures. This theoretical\u0000dissertation serves as the basis for formulating the PWDG method within the\u0000context of wave equation. The DtN operator is presented and it is used to\u0000derive a suitable boundary condition. The numerical implementation of PWDG methods is discussed in detail,\u0000emphasizing key aspects such as basis function selection and boundary\u0000conditions. The algorithm's efficiency is assessed through numerical\u0000experiments. We then present the DtN-PWDG method, which is discussed in detail and is used\u0000to derive numerical solutions of the scattering problem. A comparison with the\u0000finite element method (FEM) is presented. In conclusion, this thesis demonstrates that PWDG methods are a powerful tool\u0000for simulating electromagnetic scattering by periodic structures.","PeriodicalId":501061,"journal":{"name":"arXiv - CS - Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138818636","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 exact divergence-free spectral method for incompressible and resistive magneto-hydrodynamic equations in two and three dimensions 二维和三维不可压缩和电阻磁流体动力学方程的精确无发散谱法

arXiv - CS - Numerical Analysis Pub Date : 2023-12-19 DOI: arxiv-2312.12218

Lechang Qin, Huiyuan Li, Zhiguo Yang

{"title":"An exact divergence-free spectral method for incompressible and resistive magneto-hydrodynamic equations in two and three dimensions","authors":"Lechang Qin, Huiyuan Li, Zhiguo Yang","doi":"arxiv-2312.12218","DOIUrl":"https://doi.org/arxiv-2312.12218","url":null,"abstract":"In this paper, we present exact divergence-free spectral method for solving\u0000the incompressible and resistive magneto-hydrodynamic (MHD) equations in two\u0000and three dimensions, as well as the efficient solution algorithm and\u0000unconditionally energy-stable fully-discretized numerical schemes. We introduce\u0000new ideas of constructing two families of exact divergence-free vectorial\u0000spectral basis functions on domains diffeomorphic to squares or cubes. These\u0000bases are obtained with the help of orthogonality and derivative relation of\u0000generalised Jacobi polynomials, several de Rham complexes, as well as the\u0000property of contravariant Piola transformation. They are well-suited for\u0000discretizing the velocity and magnetic fields, respectively, thereby ensuring\u0000point-wise preservation of the incompressibility condition and the magnetic\u0000Gauss's law. With the aid of these bases, we propose a family of exact\u0000divergence-free implicit-explicit $k$-step backward differentiation formula\u0000(DF-BDF-$k$) fully-discretized schemes for the MHD system. These schemes\u0000naturally decouple the pressure field from the velocity field. Consequently,\u0000the stability of the space-time fully-discretized numerical schemes based on\u0000these bases are significantly enhanced. These schemes exhibit unconditional\u0000stability for $k=1,2$, and demonstrate exceptional stability and accuracy for\u0000$k=3,4$, verified with extensive numerical results for long time simulations\u0000using large time step sizes. Furthermore, we present efficient solution\u0000algorithms for these two decoupled equations for the velocity and magnetic\u0000fields, respectively, by exploiting the sparsity and structure of the resultant\u0000linear algebraic systems. Ample numerical examples in two and three dimensions\u0000are provided to demonstrate the distinctive accuracy, efficiency and stability\u0000of our proposed method.","PeriodicalId":501061,"journal":{"name":"arXiv - CS - Numerical Analysis","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138818588","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

Revisiting Diffusive Representations for Enhanced Numerical Approximation of Fractional Integrals 重新审视用于增强分数积分数值逼近的扩散表示法

arXiv - CS - Numerical Analysis Pub Date : 2023-12-18 DOI: arxiv-2312.11590

Renu Chaudhary, Kai Diethelm

引用次数: 0

An abstract framework for heterogeneous coupling: stability, approximation and applications 异质耦合的抽象框架：稳定性、近似性和应用

arXiv - CS - Numerical Analysis Pub Date : 2023-12-18 DOI: arxiv-2312.11733

Silvia Bertoluzza, Erik Burman

引用次数: 0

Learned Regularization for Inverse Problems: Insights from a Spectral Model 逆问题的学习正则化：光谱模型的启示

arXiv - CS - Numerical Analysis Pub Date : 2023-12-15 DOI: arxiv-2312.09845

Martin Burger, Samira Kabri

引用次数: 0

New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties 具有不确定性的非线性 PDE 双曲系统的新型高阶数值方法

arXiv - CS - Numerical Analysis Pub Date : 2023-12-13 DOI: arxiv-2312.08280

Alina Chertock, Michael Herty, Arsen S. Iskhakov, Safa Janajra, Alexander Kurganov, Maria Lukacova-Medvidova

引用次数: 0

A hybrid finite element method for moving-habitat models in two spatial dimensions 二维移动栖息地模型的混合有限元方法

arXiv - CS - Numerical Analysis Pub Date : 2023-12-13 DOI: arxiv-2312.07842

Jane Shaw MacDonald, Yves Bourgault, Frithjof Lutscher

{"title":"A hybrid finite element method for moving-habitat models in two spatial dimensions","authors":"Jane Shaw MacDonald, Yves Bourgault, Frithjof Lutscher","doi":"arxiv-2312.07842","DOIUrl":"https://doi.org/arxiv-2312.07842","url":null,"abstract":"Moving-habitat models track the density of a population whose suitable\u0000habitat shifts as a consequence of climate change. Whereas most previous\u0000studies in this area consider 1-dimensional space, we derive and study a\u0000spatially 2-dimensional moving-habitat model via reaction-diffusion equations.\u0000The population inhabits the whole space. The suitable habitat is a bounded\u0000region where population growth is positive; the unbounded complement of its\u0000closure is unsuitable with negative growth. The interface between the two\u0000habitat types moves, depicting the movement of the suitable habitat poleward.\u0000Detailed modelling of individual movement behaviour induces a nonstandard\u0000discontinuity in the density across the interface. For the corresponding\u0000semi-discretised system we prove well-posedness for a constant shifting\u0000velocity before constructing an implicit-explicit hybrid finite element method.\u0000In this method, a Lagrange multiplier weakly imposes the jump discontinuity\u0000across the interface. For a stationary interface, we derive optimal a priori\u0000error estimates over a conformal mesh with nonconformal discretisation. We\u0000demonstrate with numerical convergence tests that these results hold for the\u0000moving interface. Finally, we demonstrate the strength of our hybrid finite\u0000element method with two biologically motivated cases, one for a domain with a\u0000curved boundary and the other for non-constant shifting velocity.","PeriodicalId":501061,"journal":{"name":"arXiv - CS - Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138630640","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 a posteriori error estimate for a 0D/2D coupled model 0D/2D 耦合模型的后验误差估计

arXiv - CS - Numerical Analysis Pub Date : 2023-12-13 DOI: arxiv-2312.07959

Hussein Albazzal, Alexei Lozinski, Roberta Tittarelli

{"title":"An a posteriori error estimate for a 0D/2D coupled model","authors":"Hussein Albazzal, Alexei Lozinski, Roberta Tittarelli","doi":"arxiv-2312.07959","DOIUrl":"https://doi.org/arxiv-2312.07959","url":null,"abstract":"This work is motivated by the need of efficient numerical simulations of gas\u0000flows in the serpentine channels used in proton-exchange membrane fuel cells.\u0000In particular, we consider the Poisson problem in a 2D domain composed of\u0000several long straight rectangular sections and of several bends corners. In\u0000order to speed up the resolution, we propose a 0D model in the rectangular\u0000parts of the channel and a Finite Element resolution in the bends. To find a\u0000good compromise between precision and time consuming, the challenge is double:\u0000how to choose a suitable position of the interface between the 0D and the 2D\u0000models and how to control the discretization error in the bends. We shall\u0000present an textit{a posteriori} error estimator based on an equilibrated flux\u0000reconstruction in the subdomains where the Finite Element method is applied.\u0000The estimates give a global upper bound on the error measured in the energy\u0000norm of the difference between the exact and approximate solutions on the whole\u0000domain. They are guaranteed, meaning that they feature no undetermined\u0000constants. (global) Lower bounds for the error are also derived. An adaptive\u0000algorithm is proposed to use smartly the estimator for aforementioned double\u0000challenge. A numerical validation of the estimator and the algorithm completes\u0000the work. end{abstract}","PeriodicalId":501061,"journal":{"name":"arXiv - CS - Numerical Analysis","volume":"260 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138630642","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 reconstruction of functions with rough edges from discrete Radon data in $mathbb R^2$ 从 $mathbb R^2$ 中的离散拉顿数据重建具有粗糙边缘的函数的分析

arXiv - CS - Numerical Analysis Pub Date : 2023-12-13 DOI: arxiv-2312.08259

Alexander Katsevich

引用次数: 0