Advances in Computational Mathematics最新文献_第5页

SVD-based algorithms for tensor wheel decomposition 基于 SVD 的张量轮分解算法

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-09-05 DOI: 10.1007/s10444-024-10194-9

Mengyu Wang, Honghua Cui, Hanyu Li

引用次数: 0

Eigenvalue analysis and applications of the Legendre dual-Petrov-Galerkin methods for initial value problems 初值问题的特征值分析和 Legendre 双-Petrov-Galerkin 方法的应用

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-09-02 DOI: 10.1007/s10444-024-10190-z

Desong Kong, Jie Shen, Li-Lian Wang, Shuhuang Xiang

{"title":"Eigenvalue analysis and applications of the Legendre dual-Petrov-Galerkin methods for initial value problems","authors":"Desong Kong, Jie Shen, Li-Lian Wang, Shuhuang Xiang","doi":"10.1007/s10444-024-10190-z","DOIUrl":"10.1007/s10444-024-10190-z","url":null,"abstract":"<div>In this paper, we show that the eigenvalues and eigenvectors of the spectral discretisation matrices resulting from the Legendre dual-Petrov-Galerkin (LDPG) method for the mth-order initial value problem (IVP): (u^{(m)}(t)=sigma u(t),, tin (-1,1)) with constant (sigma not =0) and usual initial conditions at t(=-1,) are associated with the generalised Bessel polynomials (GBPs). In particular, we derive analytical formulae for the eigenvalues and eigenvectors in the cases m(=1,2). As a by-product, we are able to answer some open questions related to the collocation method at Legendre points (extensively studied in the 1980s) for the first-order IVP, by reformulating it into a Petrov-Galerkin formulation. Our results have direct bearing on the CFL conditions of time-stepping schemes with spectral or spectral-element discretisation in space. Moreover, we present two stable algorithms for computing zeros of the GBPs and develop a general space-time method for evolutionary PDEs. We provide ample numerical results to demonstrate the high accuracy and robustness of the space-time methods for some interesting examples of linear and nonlinear wave problems.</div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 5","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142123679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On the latent dimension of deep autoencoders for reduced order modeling of PDEs parametrized by random fields 论深度自动编码器的潜在维度，用于对随机场参数化的 PDE 进行降阶建模

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-08-28 DOI: 10.1007/s10444-024-10189-6

Nicola Rares Franco, Daniel Fraulin, Andrea Manzoni, Paolo Zunino

{"title":"On the latent dimension of deep autoencoders for reduced order modeling of PDEs parametrized by random fields","authors":"Nicola Rares Franco, Daniel Fraulin, Andrea Manzoni, Paolo Zunino","doi":"10.1007/s10444-024-10189-6","DOIUrl":"10.1007/s10444-024-10189-6","url":null,"abstract":"<div>Deep Learning is having a remarkable impact on the design of Reduced Order Models (ROMs) for Partial Differential Equations (PDEs), where it is exploited as a powerful tool for tackling complex problems for which classical methods might fail. In this respect, deep autoencoders play a fundamental role, as they provide an extremely flexible tool for reducing the dimensionality of a given problem by leveraging on the nonlinear capabilities of neural networks. Indeed, starting from this paradigm, several successful approaches have already been developed, which are here referred to as Deep Learning-based ROMs (DL-ROMs). Nevertheless, when it comes to stochastic problems parameterized by random fields, the current understanding of DL-ROMs is mostly based on empirical evidence: in fact, their theoretical analysis is currently limited to the case of PDEs depending on a finite number of (deterministic) parameters. The purpose of this work is to extend the existing literature by providing some theoretical insights about the use of DL-ROMs in the presence of stochasticity generated by random fields. In particular, we derive explicit error bounds that can guide domain practitioners when choosing the latent dimension of deep autoencoders. We evaluate the practical usefulness of our theory by means of numerical experiments, showing how our analysis can significantly impact the performance of DL-ROMs.</div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 5","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10189-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142090103","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

Families of annihilating skew-selfadjoint operators and their connection to Hilbert complexes 湮灭偏自交算子族及其与希尔伯特复数的联系

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-08-27 DOI: 10.1007/s10444-024-10184-x

Dirk Pauly, Rainer Picard

引用次数: 0

Computing eigenvalues of quasi-rational Said–Ball–Vandermonde matrices 计算准有理 Said-Ball-Vandermonde 矩阵的特征值

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-08-22 DOI: 10.1007/s10444-024-10191-y

Xiaoxiao Ma, Yingqing Xiao

引用次数: 0

Morley type virtual element method for von Kármán equations 用于 von Kármán 方程的莫雷型虚拟元素法

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-08-22 DOI: 10.1007/s10444-024-10158-z

Devika Shylaja, Sarvesh Kumar

引用次数: 0

Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems 用于涡流问题等距分析的同调发生器的任意阶样条表示法

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-08-19 DOI: 10.1007/s10444-024-10181-0

Bernard Kapidani, Melina Merkel, Sebastian Schöps, Rafael Vázquez

引用次数: 0

Convergence analysis of the Dirichlet-Neumann Waveform Relaxation algorithm for time fractional sub-diffusion and diffusion-wave equations in heterogeneous media 异质介质中时间分数次扩散和扩散波方程的迪里夏-诺伊曼波形松弛算法收敛性分析

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-08-14 DOI: 10.1007/s10444-024-10185-w

Soura Sana, Bankim C Mandal

引用次数: 0

Frame-normalizable sequences 帧正则序列

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-08-09 DOI: 10.1007/s10444-024-10182-z

Pu-Ting Yu

{"title":"Frame-normalizable sequences","authors":"Pu-Ting Yu","doi":"10.1007/s10444-024-10182-z","DOIUrl":"10.1007/s10444-024-10182-z","url":null,"abstract":"<div>Let H be a separable Hilbert space and let ({x_{n}}) be a sequence in H that does not contain any zero elements. We say that ({x_{n}}) is a Bessel-normalizable or frame-normalizable sequence if the normalized sequence ({bigl {frac{x_n}{Vert x_nVert }bigr }}) is a Bessel sequence or a frame for H, respectively. In this paper, several necessary and sufficient conditions for sequences to be frame-normalizable and not frame-normalizable are proved. Perturbation theorems for frame-normalizable sequences are also proved. As applications, we show that the Balazs–Stoeva conjecture holds for Bessel-normalizable sequences. Finally, we apply our results to partially answer the open question raised by Aldroubi et al. as to whether the iterative system (bigl {frac{A^{n} x}{Vert A^{n}xVert }bigr }_{nge 0,, xin S}) associated with a normal operator (A:Hrightarrow H) and a countable subset S of H, is a frame for H. In particular, if S is finite, then we are able to show that (bigl {frac{A^{n} x}{Vert A^{n}xVert }bigr }_{nge 0,, xin S}) is not a frame for H whenever ({A^{n}x}_{nge 0,,xin S}) is a frame for H.</div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 4","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141908893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

SlabLU: a two-level sparse direct solver for elliptic PDEs SlabLU：椭圆 PDE 的两级稀疏直接求解器

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-08-09 DOI: 10.1007/s10444-024-10176-x

Anna Yesypenko, Per-Gunnar Martinsson

{"title":"SlabLU: a two-level sparse direct solver for elliptic PDEs","authors":"Anna Yesypenko, Per-Gunnar Martinsson","doi":"10.1007/s10444-024-10176-x","DOIUrl":"10.1007/s10444-024-10176-x","url":null,"abstract":"<div>The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two-dimensional domain. The scheme decomposes the domain into thin subdomains, or “slabs” and uses a two-level approach that is designed with parallelization in mind. The scheme takes advantage of (varvec{mathcal {H}}^textbf{2})-matrix structure emerging during factorization and utilizes randomized algorithms to efficiently recover this structure. As opposed to multi-level nested dissection schemes that incorporate the use of (varvec{mathcal {H}}) or (varvec{mathcal {H}}^textbf{2}) matrices for a hierarchy of front sizes, SlabLU is a two-level scheme which only uses (varvec{mathcal {H}}^textbf{2})-matrix algebra for fronts of roughly the same size. The simplicity allows the scheme to be easily tuned for performance on modern architectures and GPUs. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its performance for regular discretizations of rectangular and curved geometries. The technique becomes particularly efficient when combined with very high-order accurate multidomain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size (textbf{1000} varvec{lambda } times textbf{1000} varvec{lambda }) (for which (varvec{N}~mathbf {=100} textbf{M})) is solved in 15 min to 6 correct digits on a high-powered desktop with GPU acceleration.</div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 4","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141909211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0