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Artificial neural networks with uniform norm-based loss functions 基于统一规范损失函数的人工神经网络
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-04-23 DOI: 10.1007/s10444-024-10124-9
Vinesha Peiris, Vera Roshchina, Nadezda Sukhorukova
{"title":"Artificial neural networks with uniform norm-based loss functions","authors":"Vinesha Peiris, Vera Roshchina, Nadezda Sukhorukova","doi":"10.1007/s10444-024-10124-9","DOIUrl":"https://doi.org/10.1007/s10444-024-10124-9","url":null,"abstract":"<p>We explore the potential for using a nonsmooth loss function based on the max-norm in the training of an artificial neural network without hidden layers. We hypothesise that this may lead to superior classification results in some special cases where the training data are either very small or the class size is disproportional. Our numerical experiments performed on a simple artificial neural network with no hidden layer appear to confirm our hypothesis.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140637705","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
The improvement of the truncated Euler-Maruyama method for non-Lipschitz stochastic differential equations 非 Lipschitz 随机微分方程的截断 Euler-Maruyama 方法的改进
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-04-22 DOI: 10.1007/s10444-024-10131-w
Weijun Zhan, Yuyuan Li
{"title":"The improvement of the truncated Euler-Maruyama method for non-Lipschitz stochastic differential equations","authors":"Weijun Zhan, Yuyuan Li","doi":"10.1007/s10444-024-10131-w","DOIUrl":"https://doi.org/10.1007/s10444-024-10131-w","url":null,"abstract":"<p>This paper is concerned with the numerical approximations for stochastic differential equations with non-Lipschitz drift or diffusion coefficients. A modified truncated Euler-Maruyama discretization scheme is developed. Moreover, by establishing the criteria on stochastic C-stability and B-consistency of the truncated Euler-Maruyama method, we obtain the strong convergence and the convergence rate of the numerical method. Finally, numerical examples are given to illustrate our theoretical results.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140632242","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
The adjoint double layer potential on smooth surfaces in $$mathbb {R}^3$$ and the Neumann problem $$mathbb{R}^3$$中光滑表面上的邻接双层势和诺伊曼问题
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-04-19 DOI: 10.1007/s10444-024-10111-0
J. Thomas Beale, Michael Storm, Svetlana Tlupova
{"title":"The adjoint double layer potential on smooth surfaces in $$mathbb {R}^3$$ and the Neumann problem","authors":"J. Thomas Beale, Michael Storm, Svetlana Tlupova","doi":"10.1007/s10444-024-10111-0","DOIUrl":"https://doi.org/10.1007/s10444-024-10111-0","url":null,"abstract":"<p>We present a simple yet accurate method to compute the adjoint double layer potential, which is used to solve the Neumann boundary value problem for Laplace’s equation in three dimensions. An expansion in curvilinear coordinates leads us to modify the expression for the adjoint double layer so that the singularity is reduced when evaluating the integral on the surface. Then, to regularize the integral, we multiply the Green’s function by a radial function with length parameter <span>(delta )</span> chosen so that the product is smooth. We show that a natural regularization has error <span>(O(delta ^3))</span>, and a simple modification improves the error to <span>(O(delta ^5))</span>. The integral is evaluated numerically without the need of special coordinates. We use this treatment of the adjoint double layer to solve the classical integral equation for the interior Neumann problem, altered to account for the solvability condition, and evaluate the solution on the boundary. Choosing <span>(delta = ch^{4/5})</span>, we find about <span>(O(h^4))</span> convergence in our examples, where <i>h</i> is the spacing in a background grid.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140620315","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
$$mathcal {H}_2$$ optimal rational approximation on general domains 一般域上的 $$mathcal {H}_2$$ 最佳理性逼近
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-04-18 DOI: 10.1007/s10444-024-10125-8
Alessandro Borghi, Tobias Breiten
{"title":"$$mathcal {H}_2$$ optimal rational approximation on general domains","authors":"Alessandro Borghi, Tobias Breiten","doi":"10.1007/s10444-024-10125-8","DOIUrl":"https://doi.org/10.1007/s10444-024-10125-8","url":null,"abstract":"<p>Optimal model reduction for large-scale linear dynamical systems is studied. In contrast to most existing works, the systems under consideration are not required to be stable, neither in discrete nor in continuous time. As a consequence, the underlying rational transfer functions are allowed to have poles in general domains in the complex plane. In particular, this covers the case of specific conservative partial differential equations such as the linear Schrödinger and the undamped linear wave equation with spectra on the imaginary axis. By an appropriate modification of the classical continuous time Hardy space <span>(varvec{mathcal {H}}_{varvec{2}})</span>, a new <span>(varvec{mathcal {H}}_{varvec{2}})</span>-like optimal model reduction problem is introduced and first-order optimality conditions are derived. As in the classical <span>(varvec{mathcal {H}}_{varvec{2}})</span> case, these conditions exhibit a rational Hermite interpolation structure for which an iterative model reduction algorithm is proposed. Numerical examples demonstrate the effectiveness of the new method.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140607911","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
Structured barycentric forms for interpolation-based data-driven reduced modeling of second-order systems 基于插值的数据驱动二阶系统还原建模的结构化巴里心形式
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-04-11 DOI: 10.1007/s10444-024-10118-7
Ion Victor Gosea, Serkan Gugercin, Steffen W. R. Werner
{"title":"Structured barycentric forms for interpolation-based data-driven reduced modeling of second-order systems","authors":"Ion Victor Gosea, Serkan Gugercin, Steffen W. R. Werner","doi":"10.1007/s10444-024-10118-7","DOIUrl":"https://doi.org/10.1007/s10444-024-10118-7","url":null,"abstract":"<p>An essential tool in data-driven modeling of dynamical systems from frequency response measurements is the barycentric form of the underlying rational transfer function. In this work, we propose structured barycentric forms for modeling dynamical systems with second-order time derivatives using their frequency domain input-output data. By imposing a set of interpolation conditions, the systems’ transfer functions are rewritten in different barycentric forms using different parametrizations. Loewner-like algorithms are developed for the explicit computation of second-order systems from data based on the developed barycentric forms. Numerical experiments show the performance of these new structured data-driven modeling methods compared to other interpolation-based data-driven modeling techniques from the literature.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140547669","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
Computing equivariant matrices on homogeneous spaces for geometric deep learning and automorphic Lie algebras 计算同质空间上的等变矩阵，用于几何深度学习和非定态李代数
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-04-11 DOI: 10.1007/s10444-024-10126-7
Vincent Knibbeler
{"title":"Computing equivariant matrices on homogeneous spaces for geometric deep learning and automorphic Lie algebras","authors":"Vincent Knibbeler","doi":"10.1007/s10444-024-10126-7","DOIUrl":"https://doi.org/10.1007/s10444-024-10126-7","url":null,"abstract":"<p>We develop an elementary method to compute spaces of equivariant maps from a homogeneous space <i>G</i>/<i>H</i> of a Lie group <i>G</i> to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant sections in homogeneous vector bundles, and take a special interest in the case where the fibres are algebras. These latter cases have a natural global algebra structure. We classify these automorphic algebras for the case where the homogeneous space has compact stabilisers. This work has applications in the theoretical development of geometric deep learning and also in the theory of automorphic Lie algebras.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140547642","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 simulation of resistance furnaces by using distributed and lumped models 利用分布式和集合式模型对电阻炉进行数值模拟
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-04-10 DOI: 10.1007/s10444-024-10120-z
A. Bermúdez, D. Gómez, D. González
{"title":"Numerical simulation of resistance furnaces by using distributed and lumped models","authors":"A. Bermúdez, D. Gómez, D. González","doi":"10.1007/s10444-024-10120-z","DOIUrl":"https://doi.org/10.1007/s10444-024-10120-z","url":null,"abstract":"<p>This work proposes a methodology that combines distributed and lumped models to simulate the current distribution within an indirect heat resistance furnace and, in particular, to calculate the current to be supplied for achieving a desired power output. The distributed model is a time-harmonic eddy current problem, which is solved numerically using the finite element method. The lumped model relies on calculating a reduced impedance associated with an equivalent circuit model. Numerical simulations and plant measurements demonstrate the effectiveness of this approach. The good correlation between the results indicates that this approximation is well-suited to support the design and improve the efficiency of the furnace in a short time.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140541673","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
Robust space-time finite element methods for parabolic distributed optimal control problems with energy regularization 带能量正则化的抛物分布式最优控制问题的鲁棒时空有限元方法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-04-10 DOI: 10.1007/s10444-024-10123-w
Ulrich Langer, Olaf Steinbach, Huidong Yang
{"title":"Robust space-time finite element methods for parabolic distributed optimal control problems with energy regularization","authors":"Ulrich Langer, Olaf Steinbach, Huidong Yang","doi":"10.1007/s10444-024-10123-w","DOIUrl":"https://doi.org/10.1007/s10444-024-10123-w","url":null,"abstract":"<p>As in our previous work (<i>SINUM</i> 59(2):660–674, 2021) we consider space-time tracking optimal control problems for linear parabolic initial boundary value problems that are given in the space-time cylinder <span>(Q = Omega times (0,T))</span>, and that are controlled by the right-hand side <span>(z_varrho )</span> from the Bochner space <span>(L^2(0,T;H^{-1}(Omega )))</span>. So it is natural to replace the usual <span>(L^2(Q))</span> norm regularization by the energy regularization in the <span>(L^2(0,T;H^{-1}(Omega )))</span> norm. We derive new a priori estimates for the error <span>(Vert widetilde{u}_{varrho h} - overline{u}Vert _{L^2(Q)})</span> between the computed state <span>(widetilde{u}_{varrho h})</span> and the desired state <span>(overline{u})</span> in terms of the regularization parameter <span>(varrho )</span> and the space-time finite element mesh size <i>h</i>, and depending on the regularity of the desired state <span>(overline{u})</span>. These new estimates lead to the optimal choice <span>(varrho = h^2)</span>. The approximate state <span>(widetilde{u}_{varrho h})</span> is computed by means of a space-time finite element method using piecewise linear and continuous basis functions on completely unstructured simplicial meshes for <i>Q</i>. The theoretical results are quantitatively illustrated by a series of numerical examples in two and three space dimensions. We also provide performance studies for different solvers.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140541246","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
Robust low tubal rank tensor recovery using discrete empirical interpolation method with optimized slice/feature selection 利用优化切片/特征选择的离散经验插值法进行稳健的低管阶张量恢复
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-04-06 DOI: 10.1007/s10444-024-10117-8
Salman Ahmadi-Asl, Anh-Huy Phan, Cesar F. Caiafa, Andrzej Cichocki
{"title":"Robust low tubal rank tensor recovery using discrete empirical interpolation method with optimized slice/feature selection","authors":"Salman Ahmadi-Asl, Anh-Huy Phan, Cesar F. Caiafa, Andrzej Cichocki","doi":"10.1007/s10444-024-10117-8","DOIUrl":"https://doi.org/10.1007/s10444-024-10117-8","url":null,"abstract":"<p>In this paper, we extend the Discrete Empirical Interpolation Method (DEIM) to the third-order tensor case based on the t-product and use it to select important/significant lateral and horizontal slices/features. The proposed Tubal DEIM (TDEIM) is investigated both theoretically and numerically. In particular, the details of the error bounds of the proposed TDEIM method are derived. The experimental results show that the TDEIM can provide more accurate approximations than the existing methods. An application of the proposed method to the supervised classification task is also presented.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140352136","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 block-randomized stochastic method with importance sampling for CP tensor decomposition 用于 CP 张量分解的带重要性采样的分块随机方法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-03-25 DOI: 10.1007/s10444-024-10119-6
Yajie Yu, Hanyu Li
{"title":"A block-randomized stochastic method with importance sampling for CP tensor decomposition","authors":"Yajie Yu, Hanyu Li","doi":"10.1007/s10444-024-10119-6","DOIUrl":"https://doi.org/10.1007/s10444-024-10119-6","url":null,"abstract":"<p>One popular way to compute the CANDECOMP/PARAFAC (CP) decomposition of a tensor is to transform the problem into a sequence of overdetermined least squares subproblems with Khatri-Rao product (KRP) structure involving factor matrices. In this work, based on choosing the factor matrix randomly, we propose a mini-batch stochastic gradient descent method with importance sampling for those special least squares subproblems. Two different sampling strategies are provided. They can avoid forming the full KRP explicitly and computing the corresponding probabilities directly. The adaptive step size version of the method is also given. For the proposed method, we present its theoretical properties and comprehensive numerical performance. The results on synthetic and real data show that our method is effective and efficient, and for unevenly distributed data, it performs better than the corresponding one in the literature.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140209678","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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