Advances in Computational Mathematics最新文献_第5页

Extrapolated regularization of nearly singular integrals on surfaces 曲面上近奇异积分的外推正则化

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-07-01 DOI: 10.1007/s10444-024-10161-4

J. Thomas Beale, Svetlana Tlupova

{"title":"Extrapolated regularization of nearly singular integrals on surfaces","authors":"J. Thomas Beale, Svetlana Tlupova","doi":"10.1007/s10444-024-10161-4","DOIUrl":"https://doi.org/10.1007/s10444-024-10161-4","url":null,"abstract":"We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral equation when one surface is close to another or to obtain values at grid points. We replace the singular kernel with a regularized version having a length parameter (delta ) in order to control discretization error. Analysis near the singularity leads to an expression for the error due to regularization which has terms with unknown coefficients multiplying known quantities. By computing the integral with three choices of (delta ), we can solve for an extrapolated value that has regularization error reduced to (O(delta ^5)), uniformly for target points on or near the surface. In examples with (delta /h) constant and moderate resolution, we observe total error about (O(h^5)) close to the surface. For convergence as (h rightarrow 0), we can choose (delta ) proportional to (h^q) with (q < 1) to ensure the discretization error is dominated by the regularization error. With (q = 4/5), we find errors about (O(h^4)). For harmonic potentials, we extend the approach to a version with (O(delta ^7)) regularization; it typically has smaller errors, but the order of accuracy is less predictable.","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141489637","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

Stochastic modeling of stationary scalar Gaussian processes in continuous time from autocorrelation data 根据自相关数据建立连续时间内静止标量高斯过程的随机模型

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-06-24 DOI: 10.1007/s10444-024-10150-7

Martin Hanke

{"title":"Stochastic modeling of stationary scalar Gaussian processes in continuous time from autocorrelation data","authors":"Martin Hanke","doi":"10.1007/s10444-024-10150-7","DOIUrl":"https://doi.org/10.1007/s10444-024-10150-7","url":null,"abstract":"We consider the problem of constructing a vector-valued linear Markov process in continuous time, such that its first coordinate is in good agreement with given samples of the scalar autocorrelation function of an otherwise unknown stationary Gaussian process. This problem has intimate connections to the computation of a passive reduced model of a deterministic time-invariant linear system from given output data in the time domain. We construct the stochastic model in two steps. First, we employ the AAA algorithm to determine a rational function which interpolates the z-transform of the discrete data on the unit circle and use this function to assign the poles of the transfer function of the reduced model. Second, we choose the associated residues as the minimizers of a linear inequality constrained least squares problem which ensures the positivity of the transfer function’s real part for large frequencies. We apply this method to compute extended Markov models for stochastic processes obtained from generalized Langevin dynamics in statistical physics. Numerical examples demonstrate that the algorithm succeeds in determining passive reduced models and that the associated Markov processes provide an excellent match of the given data.","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141444892","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

On relaxed inertial projection and contraction algorithms for solving monotone inclusion problems 论解决单调包含问题的松弛惯性投影和收缩算法

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-06-18 DOI: 10.1007/s10444-024-10156-1

Bing Tan, Xiaolong Qin

引用次数: 0

Numerical methods for forward fractional Feynman–Kac equation 前向分数费曼-卡克方程的数值方法

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-06-10 DOI: 10.1007/s10444-024-10152-5

Daxin Nie, Jing Sun, Weihua Deng

引用次数: 0

An efficient rotational pressure-correction schemes for 2D/3D closed-loop geothermal system 二维/三维闭环地热系统的高效旋转压力校正方案

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-06-10 DOI: 10.1007/s10444-024-10154-3

Jian Li, Jiawei Gao, Yi Qin

引用次数: 0

Semi-active damping optimization of vibrational systems using the reduced basis method 使用还原法优化振动系统的半主动阻尼

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-05-31 DOI: 10.1007/s10444-024-10141-8

Jennifer Przybilla, Igor Pontes Duff, Peter Benner

{"title":"Semi-active damping optimization of vibrational systems using the reduced basis method","authors":"Jennifer Przybilla, Igor Pontes Duff, Peter Benner","doi":"10.1007/s10444-024-10141-8","DOIUrl":"https://doi.org/10.1007/s10444-024-10141-8","url":null,"abstract":"In this article, we consider vibrational systems with semi-active damping that are described by a second-order model. In order to minimize the influence of external inputs to the system response, we are optimizing some damping values. As minimization criterion, we evaluate the energy response, that is the (mathcal {H}_2)-norm of the corresponding transfer function of the system. Computing the energy response includes solving Lyapunov equations for different damping parameters. Hence, the minimization process leads to high computational costs if the system is of large dimension. We present two techniques that reduce the optimization problem by applying the reduced basis method to the corresponding parametric Lyapunov equations. In the first method, we determine a reduced solution space on which the Lyapunov equations and hence the resulting energy response values are computed approximately in a reasonable time. The second method includes the reduced basis method in the minimization process. To evaluate the quality of the approximations, we introduce error estimators that evaluate the error in the controllability Gramians and the energy response. Finally, we illustrate the advantages of our methods by applying them to two different examples.","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141185288","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 rotational pressure-correction discontinuous Galerkin scheme for the Cahn-Hilliard-Darcy-Stokes system 卡恩-希利亚德-达西-斯托克斯系统的旋转压力校正非连续伽勒金方案

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-05-30 DOI: 10.1007/s10444-024-10151-6

Meiting Wang, Guang-an Zou, Jian Li

引用次数: 0

Approximation in the extended functional tensor train format 扩展功能张量列车格式的近似值

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-05-28 DOI: 10.1007/s10444-024-10140-9

Christoph Strössner, Bonan Sun, Daniel Kressner

引用次数: 0

Variable transformations in combination with wavelets and ANOVA for high-dimensional approximation 将变量变换与小波和方差分析结合起来进行高维逼近

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-05-23 DOI: 10.1007/s10444-024-10147-2

Daniel Potts, Laura Weidensager

{"title":"Variable transformations in combination with wavelets and ANOVA for high-dimensional approximation","authors":"Daniel Potts, Laura Weidensager","doi":"10.1007/s10444-024-10147-2","DOIUrl":"https://doi.org/10.1007/s10444-024-10147-2","url":null,"abstract":"We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low-dimensional variable interactions. Compactly supported periodic Chui-Wang wavelets are used for the tensorized hyperbolic wavelet basis on the torus. With a variable transformation, we are able to transform the approximation rates and fast algorithms from the torus to other domains. We perform and analyze scattered data approximation for smooth but arbitrary density functions by using a least squares method. The corresponding system matrix is sparse due to the compact support of the wavelets, which leads to a significant acceleration of the matrix vector multiplication. For non-periodic functions, we propose a new extension method. A proper choice of the extension parameter together with the piecewise polynomial Chui-Wang wavelets extends the functions appropriately. In every case, we are able to bound the approximation error with high probability. Additionally, if the function has a low effective dimension (i.e., only interactions of a few variables), we qualitatively determine the variable interactions and omit ANOVA terms with low variance in a second step in order to decrease the approximation error. This allows us to suggest an adapted model for the approximation. Numerical results show the efficiency of the proposed method.","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141085204","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

An optimal control framework for adaptive neural ODEs 自适应神经 ODE 的优化控制框架

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2024-05-23 DOI: 10.1007/s10444-024-10149-0

Joubine Aghili, Olga Mula

{"title":"An optimal control framework for adaptive neural ODEs","authors":"Joubine Aghili, Olga Mula","doi":"10.1007/s10444-024-10149-0","DOIUrl":"https://doi.org/10.1007/s10444-024-10149-0","url":null,"abstract":"In recent years, the notion of neural ODEs has connected deep learning with the field of ODEs and optimal control. In this setting, neural networks are defined as the mapping induced by the corresponding time-discretization scheme of a given ODE. The learning task consists in finding the ODE parameters as the optimal values of a sampled loss minimization problem. In the limit of infinite time steps, and data samples, we obtain a notion of continuous formulation of the problem. The practical implementation involves two discretization errors: a sampling error and a time-discretization error. In this work, we develop a general optimal control framework to analyze the interplay between the above two errors. We prove that to approximate the solution of the fully continuous problem at a certain accuracy, we not only need a minimal number of training samples, but also need to solve the control problem on the sampled loss function with some minimal accuracy. The theoretical analysis allows us to develop rigorous adaptive schemes in time and sampling, and gives rise to a notion of adaptive neural ODEs. The performance of the approach is illustrated in several numerical examples.","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141085343","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