Acta Numerica最新文献

{"title":"Numerical algebraic geometry and algebraic kinematics","authors":"C. Wampler, A. Sommese","doi":"10.1017/S0962492911000067","DOIUrl":"https://doi.org/10.1017/S0962492911000067","url":null,"abstract":"In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"20 1","pages":"469 - 567"},"PeriodicalIF":14.2,"publicationDate":"2011-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492911000067","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57444935","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}

{"title":"Variationally consistent discretization schemes and numerical algorithms for contact problems*","authors":"B. Wohlmuth","doi":"10.1017/S0962492911000079","DOIUrl":"https://doi.org/10.1017/S0962492911000079","url":null,"abstract":"We consider variationally consistent discretization schemes for mechanical contact problems. Most of the results can also be applied to other variational inequalities, such as those for phase transition problems in porous media, for plasticity or for option pricing applications from finance. The starting point is to weakly incorporate the constraint into the setting and to reformulate the inequality in the displacement in terms of a saddle-point problem. Here, the Lagrange multiplier represents the surface forces, and the constraints are restricted to the boundary of the simulation domain. Having a uniform inf-sup bound, one can then establish optimal low-order a priori convergence rates for the discretization error in the primal and dual variables. In addition to the abstract framework of linear saddle-point theory, complementarity terms have to be taken into account. The resulting inequality system is solved by rewriting it equivalently by means of the non-linear complementarity function as a system of equations. Although it is not differentiable in the classical sense, semi-smooth Newton methods, yielding super-linear convergence rates, can be applied and easily implemented in terms of a primal–dual active set strategy. Quite often the solution of contact problems has a low regularity, and the efficiency of the approach can be improved by using adaptive refinement techniques. Different standard types, such as residual- and equilibrated-based a posteriori error estimators, can be designed based on the interpretation of the dual variable as Neumann boundary condition. For the fully dynamic setting it is of interest to apply energy-preserving time-integration schemes. However, the differential algebraic character of the system can result in high oscillations if standard methods are applied. A possible remedy is to modify the fully discretized system by a local redistribution of the mass. Numerical results in two and three dimensions illustrate the wide range of possible applications and show the performance of the space discretization scheme, non-linear solver, adaptive refinement process and time integration.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"20 1","pages":"569 - 734"},"PeriodicalIF":14.2,"publicationDate":"2011-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492911000079","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57444952","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}

{"title":"Mathematical and computational methods for semiclassical Schrödinger equations*","authors":"Shi Jin, P. Markowich, Christof Sparber","doi":"10.1017/S0962492911000031","DOIUrl":"https://doi.org/10.1017/S0962492911000031","url":null,"abstract":"We consider time-dependent (linear and nonlinear) Schrödinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schrödinger equation in the semiclassical regime.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"20 1","pages":"121 - 209"},"PeriodicalIF":14.2,"publicationDate":"2011-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492911000031","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57444846","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}

{"title":"Finite element systems of differential forms","authors":"S. Christiansen","doi":"10.1017/S096249291100002X","DOIUrl":"https://doi.org/10.1017/S096249291100002X","url":null,"abstract":"We develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells corresponds to pullback of forms. The theory covers for instance composite piecewise polynomial finite elements of variable order over polyhedral grids. Under natural algebraic and metric conditions, interpolators and smoothers are constructed, which commute with the exterior derivative and whose product is uniformly stable in Lebesgue spaces. As a consequence we obtain not only eigenpair approximation for the Hodge-Laplacian in mixed form, but also variants of Sobolev injections and translation estimates adapted to variational discretizations.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"1 1","pages":""},"PeriodicalIF":14.2,"publicationDate":"2010-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S096249291100002X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57444811","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}

{"title":"ANU volume 19 Cover and Front matter","authors":"Awad H. Al-Mohy","doi":"10.1017/s0962492910999984","DOIUrl":"https://doi.org/10.1017/s0962492910999984","url":null,"abstract":"","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"19 1","pages":"f1 - f6"},"PeriodicalIF":14.2,"publicationDate":"2010-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/s0962492910999984","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57444730","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}

{"title":"ANU volume 19 Cover and Back matter","authors":"","doi":"10.1017/s0962492910999996","DOIUrl":"https://doi.org/10.1017/s0962492910999996","url":null,"abstract":"","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"19 1","pages":"b1 - b1"},"PeriodicalIF":14.2,"publicationDate":"2010-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/s0962492910999996","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57444748","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}

{"title":"Computing matrix functions","authors":"N. Higham, Awad H. Al-Mohy","doi":"10.1017/S0962492910000036","DOIUrl":"https://doi.org/10.1017/S0962492910000036","url":null,"abstract":"The need to evaluate a function f(A) ∈ ℂn×n of a matrix A ∈ ℂn×n arises in a wide and growing number of applications, ranging from the numerical solution of differential equations to measures of the complexity of networks. We give a survey of numerical methods for evaluating matrix functions, along with a brief treatment of the underlying theory and a description of two recent applications. The survey is organized by classes of methods, which are broadly those based on similarity transformations, those employing approximation by polynomial or rational functions, and matrix iterations. Computation of the Fréchet derivative, which is important for condition number estimation, is also treated, along with the problem of computing f(A)b without computing f(A). A summary of available software completes the survey.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"19 1","pages":"159 - 208"},"PeriodicalIF":14.2,"publicationDate":"2010-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492910000036","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57444554","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}

{"title":"Binary separation and training support vector machines*","authors":"R. Fletcher, G. Zanghirati","doi":"10.1017/S0962492910000024","DOIUrl":"https://doi.org/10.1017/S0962492910000024","url":null,"abstract":"We introduce basic ideas of binary separation by a linear hyperplane, which is a technique exploited in the support vector machine (SVM) concept. This is a decision-making tool for pattern recognition and related problems. We describe a fundamental standard problem (SP) and show how this is used in most existing research to develop a dual-based algorithm for its solution. This algorithm is shown to be deficient in certain aspects, and we develop a new primal-based SQP-like algorithm, which has some interesting features. Most practical SVM problems are not adequately handled by a linear hyperplane. We describe the nonlinear SVM technique, which enables a nonlinear separating surface to be computed, and we propose a new primal algorithm based on the use of low-rank Cholesky factors. It may be, however, that exact separation is not desirable due to the presence of uncertain or mislabelled data. Dealing with this situation is the main challenge in developing suitable algorithms. Existing dual-based algorithms use the idea of L1 penalties, which has merit. We suggest how penalties can be incorporated into a primal-based algorithm. Another aspect of practical SVM problems is often the huge size of the data set, which poses severe challenges both for software package development and for control of ill-conditioning. We illustrate some of these issues with numerical experiments on a range of problems.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"19 1","pages":"121 - 158"},"PeriodicalIF":14.2,"publicationDate":"2010-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492910000024","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57444417","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}

{"title":"Inverse problems: A Bayesian perspective","authors":"A. Stuart","doi":"10.1017/S0962492910000061","DOIUrl":"https://doi.org/10.1017/S0962492910000061","url":null,"abstract":"The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"410 1","pages":"451 - 559"},"PeriodicalIF":14.2,"publicationDate":"2010-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492910000061","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57444651","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}

{"title":"Recent trends in the numerical solution of retarded functional differential equations*","authors":"A. Bellen, S. Maset, M. Zennaro, N. Guglielmi","doi":"10.1017/S0962492906390010","DOIUrl":"https://doi.org/10.1017/S0962492906390010","url":null,"abstract":"Retarded functional differential equations (RFDEs) form a wide class of evolution equations which share the property that, at any point, the rate of the solution depends on a discrete or distributed set of values attained by the solution itself in the past. Thus the initial problem for RFDEs is an infinite-dimensional problem, taking its theoretical and numerical analysis beyond the classical schemes developed for differential equations with no functional elements. In particular, numerically solving initial problems for RFDEs is a diffcult task that cannot be founded on the mere adaptation of well-known methods for ordinary, partial or integro-differential equations to the presence of retarded arguments. Indeed, efficient codes for their numerical integration need speciffc approaches designed according to the nature of the equation and the behaviour of the solution. By defining the numerical method as a suitable approximation of the solution map of the given equation, we present an original and unifying theory for the convergence and accuracy analysis of the approximate solution. Two particular approaches, both inspired by Runge–Kutta methods, are described. Despite being apparently similar, they are intrinsically different. Indeed, in the presence of speciffc types of functionals on the right-hand side, only one of them can have an explicit character, whereas the other gives rise to an overall procedure which is implicit in any case, even for non-stiff problems. In the panorama of numerical RFDEs, some critical situations have been recently investigated in connection to speciffc classes of equations, such as the accurate location of discontinuity points, the termination and bifurcation of the solutions of neutral equations, with state-dependent delays, the regularization of the equation and the generalization of the solution behind possible termination points, and the treatment of equations stated in the implicit form, which include singularly perturbed problems and delay differential-algebraic equations as well. All these issues are tackled in the last three sections. In this paper we have not considered the important issue of stability, for which we refer the interested reader to the comprehensive book by Bellen and Zennaro (2003).","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"18 1","pages":"1 - 110"},"PeriodicalIF":14.2,"publicationDate":"2009-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492906390010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57444070","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}