Pub Date : 2013-04-19DOI: 10.1017/S096249291500001X
M. Giles
Monte Carlo methods are a very general and useful approach for the estimation of expectations arising from stochastic simulation. However, they can be computationally expensive, particularly when the cost of generating individual stochastic samples is very high, as in the case of stochastic PDEs. Multilevel Monte Carlo is a recently developed approach which greatly reduces the computational cost by performing most simulations with low accuracy at a correspondingly low cost, with relatively few simulations being performed at high accuracy and a high cost. In this article, we review the ideas behind the multilevel Monte Carlo method, and various recent generalizations and extensions, and discuss a number of applications which illustrate the flexibility and generality of the approach and the challenges in developing more efficient implementations with a faster rate of convergence of the multilevel correction variance.
{"title":"Multilevel Monte Carlo methods","authors":"M. Giles","doi":"10.1017/S096249291500001X","DOIUrl":"https://doi.org/10.1017/S096249291500001X","url":null,"abstract":"Monte Carlo methods are a very general and useful approach for the estimation of expectations arising from stochastic simulation. However, they can be computationally expensive, particularly when the cost of generating individual stochastic samples is very high, as in the case of stochastic PDEs. Multilevel Monte Carlo is a recently developed approach which greatly reduces the computational cost by performing most simulations with low accuracy at a correspondingly low cost, with relatively few simulations being performed at high accuracy and a high cost. In this article, we review the ideas behind the multilevel Monte Carlo method, and various recent generalizations and extensions, and discuss a number of applications which illustrate the flexibility and generality of the approach and the challenges in developing more efficient implementations with a faster rate of convergence of the multilevel correction variance.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"24 1","pages":"259 - 328"},"PeriodicalIF":14.2,"publicationDate":"2013-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S096249291500001X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57446239","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 : 2013-04-02DOI: 10.1017/S0962492913000056
G. Dziuk, C. M. Elliott
In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples.
{"title":"Finite element methods for surface PDEs*","authors":"G. Dziuk, C. M. Elliott","doi":"10.1017/S0962492913000056","DOIUrl":"https://doi.org/10.1017/S0962492913000056","url":null,"abstract":"In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"91 1","pages":"289 - 396"},"PeriodicalIF":14.2,"publicationDate":"2013-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492913000056","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445149","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 : 2013-04-02DOI: 10.1017/S0962492913000044
J. Dick, F. Kuo, I. Sloan
This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s, where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.
{"title":"High-dimensional integration: The quasi-Monte Carlo way*†","authors":"J. Dick, F. Kuo, I. Sloan","doi":"10.1017/S0962492913000044","DOIUrl":"https://doi.org/10.1017/S0962492913000044","url":null,"abstract":"This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s, where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"22 1","pages":"133 - 288"},"PeriodicalIF":14.2,"publicationDate":"2013-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492913000044","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445076","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 : 2013-04-02DOI: 10.1017/S096249291300007X
Y. Nesterov, A. Nemirovski
In the past decade, problems related to l1/nuclear norm minimization have attracted much attention in the signal processing, machine learning and optimization communities. In this paper, devoted to l1/nuclear norm minimization as ‘optimization beasts’, we give a detailed description of two attractive first-order optimization techniques for solving problems of this type. The first one, aimed primarily at lasso-type problems, comprises fast gradient methods applied to composite minimization formulations. The second approach, aimed at Dantzig-selector-type problems, utilizes saddle-point first-order algorithms and reformulation of the problem of interest as a generalized bilinear saddle-point problem. For both approaches, we give complete and detailed complexity analyses and discuss the application domains.
{"title":"On first-order algorithms for l1/nuclear norm minimization","authors":"Y. Nesterov, A. Nemirovski","doi":"10.1017/S096249291300007X","DOIUrl":"https://doi.org/10.1017/S096249291300007X","url":null,"abstract":"In the past decade, problems related to l1/nuclear norm minimization have attracted much attention in the signal processing, machine learning and optimization communities. In this paper, devoted to l1/nuclear norm minimization as ‘optimization beasts’, we give a detailed description of two attractive first-order optimization techniques for solving problems of this type. The first one, aimed primarily at lasso-type problems, comprises fast gradient methods applied to composite minimization formulations. The second approach, aimed at Dantzig-selector-type problems, utilizes saddle-point first-order algorithms and reformulation of the problem of interest as a generalized bilinear saddle-point problem. For both approaches, we give complete and detailed complexity analyses and discuss the application domains.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"22 1","pages":"509 - 575"},"PeriodicalIF":14.2,"publicationDate":"2013-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S096249291300007X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445259","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 : 2013-04-02DOI: 10.1017/S0962492913000032
P. Belotti, C. Kirches, S. Leyffer, Jeff T. Linderoth, James R. Luedtke, Ashutosh Mahajan
Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems. Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques. Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations. We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.
{"title":"Mixed-integer nonlinear optimization*†","authors":"P. Belotti, C. Kirches, S. Leyffer, Jeff T. Linderoth, James R. Luedtke, Ashutosh Mahajan","doi":"10.1017/S0962492913000032","DOIUrl":"https://doi.org/10.1017/S0962492913000032","url":null,"abstract":"Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems. Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques. Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations. We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"22 1","pages":"1 - 131"},"PeriodicalIF":14.2,"publicationDate":"2013-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492913000032","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445475","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 : 2013-04-02DOI: 10.1017/S0962492913000068
M. Luskin, C. Ortner
Atomistic-to-continuum (a/c) coupling methods are a class of computational multiscale schemes that combine the accuracy of atomistic models with the efficiency of continuum elasticity. They are increasingly being utilized in materials science to study the fundamental mechanisms of material failure such as crack propagation and plasticity, which are governed by the interaction between crystal defects and long-range elastic fields. In the construction of a/c coupling methods, various approximation errors are committed. A rigorous numerical analysis approach that classifies and quantifies these errors can give confidence in the simulation results, as well as enable optimization of the numerical methods for accuracy and computational cost. In this article, we present such a numerical analysis framework, which is inspired by recent research activity.
{"title":"Atomistic-to-continuum coupling","authors":"M. Luskin, C. Ortner","doi":"10.1017/S0962492913000068","DOIUrl":"https://doi.org/10.1017/S0962492913000068","url":null,"abstract":"Atomistic-to-continuum (a/c) coupling methods are a class of computational multiscale schemes that combine the accuracy of atomistic models with the efficiency of continuum elasticity. They are increasingly being utilized in materials science to study the fundamental mechanisms of material failure such as crack propagation and plasticity, which are governed by the interaction between crystal defects and long-range elastic fields. In the construction of a/c coupling methods, various approximation errors are committed. A rigorous numerical analysis approach that classifies and quantifies these errors can give confidence in the simulation results, as well as enable optimization of the numerical methods for accuracy and computational cost. In this article, we present such a numerical analysis framework, which is inspired by recent research activity.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"22 1","pages":"397 - 508"},"PeriodicalIF":14.2,"publicationDate":"2013-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492913000068","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445235","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 : 2012-04-19DOI: 10.1017/s0962492911999978
S. Chandler-Wilde, I. Graham, S. Langdon, O. Pironneau
{"title":"ANU volume 21 Cover and Front matter","authors":"S. Chandler-Wilde, I. Graham, S. Langdon, O. Pironneau","doi":"10.1017/s0962492911999978","DOIUrl":"https://doi.org/10.1017/s0962492911999978","url":null,"abstract":"","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"21 1","pages":"f1 - f6"},"PeriodicalIF":14.2,"publicationDate":"2012-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/s0962492911999978","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57444997","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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