Pub Date : 2022-02-08DOI: 10.1017/S0962492922000046
Borjan Geshkovski, E. Zuazua
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.
{"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}
Pub Date : 2022-01-17DOI: 10.1017/S0962492922000083
V. Mehrmann, B. Unger
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.
{"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}
Pub Date : 2021-12-11DOI: 10.1017/S0962492922000010
Shi Jin
We present the asymptotic transitions from microscopic to macroscopic physics, their computational challenges and the asymptotic-preserving (AP) strategies to compute multiscale physical problems efficiently. Specifically, we will first study the asymptotic transition from quantum to classical mechanics, from classical mechanics to kinetic theory, and then from kinetic theory to hydrodynamics. We then review some representative AP schemes that mimic these asymptotic transitions at the discrete level, and hence can be used crossing scales and, in particular, capture the macroscopic behaviour without resolving the microscopic physical scale numerically.
{"title":"Asymptotic-preserving schemes for multiscale physical problems","authors":"Shi Jin","doi":"10.1017/S0962492922000010","DOIUrl":"https://doi.org/10.1017/S0962492922000010","url":null,"abstract":"We present the asymptotic transitions from microscopic to macroscopic physics, their computational challenges and the asymptotic-preserving (AP) strategies to compute multiscale physical problems efficiently. Specifically, we will first study the asymptotic transition from quantum to classical mechanics, from classical mechanics to kinetic theory, and then from kinetic theory to hydrodynamics. We then review some representative AP schemes that mimic these asymptotic transitions at the discrete level, and hence can be used crossing scales and, in particular, capture the macroscopic behaviour without resolving the microscopic physical scale numerically.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"31 1","pages":"415 - 489"},"PeriodicalIF":14.2,"publicationDate":"2021-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43359804","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}
Pub Date : 2021-05-01DOI: 10.1017/s096249292100009x
R. Altmann, P. Henning, D. Peterseim, P. Bartlett, A. Montanari, A. Rakhlin, Ronald A. DeVore, B. Hanin
{"title":"ANU volume 30 Cover and Front matter","authors":"R. Altmann, P. Henning, D. Peterseim, P. Bartlett, A. Montanari, A. Rakhlin, Ronald A. DeVore, B. Hanin","doi":"10.1017/s096249292100009x","DOIUrl":"https://doi.org/10.1017/s096249292100009x","url":null,"abstract":"","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"30 1","pages":"f1 - f6"},"PeriodicalIF":14.2,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47797032","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}
Pub Date : 2021-05-01DOI: 10.1017/S0962492921000015
R. Altmann, P. Henning, D. Peterseim
Numerical homogenization is a methodology for the computational solution of multiscale partial differential equations. It aims at reducing complex large-scale problems to simplified numerical models valid on some target scale of interest, thereby accounting for the impact of features on smaller scales that are otherwise not resolved. While constructive approaches in the mathematical theory of homogenization are restricted to problems with a clear scale separation, modern numerical homogenization methods can accurately handle problems with a continuum of scales. This paper reviews such approaches embedded in a historical context and provides a unified variational framework for their design and numerical analysis. Apart from prototypical elliptic model problems, the class of partial differential equations covered here includes wave scattering in heterogeneous media and serves as a template for more general multi-physics problems.
{"title":"Numerical homogenization beyond scale separation","authors":"R. Altmann, P. Henning, D. Peterseim","doi":"10.1017/S0962492921000015","DOIUrl":"https://doi.org/10.1017/S0962492921000015","url":null,"abstract":"Numerical homogenization is a methodology for the computational solution of multiscale partial differential equations. It aims at reducing complex large-scale problems to simplified numerical models valid on some target scale of interest, thereby accounting for the impact of features on smaller scales that are otherwise not resolved. While constructive approaches in the mathematical theory of homogenization are restricted to problems with a clear scale separation, modern numerical homogenization methods can accurately handle problems with a continuum of scales. This paper reviews such approaches embedded in a historical context and provides a unified variational framework for their design and numerical analysis. Apart from prototypical elliptic model problems, the class of partial differential equations covered here includes wave scattering in heterogeneous media and serves as a template for more general multi-physics problems.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"30 1","pages":"1 - 86"},"PeriodicalIF":14.2,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48106578","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}
Pub Date : 2021-05-01DOI: 10.1017/S0962492921000040
J. Benamou
We present an overviewof the basic theory, modern optimal transportation extensions and recent algorithmic advances. Selected modelling and numerical applications illustrate the impact of optimal transportation in numerical analysis.
{"title":"Optimal transportation, modelling and numerical simulation","authors":"J. Benamou","doi":"10.1017/S0962492921000040","DOIUrl":"https://doi.org/10.1017/S0962492921000040","url":null,"abstract":"We present an overviewof the basic theory, modern optimal transportation extensions and recent algorithmic advances. Selected modelling and numerical applications illustrate the impact of optimal transportation in numerical analysis.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"30 1","pages":"249 - 325"},"PeriodicalIF":14.2,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41848083","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}
Pub Date : 2021-05-01DOI: 10.1017/S0962492921000076
Lek-Heng Lim
The notion of a tensor captures three great ideas: equivariance, multilinearity, separability. But trying to be three things at once makes the notion difficult to understand. We will explain tensors in an accessible and elementary way through the lens of linear algebra and numerical linear algebra, elucidated with examples from computational and applied mathematics.
{"title":"Tensors in computations","authors":"Lek-Heng Lim","doi":"10.1017/S0962492921000076","DOIUrl":"https://doi.org/10.1017/S0962492921000076","url":null,"abstract":"The notion of a tensor captures three great ideas: equivariance, multilinearity, separability. But trying to be three things at once makes the notion difficult to understand. We will explain tensors in an accessible and elementary way through the lens of linear algebra and numerical linear algebra, elucidated with examples from computational and applied mathematics.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"30 1","pages":"555 - 764"},"PeriodicalIF":14.2,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44930203","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}
Pub Date : 2021-05-01DOI: 10.1017/S0962492921000064
O. Ghattas, K. Willcox
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.
{"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}
Pub Date : 2021-05-01DOI: 10.1017/S0962492921000039
M. Belkin
In the past decade the mathematical theory of machine learning has lagged far behind the triumphs of deep neural networks on practical challenges. However, the gap between theory and practice is gradually starting to close. In this paper I will attempt to assemble some pieces of the remarkable and still incomplete mathematical mosaic emerging from the efforts to understand the foundations of deep learning. The two key themes will be interpolation and its sibling over-parametrization. Interpolation corresponds to fitting data, even noisy data, exactly. Over-parametrization enables interpolation and provides flexibility to select a suitable interpolating model. As we will see, just as a physical prism separates colours mixed within a ray of light, the figurative prism of interpolation helps to disentangle generalization and optimization properties within the complex picture of modern machine learning. This article is written in the belief and hope that clearer understanding of these issues will bring us a step closer towards a general theory of deep learning and machine learning.
{"title":"Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation","authors":"M. Belkin","doi":"10.1017/S0962492921000039","DOIUrl":"https://doi.org/10.1017/S0962492921000039","url":null,"abstract":"In the past decade the mathematical theory of machine learning has lagged far behind the triumphs of deep neural networks on practical challenges. However, the gap between theory and practice is gradually starting to close. In this paper I will attempt to assemble some pieces of the remarkable and still incomplete mathematical mosaic emerging from the efforts to understand the foundations of deep learning. The two key themes will be interpolation and its sibling over-parametrization. Interpolation corresponds to fitting data, even noisy data, exactly. Over-parametrization enables interpolation and provides flexibility to select a suitable interpolating model. As we will see, just as a physical prism separates colours mixed within a ray of light, the figurative prism of interpolation helps to disentangle generalization and optimization properties within the complex picture of modern machine learning. This article is written in the belief and hope that clearer understanding of these issues will bring us a step closer towards a general theory of deep learning and machine learning.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"30 1","pages":"203 - 248"},"PeriodicalIF":14.2,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46793783","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}
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