Acta Numerica最新文献_第3页

Turnpike in optimal control of PDEs, ResNets, and beyond PDE、ResNets及其他优化控制中的收费公路

IF 14.2 1区数学

Acta Numerica Pub Date : 2022-02-08 DOI: 10.1017/S0962492922000046

Borjan Geshkovski, E. Zuazua

{"title":"Turnpike in optimal control of PDEs, ResNets, and beyond","authors":"Borjan Geshkovski, E. Zuazua","doi":"10.1017/S0962492922000046","DOIUrl":"https://doi.org/10.1017/S0962492922000046","url":null,"abstract":"The turnpike property in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized mathematically, the turnpike theory corroborates insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states are close (often exponentially) most of the time, except near the initial and final times, to the optimal control and the corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corresponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade – the controllability of the underlying system is an important ingredient, and can even be used to devise simple turnpike-like strategies which are nearly optimal – and present several novel applications, including, among many others, the characterization of Hamilton–Jacobi–Bellman asymptotics, and stability estimates in deep learning via residual neural networks.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"31 1","pages":"135 - 263"},"PeriodicalIF":14.2,"publicationDate":"2022-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46313032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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: 14

Control of port-Hamiltonian differential-algebraic systems and applications 波特-哈密顿微分代数系统的控制及其应用

IF 14.2 1区数学

Acta Numerica Pub Date : 2022-01-17 DOI: 10.1017/S0962492922000083

V. Mehrmann, B. Unger

{"title":"Control of port-Hamiltonian differential-algebraic systems and applications","authors":"V. Mehrmann, B. Unger","doi":"10.1017/S0962492922000083","DOIUrl":"https://doi.org/10.1017/S0962492922000083","url":null,"abstract":"We discuss the modelling framework of port-Hamiltonian descriptor systems and their use in numerical simulation and control. The structure is ideal for automated network-based modelling since it is invariant under power-conserving interconnection, congruence transformations and Galerkin projection. Moreover, stability and passivity properties are easily shown. Condensed forms under orthogonal transformations present easy analysis tools for existence, uniqueness, regularity and numerical methods to check these properties. After recalling the concepts for general linear and nonlinear descriptor systems, we demonstrate that many difficulties that arise in general descriptor systems can be easily overcome within the port-Hamiltonian framework. The properties of port-Hamiltonian descriptor systems are analysed, and time discretization and numerical linear algebra techniques are discussed. Structure-preserving regularization procedures for descriptor systems are presented to make them suitable for simulation and control. Model reduction techniques that preserve the structure and stabilization and optimal control techniques are discussed. The properties of port-Hamiltonian descriptor systems and their use in modelling simulation and control methods are illustrated with several examples from different physical domains. The survey concludes with open problems and research topics that deserve further attention.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"32 1","pages":"395 - 515"},"PeriodicalIF":14.2,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46524517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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: 36

Asymptotic-preserving schemes for multiscale physical problems 多尺度物理问题的渐近保持格式

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-12-11 DOI: 10.1017/S0962492922000010

Shi Jin

引用次数: 25

ANU volume 30 Cover and Front matter 澳大利亚国立大学第30卷封面和封面问题

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-05-01 DOI: 10.1017/s096249292100009x

R. Altmann, P. Henning, D. Peterseim, P. Bartlett, A. Montanari, A. Rakhlin, Ronald A. DeVore, B. Hanin

引用次数: 0

Numerical homogenization beyond scale separation 超越尺度分离的数值均质化

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-05-01 DOI: 10.1017/S0962492921000015

R. Altmann, P. Henning, D. Peterseim

引用次数: 44

Optimal transportation, modelling and numerical simulation 最优运输，建模和数值模拟

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-05-01 DOI: 10.1017/S0962492921000040

J. Benamou

引用次数: 13

Tensors in computations 计算中的张量

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-05-01 DOI: 10.1017/S0962492921000076

Lek-Heng Lim

引用次数: 23

Learning physics-based models from data: perspectives from inverse problems and model reduction 从数据中学习基于物理的模型:从反问题和模型简化的角度

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-05-01 DOI: 10.1017/S0962492921000064

O. Ghattas, K. Willcox

{"title":"Learning physics-based models from data: perspectives from inverse problems and model reduction","authors":"O. Ghattas, K. Willcox","doi":"10.1017/S0962492921000064","DOIUrl":"https://doi.org/10.1017/S0962492921000064","url":null,"abstract":"This article addresses the inference of physics models from data, from the perspectives of inverse problems and model reduction. These fields develop formulations that integrate data into physics-based models while exploiting the fact that many mathematical models of natural and engineered systems exhibit an intrinsically low-dimensional solution manifold. In inverse problems, we seek to infer uncertain components of the inputs from observations of the outputs, while in model reduction we seek low-dimensional models that explicitly capture the salient features of the input–output map through approximation in a low-dimensional subspace. In both cases, the result is a predictive model that reflects data-driven learning yet deeply embeds the underlying physics, and thus can be used for design, control and decision-making, often with quantified uncertainties. We highlight recent developments in scalable and efficient algorithms for inverse problems and model reduction governed by large-scale models in the form of partial differential equations. Several illustrative applications to large-scale complex problems across different domains of science and engineering are provided.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"30 1","pages":"445 - 554"},"PeriodicalIF":14.2,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47260397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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: 69

Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation 无所畏惧:通过插补棱镜的深度学习的非凡数学现象

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-05-01 DOI: 10.1017/S0962492921000039

M. Belkin

引用次数: 116

ANU volume 30 Cover and Back matter 澳大利亚国立大学第30卷封面和封底

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-05-01 DOI: 10.1017/s0962492921000106

引用次数: 0