Acta NumericaPub Date : 2015-02-24DOI: 10.1017/S0962492915000033
A. Cohen, R. DeVore
{"title":"Approximation of high-dimensional parametric PDEs *","authors":"A. Cohen, R. DeVore","doi":"10.1017/S0962492915000033","DOIUrl":"https://doi.org/10.1017/S0962492915000033","url":null,"abstract":"Parametrized families of PDEs arise in various contexts such as inverse problems, control and optimization, risk assessment, and uncertainty quantification. In most of these applications, the number of parameters is large or perhaps even infinite. Thus, the development of numerical methods for these parametric problems is faced with the possible curse of dimensionality. This article is directed at (i) identifying and understanding which properties of parametric equations allow one to avoid this curse and (ii) developing and analysing effective numerical methods which fully exploit these properties and, in turn, are immune to the growth in dimensionality. Part I of this article studies the smoothness and approximability of the solution map, that is, the map $amapsto u(a)$ , where $a$ is the parameter value and $u(a)$ is the corresponding solution to the PDE. It is shown that for many relevant parametric PDEs, the parametric smoothness of this map is typically holomorphic and also highly anisotropic, in that the relevant parameters are of widely varying importance in describing the solution. These two properties are then exploited to establish convergence rates of $n$ -term approximations to the solution map, for which each term is separable in the parametric and physical variables. These results reveal that, at least on a theoretical level, the solution map can be well approximated by discretizations of moderate complexity, thereby showing how the curse of dimensionality is broken. This theoretical analysis is carried out through concepts of approximation theory such as best $n$ -term approximation, sparsity, and $n$ -widths. These notions determine a priori the best possible performance of numerical methods and thus serve as a benchmark for concrete algorithms. Part II of this article turns to the development of numerical algorithms based on the theoretically established sparse separable approximations. The numerical methods studied fall into two general categories. The first uses polynomial expansions in terms of the parameters to approximate the solution map. The second one searches for suitable low-dimensional spaces for simultaneously approximating all members of the parametric family. The numerical implementation of these approaches is carried out through adaptive and greedy algorithms. An a priori analysis of the performance of these algorithms establishes how well they meet the theoretical benchmarks.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"24 1","pages":"1 - 159"},"PeriodicalIF":14.2,"publicationDate":"2015-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492915000033","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57446312","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}
Acta NumericaPub Date : 2015-01-01DOI: 10.1017/S0962492915000057
{"title":"Contents List and front matter/prelims","authors":"","doi":"10.1017/S0962492915000057","DOIUrl":"https://doi.org/10.1017/S0962492915000057","url":null,"abstract":"","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"23 1","pages":"f1-f6"},"PeriodicalIF":14.2,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74242543","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}
Acta NumericaPub Date : 2014-05-01DOI: 10.1017/S0962492914000087
W. Hackbusch
{"title":"Numerical tensor calculus*","authors":"W. Hackbusch","doi":"10.1017/S0962492914000087","DOIUrl":"https://doi.org/10.1017/S0962492914000087","url":null,"abstract":"The usual large-scale discretizations are applied to two or three spatial dimensions. The standard methods fail for higher dimensions because the data size increases exponentially with the dimension. In the case of a regular grid with n grid points per direction, a spatial dimension d yields nd grid points. A grid function defined on such a grid is an example of a tensor of order d. Here, suitable tensor formats help, since they try to approximate these huge objects by a much smaller number of parameters, which increases only linearly in d. In this way, data of size nd = 10001000 can also be treated. This paper introduces the algebraic and analytical aspects of tensor spaces. The main part concerns the numerical representation of tensors and the numerical performance of tensor operations.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"54 29 1","pages":"651 - 742"},"PeriodicalIF":14.2,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492914000087","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445819","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}
Acta NumericaPub Date : 2014-05-01DOI: 10.1017/S0962492914000038
Grey Ballard, E. Carson, J. Demmel, M. Hoemmen, Nicholas Knight, O. Schwartz
{"title":"Communication lower bounds and optimal algorithms for numerical linear algebra*†","authors":"Grey Ballard, E. Carson, J. Demmel, M. Hoemmen, Nicholas Knight, O. Schwartz","doi":"10.1017/S0962492914000038","DOIUrl":"https://doi.org/10.1017/S0962492914000038","url":null,"abstract":"The traditional metric for the efficiency of a numerical algorithm has been the number of arithmetic operations it performs. Technological trends have long been reducing the time to perform an arithmetic operation, so it is no longer the bottleneck in many algorithms; rather, communication, or moving data, is the bottleneck. This motivates us to seek algorithms that move as little data as possible, either between levels of a memory hierarchy or between parallel processors over a network. In this paper we summarize recent progress in three aspects of this problem. First we describe lower bounds on communication. Some of these generalize known lower bounds for dense classical (O(n3)) matrix multiplication to all direct methods of linear algebra, to sequential and parallel algorithms, and to dense and sparse matrices. We also present lower bounds for Strassen-like algorithms, and for iterative methods, in particular Krylov subspace methods applied to sparse matrices. Second, we compare these lower bounds to widely used versions of these algorithms, and note that these widely used algorithms usually communicate asymptotically more than is necessary. Third, we identify or invent new algorithms for most linear algebra problems that do attain these lower bounds, and demonstrate large speed-ups in theory and practice.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"23 1","pages":"1 - 155"},"PeriodicalIF":14.2,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492914000038","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445585","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}
Acta NumericaPub Date : 2014-05-01DOI: 10.1017/S096249291400004X
L. Veiga, A. Buffa, G. Sangalli, Rafael Vázquez Hernández
{"title":"Mathematical analysis of variational isogeometric methods*","authors":"L. Veiga, A. Buffa, G. Sangalli, Rafael Vázquez Hernández","doi":"10.1017/S096249291400004X","DOIUrl":"https://doi.org/10.1017/S096249291400004X","url":null,"abstract":"This review paper collects several results that form part of the theoretical foundation of isogeometric methods. We analyse variational techniques for the numerical resolution of PDEs based on splines or NURBS and we provide optimal approximation and error estimates in several cases of interest. The theory presented also includes estimates for T-splines, which are an extension of splines allowing for local refinement. In particular, we focus our attention on elliptic and saddle point problems, and we define spline edge and face elements. Our theoretical results are demonstrated by a rich set of numerical examples. Finally, we discuss implementation and efficiency together with preconditioning issues for the final linear system.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"23 1","pages":"157 - 287"},"PeriodicalIF":14.2,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S096249291400004X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445646","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}
Acta NumericaPub Date : 2014-05-01DOI: 10.1017/S0962492914000063
G. Dimarco, L. Pareschi
{"title":"Numerical methods for kinetic equations*","authors":"G. Dimarco, L. Pareschi","doi":"10.1017/S0962492914000063","DOIUrl":"https://doi.org/10.1017/S0962492914000063","url":null,"abstract":"In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"125 8 1","pages":"369 - 520"},"PeriodicalIF":14.2,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492914000063","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445687","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}
Acta NumericaPub Date : 2014-05-01DOI: 10.1017/S0962492914000075
M. Gunzburger, C. Webster, Guannan Zhang
{"title":"Stochastic finite element methods for partial differential equations with random input data*","authors":"M. Gunzburger, C. Webster, Guannan Zhang","doi":"10.1017/S0962492914000075","DOIUrl":"https://doi.org/10.1017/S0962492914000075","url":null,"abstract":"The quantification of probabilistic uncertainties in the outputs of physical, biological, and social systems governed by partial differential equations with random inputs require, in practice, the discretization of those equations. Stochastic finite element methods refer to an extensive class of algorithms for the approximate solution of partial differential equations having random input data, for which spatial discretization is effected by a finite element method. Fully discrete approximations require further discretization with respect to solution dependences on the random variables. For this purpose several approaches have been developed, including intrusive approaches such as stochastic Galerkin methods, for which the physical and probabilistic degrees of freedom are coupled, and non-intrusive approaches such as stochastic sampling and interpolatory-type stochastic collocation methods, for which the physical and probabilistic degrees of freedom are uncoupled. All these method classes are surveyed in this article, including some novel recent developments. Details about the construction of the various algorithms and about theoretical error estimates and complexity analyses of the algorithms are provided. Throughout, numerical examples are used to illustrate the theoretical results and to provide further insights into the methodologies.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"23 1","pages":"521 - 650"},"PeriodicalIF":14.2,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492914000075","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445739","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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