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A least squares space-time approach for parabolic equations 抛物方程的最小二乘时空方法
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-08-20 DOI: 10.1007/s10444-026-10345-0
Michael Hinze, Christian Kahle, Michael Stahl
{"title":"A least squares space-time approach for parabolic equations","authors":"Michael Hinze,&nbsp;Christian Kahle,&nbsp;Michael Stahl","doi":"10.1007/s10444-026-10345-0","DOIUrl":"10.1007/s10444-026-10345-0","url":null,"abstract":"<div><p>We propose a least squares formulation for abstract parabolic equations in the natural <span>(L^2(0,T;V^star )times H)</span> norm which only relies on natural regularity assumptions on the data of the problem. The resulting bilinear form then is symmetric, coercive, and continuous and contains the <span>(V^star )</span>-norm. We propose a solution approach that uses a conforming Galerkin discretization of an equivalent saddle point problem to circumvent the evaluation of <span>(V^star )</span>-norms. We prove strong convergence of discrete solutions towards continuous solutions and provide a preconditioner for efficient numerical solution of the discrete saddle point problem. We illustrate our analytical findings by selected numerical experiments.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-026-10345-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148768976","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
On block fast dominant Hermitian splitting preconditioning methods for higher-dimensional spatial fractional diffusion equations 高维空间分数扩散方程的块快速优势厄米分裂预处理方法
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-08-19 DOI: 10.1007/s10444-026-10341-4
Yu-Hong Ran, Ao Feng
{"title":"On block fast dominant Hermitian splitting preconditioning methods for higher-dimensional spatial fractional diffusion equations","authors":"Yu-Hong Ran,&nbsp;Ao Feng","doi":"10.1007/s10444-026-10341-4","DOIUrl":"10.1007/s10444-026-10341-4","url":null,"abstract":"<div><p>The discretizations of two- and three-dimensional spatial fractional diffusion equations with the shifted finite-difference formulas of the Grünwald-Letnikov type can result in discrete linear systems whose coefficient matrices are equal to the sum of the identity matrix and two diagonal-times-block-Toeplitz with Toeplitz-block matrices or block-Toeplitz with each block being block-Toeplitz with Toeplitz-block matrices. For these discrete spatial fractional diffusion matrices, we construct two block fast dominant Hermitian splitting preconditioners based on circulant and <span>(tau )</span>-matrix approximations to further accelerate the convergence rates of Krylov subspace methods. Theoretical analyses demonstrate that most of the eigenvalues of the corresponding preconditioned matrices are clustered in a complex disk centered at 1 with a radius less than 1. In addition, the numerical results show that the constructed preconditioners can effectively solve the discrete linear systems of higher-dimensional spatial fractional diffusion equations.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148768978","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
Generalized Gearhart-Koshy acceleration is a Krylov subspace method 广义Gearhart-Koshy加速是一种Krylov子空间方法
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-08-19 DOI: 10.1007/s10444-026-10342-3
Markus Hegland, Janosch Rieger
{"title":"Generalized Gearhart-Koshy acceleration is a Krylov subspace method","authors":"Markus Hegland,&nbsp;Janosch Rieger","doi":"10.1007/s10444-026-10342-3","DOIUrl":"10.1007/s10444-026-10342-3","url":null,"abstract":"<div><p>The Kaczmarz method is a row-action method for solving consistent non-square linear systems with applications in inverse problems, and Gearhart-Koshy acceleration is a line-search that minimizes the Euclidean norm of the error along a ray in the direction of a Kaczmarz step. Recently, one of the authors generalized this procedure to a search for the point with minimal Euclidean error norm within a sequence of nested affine subspaces. In this paper, we demonstrate that this generalization can be interpreted as a Krylov subspace method for a square linear system, which is equivalent to the original system to be solved. In exact arithmetic, the method cannot break down prematurely, and it makes progress in every step. We also present a mathematically equivalent reformulation of the algorithm in terms of the Gram-Schmidt orthogonalization procedure, and we illustrate the convergence behavior of the new method with numerical experiments.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148768977","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
Neural non-canonical Hamiltonian dynamics for long-time simulations 长时间模拟的神经非正则哈密顿动力学
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-08-17 DOI: 10.1007/s10444-026-10336-1
Clémentine Courtès, Emmanuel Franck, Michael Kraus, Laurent Navoret, Léopold Trémant
{"title":"Neural non-canonical Hamiltonian dynamics for long-time simulations","authors":"Clémentine Courtès,&nbsp;Emmanuel Franck,&nbsp;Michael Kraus,&nbsp;Laurent Navoret,&nbsp;Léopold Trémant","doi":"10.1007/s10444-026-10336-1","DOIUrl":"10.1007/s10444-026-10336-1","url":null,"abstract":"<div><p>This work focuses on learning non-canonical Hamiltonian dynamics from data, where long-term predictions require the preservation of structure both in the learned model and in numerical schemes. Previous research focused on either facet, respectively with a potential-based architecture and with degenerate variational integrators, but new issues arise when combining both. In experiments, the learnt model is sometimes numerically unstable due to the gauge dependency of the scheme, rendering long-time simulations impossible. In this paper, we identify this problem and propose two different training strategies to address it, either by directly learning the vector field or by learning a time-discrete dynamics through the scheme. Several numerical test cases assess the ability of the methods to learn complex physical dynamics, like the guiding center from gyrokinetic plasma physics.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148751613","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
Newtonian potentials of Legendre polynomials on rectangles have displacement structure 矩形上的勒让德多项式的牛顿势具有位移结构
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-08-14 DOI: 10.1007/s10444-026-10327-2
Sheehan Olver
{"title":"Newtonian potentials of Legendre polynomials on rectangles have displacement structure","authors":"Sheehan Olver","doi":"10.1007/s10444-026-10327-2","DOIUrl":"10.1007/s10444-026-10327-2","url":null,"abstract":"<div><p>Particular solutions of the Poisson equation can be constructed via Newtonian potentials, integrals involving the corresponding Green’s function, which in two-dimensions has a logarithmic singularity. The singularity represents a significant challenge for computing the integrals, which is typically overcome via specially designed quadrature methods involving a large number of evaluations of the function and kernel. We present an attractive alternative: we show that Newtonian potentials (and their gradient) applied to (tensor products of) Legendre polynomials can be expressed in terms of complex integrals which satisfy simple and explicit recurrences that can be utilised to <i>exactly</i> compute singular integrals, i.e., singular integral quadrature is completely avoided. The inhomogeneous part of the recurrence has low rank structure (its rank is at most three for the Newtonian potential) and hence these recurrences have displacement structure. Using the recurrence directly is a fast approach for evaluation on or near the integration domain that remains accurate for low-degree polynomial approximations, while high-precision arithmetic allows accurate use of the approach for moderate-degree polynomials.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-026-10327-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148728729","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
Error estimate for a decoupled energy-stable scheme for smectic-A liquid crystals smic - a液晶解耦能量稳定方案的误差估计
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-08-06 DOI: 10.1007/s10444-026-10340-5
Ning Lu, Guanghua Ji
{"title":"Error estimate for a decoupled energy-stable scheme for smectic-A liquid crystals","authors":"Ning Lu,&nbsp;Guanghua Ji","doi":"10.1007/s10444-026-10340-5","DOIUrl":"10.1007/s10444-026-10340-5","url":null,"abstract":"<div><p>We study a numerical scheme for a hydrodynamic model of smectic-A liquid crystals, which incorporates velocity–pressure variables <span>((varvec{u},p))</span> and the order parameter of smectic-A. The model is a highly nonlinear system that couples the incompressible Navier–Stokes equations with a fourth-order parabolic equation for the layer variable <span>(phi )</span>, which is endowed with periodic boundary conditions. We first introduce a first-order decoupled scalar auxiliary variable (SAV) scheme for numerically solving the smectic-A liquid crystal model and present an efficient implementation. Moreover, we prove that the scheme is uniquely solvable and energy stable. Furthermore, we carry out an error estimate for the scheme. Various numerical experiments are presented to verify our theoretical results and simulate the dynamics of the system.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687775","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
A priori error estimates for a conforming VEM for the Oseen eigenvalue problem 针对Oseen特征值问题，给出了符合向量机的先验误差估计
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-08-04 DOI: 10.1007/s10444-026-10339-y
Danilo Amigo, Felipe Lepe, Nitesh Verma
{"title":"A priori error estimates for a conforming VEM for the Oseen eigenvalue problem","authors":"Danilo Amigo,&nbsp;Felipe Lepe,&nbsp;Nitesh Verma","doi":"10.1007/s10444-026-10339-y","DOIUrl":"10.1007/s10444-026-10339-y","url":null,"abstract":"<div><p>In this paper, we propose a conforming virtual element method to approximate the eigenfunctions and eigenvalues of the two-dimensional Oseen eigenvalue problem in its classic velocity-pressure formulation. We employ divergence-conforming virtual element spaces, originally developed for the Stokes equations, to approximate the velocity, and piecewise constant functions for the pressure. Under standard mesh assumptions, we derive a priori error estimates for the proposed method using the compact operator theory. In particular, we provide a rigorous tracking of the constants involved in the analysis, with the aim of having a detailed characterization of the convergence in norm between the solution operators. We report a set of numerical tests to confirm the theoretical results.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687778","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
Recurrence relations and zeros of Gegenbauer–Sobolev orthogonal polynomials Gegenbauer-Sobolev正交多项式的递归关系和零
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-08-03 DOI: 10.1007/s10444-026-10338-z
Niel Van Buggenhout, Teresa Laudadio, Nicola Mastronardi, Francisco Marcellán, Paul Van Dooren
{"title":"Recurrence relations and zeros of Gegenbauer–Sobolev orthogonal polynomials","authors":"Niel Van Buggenhout,&nbsp;Teresa Laudadio,&nbsp;Nicola Mastronardi,&nbsp;Francisco Marcellán,&nbsp;Paul Van Dooren","doi":"10.1007/s10444-026-10338-z","DOIUrl":"10.1007/s10444-026-10338-z","url":null,"abstract":"<div><p>Gegenbauer–Sobolev polynomials are an important class of Sobolev orthogonal polynomials. We prove a novel recurrence relation for this sequence of polynomials that allows a more accurate construction of the sequence. This new recurrence relation allows us to decompose the recurrence matrix of the sequence as the product of three structured matrices. By computing the eigenvalues of the recurrence matrix, we obtain the zeros of the Gegenbauer–Sobolev polynomials. However, the eigenvalue problem formulated with this recurrence matrix is ill-conditioned and thus cannot lead to reliable approximations of the actual zeros. We deal with this ill-conditioning by reformulating the eigenvalue problem as a generalized eigenvalue problem, involving the three structured matrices that decompose the recurrence matrix, and applying a balancing technique to this generalized eigenvalue problem.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-026-10338-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148682209","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
Symplectic QTT-FEM solution of the one-dimensional acoustic wave equation in the time domain 一维声波方程的时域辛QTT-FEM解
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-07-28 DOI: 10.1007/s10444-026-10332-5
Sara Fraschini, Vladimir Kazeev, Ilaria Perugia
{"title":"Symplectic QTT-FEM solution of the one-dimensional acoustic wave equation in the time domain","authors":"Sara Fraschini,&nbsp;Vladimir Kazeev,&nbsp;Ilaria Perugia","doi":"10.1007/s10444-026-10332-5","DOIUrl":"10.1007/s10444-026-10332-5","url":null,"abstract":"<div><p>Structured finite element methods (FEMs) based on low-rank approximation in the form of the so-called <i>quantized tensor train (QTT)</i> decomposition (QTT-FEM) have been proposed and extensively studied in the case of elliptic equations. In this work, we design a QTT-FE method for time-domain acoustic wave equations, combining stable low-rank approximation in space with a suitable conservative discretization in time. For the acoustic wave equation with a homogeneous source term in a single space dimension as a model problem, we consider its reformulation as a first-order system in time. In space, we employ a low-rank QTT-FEM discretization based on continuous piecewise linear finite elements corresponding to uniformly refined nested meshes. Time integration is performed using symplectic high-order Gauss–Legendre Runge–Kutta methods. In our numerical experiments, we investigate the energy conservation and exponential convergence of the proposed method.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148612830","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
A Kalman filter-based variational method for ECG tensor denoising 一种基于卡尔曼滤波的心电张量去噪方法
IF 2.6 3区 数学
Advances in Computational Mathematics Pub Date : 2026-07-24 DOI: 10.1007/s10444-026-10337-0
Ping Cui, Zi-Han Song, Yu-Mei Huang, Zhuo-Fei Zhang
{"title":"A Kalman filter-based variational method for ECG tensor denoising","authors":"Ping Cui,&nbsp;Zi-Han Song,&nbsp;Yu-Mei Huang,&nbsp;Zhuo-Fei Zhang","doi":"10.1007/s10444-026-10337-0","DOIUrl":"10.1007/s10444-026-10337-0","url":null,"abstract":"<div><p>With the popularity of wearable devices, real-time electrocardiogram (ECG) monitoring is becoming increasingly important in heart health management. However, ECG signals acquired by these devices are highly susceptible to noises arising in signal acquisition. Diagnostic accuracy may often be lowered by these interfering noises. In this paper, by partitioning an ECG signal into heartbeat segments and establishing an ECG tensor, we propose a two-layer Kalman filter-based variational method (TLKFVM) for ECG signal denoising. In the first layer of the denoising process, we apply the <span>(l_1)</span>-norm regularized variational model to remove noises within each heartbeat, which is solved by a Kalman smoother (KS) based on the intra-heartbeat state space model (SSM). In the second layer, by utilizing the similarity of heartbeat segments, we apply the <span>(l_2)</span>-norm regularized variational model to further remove residue noises during the evolution period between consecutive heartbeats, which is solved by a Kalman filter (KF) based on the inter-heartbeat SSM. We call the first layer as the intra-heartbeat denoising and the second layer as the inter-heartbeat denoising. The experiments demonstrate the proposed method’s advantages in denoising quality, computational efficiency, and adaptivity when compared with the state-of-the-art methods for ECG denoising.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148586689","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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