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High-Accuracy Numerical Methods and Convergence Analysis for Schrödinger Equation with Incommensurate Potentials 具有不相称势垒的薛定谔方程的高精度数值方法和收敛性分析
IF 2.5 2区 数学
Journal of Scientific Computing Pub Date : 2024-08-29 DOI: 10.1007/s10915-024-02658-3
Kai Jiang, Shifeng Li, Juan Zhang
{"title":"High-Accuracy Numerical Methods and Convergence Analysis for Schrödinger Equation with Incommensurate Potentials","authors":"Kai Jiang, Shifeng Li, Juan Zhang","doi":"10.1007/s10915-024-02658-3","DOIUrl":"https://doi.org/10.1007/s10915-024-02658-3","url":null,"abstract":"<p>Numerical solving the Schrödinger equation with incommensurate potentials presents a great challenge since its solutions could be space-filling quasiperiodic structures without translational symmetry nor decay. In this paper, we propose two high-accuracy numerical methods to solve the time-dependent quasiperiodic Schrödinger equation. Concretely, we discretize the spatial variables by the quasiperiodic spectral method and the projection method, and the time variable by the second-order operator splitting method. The corresponding convergence analysis is also presented and shows that the proposed methods both have spectral convergence rates in space and second order accuracy in time, respectively. Meanwhile, we analyse the computational complexity of these numerical algorithms. One- and two-dimensional numerical results verify these convergence conclusions, and demonstrate that the projection method is more efficient.\u0000</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"65 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 Fast Algorithm for Rank-(L, M, N) Block Term Decomposition of Multi-Dimensional Data 多维数据的 Rank-(L, M, N) 块项分解快速算法
IF 2.5 2区 数学
Journal of Scientific Computing Pub Date : 2024-08-28 DOI: 10.1007/s10915-024-02653-8
Hao Zhang, Ting-Zhu Huang, Xi-Le Zhao, Maolin Che
{"title":"A Fast Algorithm for Rank-(L, M, N) Block Term Decomposition of Multi-Dimensional Data","authors":"Hao Zhang, Ting-Zhu Huang, Xi-Le Zhao, Maolin Che","doi":"10.1007/s10915-024-02653-8","DOIUrl":"https://doi.org/10.1007/s10915-024-02653-8","url":null,"abstract":"<p>Attribute to its powerful representation ability, block term decomposition (BTD) has recently attracted many views of multi-dimensional data processing, e.g., hyperspectral image unmixing and blind source separation. However, the popular alternating least squares algorithm for rank-(<i>L</i>, <i>M</i>, <i>N</i>) BTD (BTD-ALS) suffers expensive time and space costs from Kronecker products and solving low-rank approximation subproblems, hindering the deployment of BTD for real applications, especially for large-scale data. In this paper, we propose a fast sketching-based Kronecker product-free algorithm for rank-(<i>L</i>, <i>M</i>, <i>N</i>) BTD (termed as KPF-BTD), which is suitable for real-world multi-dimensional data. Specifically, we first decompose the original optimization problem into several rank-(<i>L</i>, <i>M</i>, <i>N</i>) approximation subproblems, and then we design the bilateral sketching to obtain the approximate solutions of these subproblems instead of the exact solutions, which allows us to avoid Kronecker products and rapidly solve rank-(<i>L</i>, <i>M</i>, <i>N</i>) approximation subproblems. As compared with BTD-ALS, the time and space complexities <span>(mathcal {O}{(2(p+1)(I^3LR+I^2L^2R+IL^3R)+I^3LR)})</span> and <span>(mathcal {O}{(I^3)})</span> of KPF-BTD are significantly cheaper than <span>(mathcal {O}{(I^3L^6R^2+I^3L^3R+I^3LR+I^2L^3R^2+I^2L^2R)})</span> and <span>(mathcal {O}{(I^3L^3R)})</span> of BTD-ALS, where <span>(p ll I)</span>. Moreover, we establish the theoretical error bound for KPF-BTD. Extensive synthetic and real experiments show KPF-BTD achieves substantial speedup and memory saving while maintaining accuracy (e.g., for a <span>(150times 150times 150)</span> synthetic tensor, the running time 0.2 seconds per iteration of KPF-BTD is significantly faster than 96.2 seconds per iteration of BTD-ALS while their accuracies are comparable).</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"4 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Poissonian Image Restoration Via the $$L_1/L_2$$ -Based Minimization 通过基于 L_1/L_2$$ 的最小化实现泊松图像复原
IF 2.5 2区 数学
Journal of Scientific Computing Pub Date : 2024-08-28 DOI: 10.1007/s10915-024-02657-4
Mujibur Rahman Chowdhury, Chao Wang, Yifei Lou
{"title":"Poissonian Image Restoration Via the $$L_1/L_2$$ -Based Minimization","authors":"Mujibur Rahman Chowdhury, Chao Wang, Yifei Lou","doi":"10.1007/s10915-024-02657-4","DOIUrl":"https://doi.org/10.1007/s10915-024-02657-4","url":null,"abstract":"<p>This study investigates the Poissonian image restoration problems. In particular, we propose a novel model that incorporates <span>(L_1/L_2)</span> minimization on the gradient as a regularization term combined with a box constraint and a nonlinear data fidelity term, specifically crafted to address the challenges caused by Poisson noise. We employ a splitting strategy, followed by the alternating direction method of multipliers (ADMM) to find a model solution. Furthermore, we show that under mild conditions, the sequence generated by ADMM has a sub-sequence that converges to a stationary point of the proposed model. Through numerical experiments on image deconvolution, super-resolution, and magnetic resonance imaging (MRI) reconstruction, we demonstrate superior performance made by the proposed approach over some existing gradient-based methods.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"21 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Flexible Ultra-convergence Structures for the Finite Volume Element Method 有限体积元素法的灵活超收敛结构
IF 2.5 2区 数学
Journal of Scientific Computing Pub Date : 2024-08-26 DOI: 10.1007/s10915-024-02654-7
Xiang Wang, Yuqing Zhang, Zhimin Zhang
{"title":"Flexible Ultra-convergence Structures for the Finite Volume Element Method","authors":"Xiang Wang, Yuqing Zhang, Zhimin Zhang","doi":"10.1007/s10915-024-02654-7","DOIUrl":"https://doi.org/10.1007/s10915-024-02654-7","url":null,"abstract":"<p>We introduce a novel class of ultra-convergent structures for the Finite Volume Element (FVE) method. These structures are characterized by asymmetric and optional superconvergent points. We establish a crucial relationship between ultra-convergence properties and the orthogonality condition. Remarkably, within this framework, certain FVE schemes achieve simultaneous superconvergence of both derivatives and function values at designated points, as demonstrated in Example 2. This is a phenomenon rarely observed in other numerical methods. Theoretical validation of these findings is provided through the proposed Generalized M-Decomposition (GMD). Numerical experiments effectively substantiate our results.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"1 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
An Alternating Direction Multiplier Method for the Inversion of FDEM Data 用于反演 FDEM 数据的交替方向乘法器方法
IF 2.5 2区 数学
Journal of Scientific Computing Pub Date : 2024-08-24 DOI: 10.1007/s10915-024-02652-9
Alessandro Buccini, Patricia Díaz de Alba, Federica Pes
{"title":"An Alternating Direction Multiplier Method for the Inversion of FDEM Data","authors":"Alessandro Buccini, Patricia Díaz de Alba, Federica Pes","doi":"10.1007/s10915-024-02652-9","DOIUrl":"https://doi.org/10.1007/s10915-024-02652-9","url":null,"abstract":"<p>In this paper, we focus on the numerical solution of nonlinear inverse problems in applied geophysics. Our aim is to reconstruct the structure of the soil, i.e., either its electrical conductivity or the magnetic permeability distribution, by inverting frequency domain electromagnetic data. This is a very challenging task since the problem is nonlinear and severely ill-conditioned. To solve the nonlinear inverse problem, we propose an alternating direction multiplier method (ADMM), we prove its convergence, and propose an automated strategy to determine the parameters involved. Moreover, we present two heuristic variations of the ADMM that either improve the accuracy of the computed solutions or lower the computational cost. The effectiveness of the different proposed methods is illustrated through few numerical examples.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"2019 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 Multigrid Solver for PDE-Constrained Optimization with Uncertain Inputs 用于具有不确定输入的 PDE 受限优化的多网格求解器
IF 2.5 2区 数学
Journal of Scientific Computing Pub Date : 2024-08-24 DOI: 10.1007/s10915-024-02646-7
Gabriele Ciaramella, Fabio Nobile, Tommaso Vanzan
{"title":"A Multigrid Solver for PDE-Constrained Optimization with Uncertain Inputs","authors":"Gabriele Ciaramella, Fabio Nobile, Tommaso Vanzan","doi":"10.1007/s10915-024-02646-7","DOIUrl":"https://doi.org/10.1007/s10915-024-02646-7","url":null,"abstract":"<p>In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective smoothers and collective two-level methods. The multigrid algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size is proportional to the number <i>N</i> of samples used to discretized the probability space. We show that this reduced system can be solved with optimal <i>O</i>(<i>N</i>) complexity. The multigrid method is tested both as a stationary method and as a preconditioner for GMRES on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and <span>(L^1)</span>-norm penalization on the control, in which the multigrid scheme is used as an inner solver within a semismooth Newton iteration; a risk-averse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.\u0000</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"66 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183462","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Analysis of Weak Galerkin Mixed Finite Element Method Based on the Velocity–Pseudostress Formulation for Navier–Stokes Equation on Polygonal Meshes 基于多边形网格上 Navier-Stokes 方程速度-伪应力公式的弱 Galerkin 混合有限元法分析
IF 2.5 2区 数学
Journal of Scientific Computing Pub Date : 2024-08-22 DOI: 10.1007/s10915-024-02651-w
Zeinab Gharibi, Mehdi Dehghan
{"title":"Analysis of Weak Galerkin Mixed Finite Element Method Based on the Velocity–Pseudostress Formulation for Navier–Stokes Equation on Polygonal Meshes","authors":"Zeinab Gharibi, Mehdi Dehghan","doi":"10.1007/s10915-024-02651-w","DOIUrl":"https://doi.org/10.1007/s10915-024-02651-w","url":null,"abstract":"<p>The present article introduces, mathematically analyzes, and numerically validates a new weak Galerkin mixed finite element method based on Banach spaces for the stationary Navier–Stokes equation in pseudostress–velocity formulation. Specifically, a modified pseudostress tensor, which depends on the pressure as well as the diffusive and convective terms, is introduced as an auxiliary unknown, and the incompressibility condition is then used to eliminate the pressure, which is subsequently computed using a postprocessing formula. Consequently, to discretize the resulting mixed formulation, it is sufficient to provide a tensorial weak Galerkin space for the pseudostress and a space of piecewise polynomial vectors of total degree at most ’k’ for the velocity. Moreover, the weak gradient/divergence operator is utilized to propose the weak discrete bilinear forms, whose continuous version involves the classical gradient/divergence operators. The well-posedness of the numerical solution is proven using a fixed-point approach and the discrete versions of the Babuška–Brezzi theory and the Banach–Nečas–Babuška theorem. Additionally, an a priori error estimate is derived for the proposed method. Finally, several numerical results illustrating the method’s good performance and confirming the theoretical rates of convergence are presented.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"11 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Provably Convergent Learned Inexact Descent Algorithm for Low-Dose CT Reconstruction 用于低剂量 CT 重建的可证明收敛学习型非精确下降算法
IF 2.5 2区 数学
Journal of Scientific Computing Pub Date : 2024-08-20 DOI: 10.1007/s10915-024-02638-7
Qingchao Zhang, Mehrdad Alvandipour, Wenjun Xia, Yi Zhang, Xiaojing Ye, Yunmei Chen
{"title":"Provably Convergent Learned Inexact Descent Algorithm for Low-Dose CT Reconstruction","authors":"Qingchao Zhang, Mehrdad Alvandipour, Wenjun Xia, Yi Zhang, Xiaojing Ye, Yunmei Chen","doi":"10.1007/s10915-024-02638-7","DOIUrl":"https://doi.org/10.1007/s10915-024-02638-7","url":null,"abstract":"<p>We propose an Efficient Inexact Learned Descent-type Algorithm (ELDA) for a class of nonconvex and nonsmooth variational models, where the regularization consists of a sparsity enhancing term and non-local smoothing term for learned features. The ELDA improves the performance of the LDA in Chen et al. (SIAM J Imag Sci 14(4), 1532–1564, 2021) by reducing the number of the subproblems from two to one for most of the iterations and allowing inexact gradient computation. We generate a deep neural network, whose architecture follows the algorithm exactly for low-dose CT (LDCT) reconstruction. The network inherits the convergence behavior of the algorithm and is interpretable as a solution of the varational model and parameter efficient. The experimental results from the ablation study and comparisons with several state-of-the-art deep learning approaches indicate the promising performance of the proposed method in solution accuracy and parameter efficiency.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"16 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Error Estimates and Adaptivity of the Space-Time Discontinuous Galerkin Method for Solving the Richards Equation 解决理查兹方程的时空非连续伽勒金方法的误差估计和适应性
IF 2.5 2区 数学
Journal of Scientific Computing Pub Date : 2024-08-20 DOI: 10.1007/s10915-024-02650-x
Vít Dolejší, Hyun-Geun Shin, Miloslav Vlasák
{"title":"Error Estimates and Adaptivity of the Space-Time Discontinuous Galerkin Method for Solving the Richards Equation","authors":"Vít Dolejší, Hyun-Geun Shin, Miloslav Vlasák","doi":"10.1007/s10915-024-02650-x","DOIUrl":"https://doi.org/10.1007/s10915-024-02650-x","url":null,"abstract":"<p>We present a higher-order space-time adaptive method for the numerical solution of the Richards equation that describes a flow motion through variably saturated media. The discretization is based on the space-time discontinuous Galerkin method, which provides high stability and accuracy and can naturally handle varying meshes. We derive reliable and efficient a posteriori error estimates in the residual-based norm. The estimates use well-balanced spatial and temporal flux reconstructions which are constructed locally over space-time elements or space-time patches. The accuracy of the estimates is verified by numerical experiments. Moreover, we develop the <i>hp</i>-adaptive method and demonstrate its efficiency and usefulness on a practically relevant example.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"2 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Direct Discontinuous Galerkin Method with Interface Correction for the Keller-Segel Chemotaxis Model 针对凯勒-西格尔趋化模型的带界面校正的直接非连续伽勒金方法
IF 2.5 2区 数学
Journal of Scientific Computing Pub Date : 2024-08-17 DOI: 10.1007/s10915-024-02648-5
Xinghui Zhong, Changxin Qiu, Jue Yan
{"title":"Direct Discontinuous Galerkin Method with Interface Correction for the Keller-Segel Chemotaxis Model","authors":"Xinghui Zhong, Changxin Qiu, Jue Yan","doi":"10.1007/s10915-024-02648-5","DOIUrl":"https://doi.org/10.1007/s10915-024-02648-5","url":null,"abstract":"<p>The Keller-Segel (KS) chemotaxis equation is a widely studied mathematical model for understanding the collective behavior of cells in response to chemical gradients. This paper investigates the direct discontinuous Galerkin method with interface correction (DDGIC) for one-dimensional and two-dimensional KS equations governing the cell density and chemoattractant concentration. We establish error estimates for the proposed scheme under suitable smoothness assumptions of the exact solutions. Numerical experiments are conducted to validate the theoretical results. We explore the impact of different coefficient settings in the numerical fluxes on the error of the DDGIC method on uniform and nonuniform meshes. Our findings reveal that the DDGIC method achieves optimal convergence rates with any admissible coefficients for polynomials of odd degrees, while the accuracy of the cell density is sensitive to the numerical flux coefficient in the chemoattractant concentration for polynomials of even degrees. These results hold regardless of whether the mesh is uniform or nonuniform.</p>","PeriodicalId":50055,"journal":{"name":"Journal of Scientific Computing","volume":"21 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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