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On Rayleigh quotients and characteristic equations for symmetric matrices connected with coordinate updatings 关于坐标更新的对称矩阵的瑞利商和特征方程
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-09-22 DOI: 10.1007/s10444-026-10356-x
Zhong-Zhi Bai
{"title":"On Rayleigh quotients and characteristic equations for symmetric matrices connected with coordinate updatings","authors":"Zhong-Zhi Bai","doi":"10.1007/s10444-026-10356-x","DOIUrl":"10.1007/s10444-026-10356-x","url":null,"abstract":"<div><p>For the standard symmetric eigenvalue problem, with respect to the coordinate descent/updating method we reveal an intrinsic connection between the Rayleigh quotient and the characteristic equation, and show that the coordinate descent method and the coordinate updating method actually share the same structure and formulation. We then further formulate a general framework for these two iteration methods.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149021110","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
Crank-Nicolson-type iterative decoupled algorithms for Biot’s consolidation model using total pressure 总压力下Biot固结模型的crank - nicolson型迭代解耦算法
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-09-18 DOI: 10.1007/s10444-026-10361-0
Huipeng Gu, Mingchao Cai, Jingzhi Li
{"title":"Crank-Nicolson-type iterative decoupled algorithms for Biot’s consolidation model using total pressure","authors":"Huipeng Gu,&nbsp;Mingchao Cai,&nbsp;Jingzhi Li","doi":"10.1007/s10444-026-10361-0","DOIUrl":"10.1007/s10444-026-10361-0","url":null,"abstract":"<div><p>In this work, we propose and analyze Crank-Nicolson-type iterative decoupled algorithms for a three-field formulation of Biot’s consolidation model using total pressure. These algorithms are derived from an equivalent, reformulated fully coupled system based on the Crank-Nicolson method, which is then decomposed using an iterative decoupled strategy. Two variants are introduced, differing in the solving order of temporal computation and iteration: a time-stepping and a global-in-time approach. The latter is particularly notable for its potential for parallel-in-time computing, offering an efficient approach for long-time simulations. Capitalizing on the properties of the underlying methods, both algorithms are proven to achieve second-order accuracy in time and unconditional stability. Through numerical experiments, we validate theoretical predictions and demonstrate the effectiveness and efficiency of these novel approaches.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-026-10361-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148987137","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, randomized preconditioning for kernel ridge regression 核岭回归的鲁棒随机预处理
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-09-18 DOI: 10.1007/s10444-026-10360-1
Mateo Díaz, Ethan N. Epperly, Zachary Frangella, Joel A. Tropp, Robert J. Webber
{"title":"Robust, randomized preconditioning for kernel ridge regression","authors":"Mateo Díaz,&nbsp;Ethan N. Epperly,&nbsp;Zachary Frangella,&nbsp;Joel A. Tropp,&nbsp;Robert J. Webber","doi":"10.1007/s10444-026-10360-1","DOIUrl":"10.1007/s10444-026-10360-1","url":null,"abstract":"<div><p>We investigate preconditioned conjugate gradient methods for kernel ridge regression (KRR) problems with a moderate to large number of data points (<span>(10^4 le N le 10^7)</span>). We develop and analyze two randomized preconditioners with complementary guarantees. For full-data KRR, <span>RPCholesky</span> preconditioning requires <span>(mathcal {O}(N^2))</span> arithmetic operations for fixed accuracy under sufficiently rapid eigenvalue decay of the kernel matrix. For restricted KRR with <span>(kll N)</span> centers, <span>KRILL</span> preconditioning requires <span>(mathcal {O}((N+k^2)klog k))</span> operations with no eigenvalue-decay assumption. Experiments on benchmark and scientific data sets demonstrate the robustness of both methods relative to existing preconditioners.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-026-10360-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148987136","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
Neural network-assisted refinement of traditional schemes for one-dimensional scalar conservation laws 一维标量守恒律传统格式的神经网络辅助改进
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-09-17 DOI: 10.1007/s10444-026-10357-w
Imre Fekete, Ferenc Izsák, Vendel P. Kupás
{"title":"Neural network-assisted refinement of traditional schemes for one-dimensional scalar conservation laws","authors":"Imre Fekete,&nbsp;Ferenc Izsák,&nbsp;Vendel P. Kupás","doi":"10.1007/s10444-026-10357-w","DOIUrl":"10.1007/s10444-026-10357-w","url":null,"abstract":"<div><p>A clear link is established between conventional numerical methods and neural network approximations for solving one-dimensional scalar conservation laws. The focus is on the construction of an appropriate flux term in the case of convex flux functions for improving the classical schemes. The first neural network developed here is able to rediscover Godunov’s method, while the second one emulates the behavior of a second-order slope-limiter function. In this way, by merging them, second-order reconstruction-based schemes can be developed. The networks presented here employ a minimal number of parameters, significantly reducing the complexity compared to previous approaches. These networks can also be linked consecutively to get a deep one corresponding to multiple time steps. Training them with an appropriate loss leads to stable schemes, improving even the classical methods without increasing their complexity.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-026-10357-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148986691","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
Reduced particle-in-cell method for the Vlasov–Poisson system using autoencoder and Hamiltonian neural networks 使用自编码器和哈密顿神经网络的Vlasov-Poisson系统的胞内约简粒子方法
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-09-16 DOI: 10.1007/s10444-026-10364-x
Raphaël Côte, Emmanuel Franck, Laurent Navoret, Guillaume Steimer, Vincent Vigon
{"title":"Reduced particle-in-cell method for the Vlasov–Poisson system using autoencoder and Hamiltonian neural networks","authors":"Raphaël Côte,&nbsp;Emmanuel Franck,&nbsp;Laurent Navoret,&nbsp;Guillaume Steimer,&nbsp;Vincent Vigon","doi":"10.1007/s10444-026-10364-x","DOIUrl":"10.1007/s10444-026-10364-x","url":null,"abstract":"<div><p>Hamiltonian particle–based simulations of plasma dynamics are inherently computationally intensive, primarily due to the large number of particles required to obtain accurate solutions. This challenge becomes even more acute in many-query contexts, where numerous simulations must be conducted across a range of time and parameter values. Consequently, it is essential to construct reduced-order models from such discretization to significantly lower computational costs while ensuring validity across the specified time and parameter domains. Preserving the Hamiltonian structure in these reduced models is also crucial, as it helps maintain long-term stability. In this paper, we introduce a nonlinear, non-intrusive, data-driven model order reduction method for the 1D-1V Vlasov–Poisson system, discretized using a Hamiltonian particle-in-cell scheme. Our approach relies on a two-step projection framework: an initial linear projection based on the proper symplectic decomposition, followed by a nonlinear projection learned via an autoencoder neural network. The reduced dynamics are then modeled using a Hamiltonian neural network. The offline phase of the method is split into two stages: first, constructing the linear projection using full-order model snapshots, and second, jointly training the autoencoder and the Hamiltonian neural network to simultaneously learn the encoder-decoder mappings and the reduced dynamics. We validate the proposed method on several benchmarks, including Landau damping and two-stream instability. The results show that our method has better reduction properties than standard linear Hamiltonian reduction methods.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148986249","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
Sampling, approximation, and interpolation of differential forms by admissible integral k-meshes 抽样，近似，和插值微分形式的可容许的积分k-网格
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-09-15 DOI: 10.1007/s10444-026-10343-2
Ludovico Bruni Bruno, Federico Piazzon
{"title":"Sampling, approximation, and interpolation of differential forms by admissible integral k-meshes","authors":"Ludovico Bruni Bruno,&nbsp;Federico Piazzon","doi":"10.1007/s10444-026-10343-2","DOIUrl":"10.1007/s10444-026-10343-2","url":null,"abstract":"<div><p>In this work, we introduce the concept of admissible integral <i>k</i>-mesh for sampling differential forms with continuous coefficients on a real body <span>(Esubset mathbb {R}^n)</span>, and provide two techniques for the construction of admissible integral <i>k</i>-meshes on real bodies enjoying the Markov or the Bernstein inequality. Admissible integral <i>k</i>-meshes allow for the construction of robust approximation schemes, and are used to extract interpolation sets with high stability properties. To this end, the concepts of Fekete currents and Leja sequences of currents are formalized, and a numerical scheme for their approximation is proposed.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-026-10343-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148986307","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
Data sparse multilevel covariance estimation in optimal complexity 最优复杂度下的数据稀疏多水平协方差估计
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-09-11 DOI: 10.1007/s10444-026-10359-8
Jürgen Dölz
{"title":"Data sparse multilevel covariance estimation in optimal complexity","authors":"Jürgen Dölz","doi":"10.1007/s10444-026-10359-8","DOIUrl":"10.1007/s10444-026-10359-8","url":null,"abstract":"<div><p>We consider the <span>(mathcal {H}^in )</span>-formatted compression and computational estimation of covariance functions on a compact set in <span>(mathbb {R}^d)</span>. The classical sample covariance or Monte Carlo estimator is prohibitively expensive for many practically relevant problems, where often approximation spaces with many degrees of freedom and many samples for the estimator are needed. In this article, we propose and analyze a data-sparse multilevel sample covariance estimator, i.e., a multilevel Monte Carlo estimator. For this purpose, we generalize the notion of asymptotically smooth kernel functions to a Gevrey-type class of kernels for which we derive new variable-order <span>(mathcal {H}^in )</span>-approximation rates. These variable-order <span>(mathcal {H}^in )</span>-approximations can be considered as a variant of <i>hp</i>-approximations. Our multilevel sample covariance estimator then uses an approximate multilevel hierarchy of variable-order <span>(mathcal {H}^in )</span>-approximations to compress the sample covariances on each level. The non-nestedness of the different levels makes the reduction to the final estimator nontrivial, and we present a suitable algorithm which can handle this task in linear complexity. This allows for a data-sparse multilevel estimator of Gevrey covariance kernel functions in the best possible complexity for Monte Carlo type multilevel estimators, which is quadratic. Numerical examples which estimate covariance matrices with tens of billions of entries are presented.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-026-10359-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148927332","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
hp-version Galerkin time-stepping with Laguerre spectral methods for parabolic equations on semi-unbounded domains 半无界区域上抛物方程的hp版Galerkin时间步进与Laguerre谱方法
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-09-11 DOI: 10.1007/s10444-026-10362-z
Zhe Li, Jie Luo, Lijun Yi
{"title":"hp-version Galerkin time-stepping with Laguerre spectral methods for parabolic equations on semi-unbounded domains","authors":"Zhe Li,&nbsp;Jie Luo,&nbsp;Lijun Yi","doi":"10.1007/s10444-026-10362-z","DOIUrl":"10.1007/s10444-026-10362-z","url":null,"abstract":"<div><p>We propose fully discrete numerical methods, high-order in both space and time, for solving parabolic equations on semi-unbounded domains. The spatial discretization uses a spectral method based on Laguerre functions, while the temporal discretization employs two distinct <i>hp</i>-version Galerkin time-stepping schemes: the continuous Petrov-Galerkin and the discontinuous Galerkin methods. A priori error estimates are derived in the <span>(L^2(0,T;H^1(mathbb {R}_{+})))</span>-norm, with all constants explicitly independent of the temporal and spatial discretization parameters. In the presence of singularities at <span>(t = 0)</span>, exponential rates of convergence in time are obtained by using geometrically refined time meshes together with approximation degrees that increase linearly. Numerical experiments are conducted to verify the theoretical findings.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148927643","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
Numerical analysis of energy, momentum, and angular momentum conserving algorithm for the closed-loop geothermal system 闭环地热系统能量、动量和角动量守恒算法的数值分析
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-09-09 DOI: 10.1007/s10444-026-10365-w
Yi Qin, Xi Gao, Yi Li, Jian Li
{"title":"Numerical analysis of energy, momentum, and angular momentum conserving algorithm for the closed-loop geothermal system","authors":"Yi Qin,&nbsp;Xi Gao,&nbsp;Yi Li,&nbsp;Jian Li","doi":"10.1007/s10444-026-10365-w","DOIUrl":"10.1007/s10444-026-10365-w","url":null,"abstract":"<div><p>This paper examines an energy, momentum, and angular momentum conserving (EMAC) algorithm for the closed-loop geothermal system. We use the finite element method for spatial discretisation and the Crank-Nicolson method for time discretisation, where the first nonlinear term of the geothermal system is processed by the EMAC algorithm. Theoretical analysis shows that the EMAC algorithm yields superior error estimates compared with the commonly used skew-symmetric (SKEW) algorithm: the error of the EMAC algorithm depends on the Reynolds number (<span>(varvec{Re})</span>), while the error of the SKEW algorithm depends on <span>(varvec{Re^3})</span>. Three numerical experiments verify the convergence, effectiveness, and physical conservation properties of the algorithm, and confirm that the <span>(varvec{L^2})</span> error of the velocity field <span>({textbf {u}})</span> obtained by the EMAC algorithm is better than that obtained by the SKEW algorithm as <span>(varvec{Re})</span> increases.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148926873","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
Optimal error analysis of a decoupled, linear, and fully discrete finite element scheme for the EHD system EHD系统解耦、线性和全离散有限元方案的最优误差分析
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-09-09 DOI: 10.1007/s10444-026-10358-9
Yuanqin Li, Xiaodi Zhang
{"title":"Optimal error analysis of a decoupled, linear, and fully discrete finite element scheme for the EHD system","authors":"Yuanqin Li,&nbsp;Xiaodi Zhang","doi":"10.1007/s10444-026-10358-9","DOIUrl":"10.1007/s10444-026-10358-9","url":null,"abstract":"<div><p>This paper focuses on the optimal error estimates of a fully discrete decoupled scheme for the electrohydrodynamics (EHD) model, which couples the Navier–Stokes equations for velocity and pressure with a transport equation for the charge density and a Poisson equation for the electric potential. By combining the stabilization method and some subtle implicit-explicit treatments for nonlinear coupling terms in time, and mixed finite element methods in space, we develop a linear and decoupled fully discrete scheme. The scheme is proved to be uniquely solvable, charge-conservative, and unconditionally energy stable. By deriving the <span>(L^{infty })</span>-bound of the numerical charge density, and utilizing the energy argument, we establish the optimal error estimates for all variables rigorously. Various numerical experiments are provided to demonstrate the theoretical results.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148926874","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
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