IF 1.7 2区数学

Calcolo Pub Date : 2024-03-04 DOI: 10.1007/s10092-024-00569-1

Rong Li, Bing Zheng

{"title":"Tensor completion via multi-directional partial tensor nuclear norm with total variation regularization","authors":"Rong Li, Bing Zheng","doi":"10.1007/s10092-024-00569-1","DOIUrl":"https://doi.org/10.1007/s10092-024-00569-1","url":null,"abstract":"This paper addresses the tensor completion problem, whose task is to estimate missing values with limited information. However, the crux of this problem is how to reasonably represent the low-rank structure embedded in the underlying data. In this work, we consider a new low-rank tensor completion model combined with the multi-directional partial tensor nuclear norm and the total variation (TV) regularization. Specifically, the partial sum of the tensor nuclear norm (PSTNN) is used to narrow the gap between the tensor tubal rank and its lower convex envelop [i.e. tensor nuclear norm (TNN)], and the TV regularization is adopted to maintain the smooth structure along the spatial dimension. In addition, the weighted sum of the tensor nuclear norm (WSTNN) is introduced to replace the traditional TNN to extend the PSTNN to the high-order tensor, which also can flexibly handle different correlations along different modes, resulting in an improved low d-tubal rank approximation. To tackle this new model, we develop the alternating directional method of multipliers (ADMM) algorithm tailored for the proposed optimization problem. Theoretical analysis of the ADMM is conducted to prove the Karush–Kuhn–Tucker (KKT) conditions. Numerical examples demonstrate the proposed method outperforms some state-of-the-art methods in qualitative and quantitative aspects.","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140032925","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 Gauss–Newton method for mixed least squares-total least squares problems 混合最小二乘法-总最小二乘法问题的高斯-牛顿法

IF 1.7 2区数学

Calcolo Pub Date : 2024-03-01 DOI: 10.1007/s10092-024-00568-2

Qiaohua Liu, Shan Wang, Yimin Wei

引用次数: 0

Supercloseness of the NIPG method for a singularly perturbed convection diffusion problem on Shishkin mesh Shishkin 网格上奇异扰动对流扩散问题的 NIPG 方法的超粘性

IF 1.7 2区数学

Calcolo Pub Date : 2024-02-28 DOI: 10.1007/s10092-024-00571-7

{"title":"Supercloseness of the NIPG method for a singularly perturbed convection diffusion problem on Shishkin mesh","authors":"","doi":"10.1007/s10092-024-00571-7","DOIUrl":"https://doi.org/10.1007/s10092-024-00571-7","url":null,"abstract":"<h3>Abstract</h3> Some popular stabilization techniques, such as nonsymmetric interior penalty Galerkin (NIPG) method, have important application value in computational fluid dynamics. In this paper, we analyze a NIPG method on Shishkin mesh for a singularly perturbed convection diffusion problem, which is a typical simplified fluid model. According to the characteristics of the solution, the mesh and the numerical scheme, a new interpolation is designed for convergence analysis. More specifically, Gauß Lobatto interpolation and Gauß Radau interpolation are introduced inside and outside the layer, respectively. On the basis of that, by selecting special penalty parameters at different mesh points, we establish supercloseness of almost (k+1) order in an energy norm. Here (kge 1) is the degree of piecewise polynomials. Then, a simple post-processing operator is constructed, and it is proved that the corresponding post-processing can make the numerical solution achieve higher accuracy. In this process, a new analysis is proposed for the stability analysis of this operator. Finally, superconvergence is derived under a discrete energy norm. These conclusions can be verified numerically. Furthermore, numerical experiments show that the increase of polynomial degree k and mesh parameter N, the decrease of perturbation parameter (varepsilon ) or the use of over-penalty technology may increase the condition number of linear system. Therefore, we need to cautiously consider the application of high-order algorithm.","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140007586","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

Numerical methods for the forward and backward problems of a time-space fractional diffusion equation 时空分数扩散方程前向和后向问题的数值方法

IF 1.7 2区数学

Calcolo Pub Date : 2024-02-21 DOI: 10.1007/s10092-024-00567-3

Xiaoli Feng, Xiaoyu Yuan, Meixia Zhao, Zhi Qian

引用次数: 0

Optimal learning 优化学习

IF 1.7 2区数学

Calcolo Pub Date : 2024-02-19 DOI: 10.1007/s10092-023-00564-y

Peter Binev, Andrea Bonito, Ronald DeVore, Guergana Petrova

{"title":"Optimal learning","authors":"Peter Binev, Andrea Bonito, Ronald DeVore, Guergana Petrova","doi":"10.1007/s10092-023-00564-y","DOIUrl":"https://doi.org/10.1007/s10092-023-00564-y","url":null,"abstract":"This paper studies the problem of learning an unknown function f from given data about f. The learning problem is to give an approximation ({hat{f}}) to f that predicts the values of f away from the data. There are numerous settings for this learning problem depending on (i) what additional information we have about f (known as a model class assumption), (ii) how we measure the accuracy of how well ({hat{f}}) predicts f, (iii) what is known about the data and data sites, (iv) whether the data observations are polluted by noise. A mathematical description of the optimal performance possible (the smallest possible error of recovery) is known in the presence of a model class assumption. Under standard model class assumptions, it is shown in this paper that a near optimal ({hat{f}}) can be found by solving a certain finite-dimensional over-parameterized optimization problem with a penalty term. Here, near optimal means that the error is bounded by a fixed constant times the optimal error. This explains the advantage of over-parameterization which is commonly used in modern machine learning. The main results of this paper prove that over-parameterized learning with an appropriate loss function gives a near optimal approximation ({hat{f}}) of the function f from which the data is collected. Quantitative bounds are given for how much over-parameterization needs to be employed and how the penalization needs to be scaled in order to guarantee a near optimal recovery of f. An extension of these results to the case where the data is polluted by additive deterministic noise is also given.","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139927993","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

Discrete duality finite volume scheme for a generalized Joule heating problem 广义焦耳加热问题的离散二元有限体积方案

IF 1.7 2区数学

Calcolo Pub Date : 2024-02-07 DOI: 10.1007/s10092-024-00566-4

Mustapha Bahari, El-Houssaine Quenjel, Mohamed Rhoudaf

{"title":"Discrete duality finite volume scheme for a generalized Joule heating problem","authors":"Mustapha Bahari, El-Houssaine Quenjel, Mohamed Rhoudaf","doi":"10.1007/s10092-024-00566-4","DOIUrl":"https://doi.org/10.1007/s10092-024-00566-4","url":null,"abstract":"In this paper we conceive and analyze a discrete duality finite volume (DDFV) scheme for the unsteady generalized thermistor problem, including a p-Laplacian for the diffusion and a Joule heating source. As in the continuous setting, the main difficulty in the design of the discrete model comes from the Joule heating term. To cope with this issue, the Joule heating term is replaced with an equivalent key formulation on which a fully implicit scheme is constructed. Introducing a tricky cut-off function in the proposed discretization, we are able to recover the energy estimates on the discrete temperature. Another feature of this approach is that we dispense with the discrete maximum principle on the approximate electric potential, which in essence poses restrictive constraints on the mesh shape. Then, the existence of discrete solution to the coupled scheme is established. Compactness estimates are also shown. Under general assumptions on the data and meshes, the convergence of the numerical scheme is addressed. Numerical results are finally presented to show the efficiency and accuracy of the proposed methodology as well as the behavior of the implemented nonlinear solver.\u0000","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139759786","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

Discontinuous Galerkin methods for Stokes equations under power law slip boundary condition: a priori analysis 幂律滑移边界条件下斯托克斯方程的非连续伽勒金方法：先验分析

IF 1.7 2区数学

Calcolo Pub Date : 2024-02-01 DOI: 10.1007/s10092-023-00563-z

Djoko Kamdem Jules, Gidey Hagos, Koko Jonas, Sayah Toni

引用次数: 0

Optimal approximation of infinite-dimensional holomorphic functions 无限维全貌函数的最优逼近

IF 1.7 2区数学

Calcolo Pub Date : 2024-01-29 DOI: 10.1007/s10092-023-00565-x

Ben Adcock, Nick Dexter, Sebastian Moraga

{"title":"Optimal approximation of infinite-dimensional holomorphic functions","authors":"Ben Adcock, Nick Dexter, Sebastian Moraga","doi":"10.1007/s10092-023-00565-x","DOIUrl":"https://doi.org/10.1007/s10092-023-00565-x","url":null,"abstract":"Over the several decades, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples m. Our work focuses on providing theoretical approximation guarantees for the class of so-called ((varvec{b},varepsilon ))-holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of m-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds.","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139649608","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

Analytic regularity and solution approximation for a semilinear elliptic partial differential equation in a polygon 多边形中的半线性椭圆偏微分方程的解析正则性和解法近似

IF 1.7 2区数学

Calcolo Pub Date : 2024-01-11 DOI: 10.1007/s10092-023-00562-0

Yanchen He, Christoph Schwab

引用次数: 0

The Hermite-type virtual element method with interior penalty for the fourth-order elliptic problem 四阶椭圆问题的带内部惩罚的赫米特型虚拟元素法

IF 1.7 2区数学

Calcolo Pub Date : 2024-01-03 DOI: 10.1007/s10092-023-00555-z

Jikun Zhao, Teng Chen, Bei Zhang, Xiaojing Dong

引用次数: 0

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