A generalized Fourier transform by means of change of variables within multilinear approximation

Chevreuil, Mathilde, Slama, Myriam
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Abstract

The paper deals with approximations of periodic functions that play a significant role in harmonic analysis. The approach revisits the trigonometric polynomials, seen as combinations of functions, and proposes to extend the class of models of the combined functions to a wider class of functions. The key here is to use structured functions, that have low complexity, with suitable functional representation and adapted parametrizations for the approximation. Such representation enables to approximate multivariate functions with few eventually random samples. The new parametrization is determined automatically with a greedy procedure, and a low rank format is used for the approximation associated with each new parametrization. A supervised learning algorithm is used for the approximation of a function of multiple random variables in tree-based tensor format, here the particular Tensor Train format. Adaptive strategies using statistical error estimates are proposed for the selection of the underlying tensor bases and the ranks for the Tensor-Train format. The method is applied for the estimation of the wall pressure for a flow over a cylinder for a range of low to medium Reynolds numbers for which we observe two flow regimes: a laminar flow with periodic vortex shedding and a laminar boundary layer with a turbulent wake (sub-critic regime). The automatic re-parametrization enables here to take into account the specific periodic feature of the pressure.
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在多元线性近似中采用变量变换的广义傅里叶变换
本文讨论了在谐波分析中起重要作用的周期函数的近似。该方法重新审视了三角多项式,将其视为函数的组合,并提出将组合函数的模型类扩展到更广泛的函数类。这里的关键是使用结构化函数,它具有低复杂性,具有合适的函数表示和适合的参数化近似。这样的表示可以用很少的随机样本近似多元函数。通过贪心过程自动确定新的参数化,并使用低秩格式进行与每个新参数化相关的逼近。监督学习算法用于逼近树状张量格式的多个随机变量的函数,这里是特定的张量训练格式。提出了使用统计误差估计的自适应策略来选择底层张量基和张量-训练格式的秩。该方法用于估计在低至中等雷诺数范围内流过圆柱体的壁面压力,其中我们观察到两种流动形式:具有周期性涡脱落的层流和具有湍流尾迹的层流边界层(亚临界状态)。自动重新参数化使这里能够考虑到压力的特定周期性特征。
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来源期刊
Advanced Modeling and Simulation in Engineering Sciences
Advanced Modeling and Simulation in Engineering Sciences Engineering-Engineering (miscellaneous)
CiteScore
6.80
自引率
0.00%
发文量
22
审稿时长
30 weeks
期刊介绍: The research topics addressed by Advanced Modeling and Simulation in Engineering Sciences (AMSES) cover the vast domain of the advanced modeling and simulation of materials, processes and structures governed by the laws of mechanics. The emphasis is on advanced and innovative modeling approaches and numerical strategies. The main objective is to describe the actual physics of large mechanical systems with complicated geometries as accurately as possible using complex, highly nonlinear and coupled multiphysics and multiscale models, and then to carry out simulations with these complex models as rapidly as possible. In other words, this research revolves around efficient numerical modeling along with model verification and validation. Therefore, the corresponding papers deal with advanced modeling and simulation, efficient optimization, inverse analysis, data-driven computation and simulation-based control. These challenging issues require multidisciplinary efforts – particularly in modeling, numerical analysis and computer science – which are treated in this journal.
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