Advances in Computational Mathematics最新文献

Maximal volume matrix cross approximation for image compression and least squares solution 用于图像压缩和最小二乘法求解的最大体积矩阵交叉近似法

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-09-16 DOI: 10.1007/s10444-024-10196-7

Kenneth Allen, Ming-Jun Lai, Zhaiming Shen

引用次数: 0

Multilevel approximation of Gaussian random fields: Covariance compression, estimation, and spatial prediction 高斯随机场的多级近似：协方差压缩、估计和空间预测

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-09-14 DOI: 10.1007/s10444-024-10187-8

Helmut Harbrecht, Lukas Herrmann, Kristin Kirchner, Christoph Schwab

{"title":"Multilevel approximation of Gaussian random fields: Covariance compression, estimation, and spatial prediction","authors":"Helmut Harbrecht, Lukas Herrmann, Kristin Kirchner, Christoph Schwab","doi":"10.1007/s10444-024-10187-8","DOIUrl":"https://doi.org/10.1007/s10444-024-10187-8","url":null,"abstract":"The distribution of centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded Euclidean domains or smooth, compact and orientable manifolds is determined by their covariance operators. We consider centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with an elliptic self-adjoint pseudodifferential coloring operator from the Hörmander class. This includes the Matérn class of GRFs as a special case. Using biorthogonal multiresolution analyses on the manifold, we prove that the precision and covariance operators, respectively, may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension p of this section. We prove that a tapering strategy by thresholding applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations. That is, asymptotically only linearly many nonzero matrix entries are sufficient to approximate the original section of the bi-infinite covariance or precision matrix using this tapering strategy to arbitrary precision. The locations of these nonzero matrix entries can be determined a priori. The tapered covariance or precision matrices may also be optimally diagonally preconditioned. Analysis of the relative size of the entries of the tapered covariance matrices motivates novel, multilevel Monte Carlo (MLMC) oracles for covariance estimation, in sample complexity that scales log-linearly with respect to the number p of parameters. In addition, we propose and analyze novel compressive algorithms for simulating and kriging of GRFs. The complexity (work and memory vs. accuracy) of these three algorithms scales near-optimally in terms of the number of parameters p of the sample-wise approximation of the GRF in Sobolev scales.","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142231551","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

Improved a posteriori error bounds for reduced port-Hamiltonian systems 改进的还原端口-哈密尔顿系统后验误差边界

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-09-11 DOI: 10.1007/s10444-024-10195-8

Johannes Rettberg, Dominik Wittwar, Patrick Buchfink, Robin Herkert, Jörg Fehr, Bernard Haasdonk

引用次数: 0

Interpolating refinable functions and $$n_s$$ -step interpolatory subdivision schemes 可细化函数内插和 $$n_s$$ 步内插细分方案

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-09-05 DOI: 10.1007/s10444-024-10192-x

Bin Han

{"title":"Interpolating refinable functions and $$n_s$$ -step interpolatory subdivision schemes","authors":"Bin Han","doi":"10.1007/s10444-024-10192-x","DOIUrl":"https://doi.org/10.1007/s10444-024-10192-x","url":null,"abstract":"Standard interpolatory subdivision schemes and their underlying interpolating refinable functions are of interest in CAGD, numerical PDEs, and approximation theory. Generalizing these notions, we introduce and study (n_s)-step interpolatory (textsf{M})-subdivision schemes and their interpolating (textsf{M})-refinable functions with (n_sin mathbb {N}cup {infty }) and a dilation factor (textsf{M}in mathbb {N}backslash {1}). We completely characterize (mathscr {C}^m)-convergence and smoothness of (n_s)-step interpolatory subdivision schemes and their interpolating (textsf{M})-refinable functions in terms of their masks. Inspired by (n_s)-step interpolatory stationary subdivision schemes, we further introduce the notion of r-mask quasi-stationary subdivision schemes, and then we characterize their (mathscr {C}^m)-convergence and smoothness properties using only their masks. Moreover, combining (n_s)-step interpolatory subdivision schemes with r-mask quasi-stationary subdivision schemes, we can obtain (r n_s)-step interpolatory subdivision schemes. Examples and construction procedures of convergent (n_s)-step interpolatory (textsf{M})-subdivision schemes are provided to illustrate our results with dilation factors (textsf{M}=2,3,4). In addition, for the dyadic dilation (textsf{M}=2) and (r=2,3), using r masks with only two-ring stencils, we provide examples of (mathscr {C}^r)-convergent r-step interpolatory r-mask quasi-stationary dyadic subdivision schemes.","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142138151","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

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":"https://doi.org/10.1007/s10444-024-10190-z","url":null,"abstract":"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.","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"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":"https://doi.org/10.1007/s10444-024-10189-6","url":null,"abstract":"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.","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","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