Pub Date : 2016-05-01DOI: 10.1017/S0962492916000076
T. Davis, S. Rajamanickam, Wissam M. Sid-Lakhdar
Wilkinson defined a sparse matrix as one with enough zeros that it pays to take advantage of them.1 This informal yet practical definition captures the essence of the goal of direct methods for solving sparse matrix problems. They exploit the sparsity of a matrix to solve problems economically: much faster and using far less memory than if all the entries of a matrix were stored and took part in explicit computations. These methods form the backbone of a wide range of problems in computational science. A glimpse of the breadth of applications relying on sparse solvers can be seen in the origins of matrices in published matrix benchmark collections (Duff and Reid 1979a, Duff, Grimes and Lewis 1989a, Davis and Hu 2011). The goal of this survey article is to impart a working knowledge of the underlying theory and practice of sparse direct methods for solving linear systems and least-squares problems, and to provide an overview of the algorithms, data structures, and software available to solve these problems, so that the reader can both understand the methods and know how best to use them.
威尔金森将稀疏矩阵定义为一个有足够多的零的矩阵，利用它们是值得的这个非正式但实用的定义抓住了求解稀疏矩阵问题的直接方法的本质目标。它们利用矩阵的稀疏性来经济地解决问题:比存储矩阵的所有条目并参与显式计算要快得多，占用的内存也少得多。这些方法构成了计算科学中广泛问题的主干。在已发表的矩阵基准集(Duff and Reid 1979a, Duff, Grimes and Lewis 1989a, Davis and Hu 2011)中，可以看到依赖稀疏求解器的应用的广度。这篇综述文章的目的是传授解决线性系统和最小二乘问题的稀疏直接方法的基本理论和实践的工作知识，并提供可用于解决这些问题的算法、数据结构和软件的概述，以便读者既可以理解这些方法，又知道如何最好地使用它们。
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Pub Date : 2016-03-09DOI: 10.1016/j.chom.2016.02.009
Jessica Humann, Beth Mann, Geli Gao, Philip Moresco, Joseph Ramahi, Lip Nam Loh, Arden Farr, Yunming Hu, Kelly Durick-Eder, Sophie A Fillon, Richard J Smeyne, Elaine I Tuomanen
Maternal infection during pregnancy is associated with adverse outcomes for the fetus, including postnatal cognitive disorders. However, the underlying mechanisms are obscure. We find that bacterial cell wall peptidoglycan (CW), a universal PAMP for TLR2, traverses the murine placenta into the developing fetal brain. In contrast to adults, CW-exposed fetal brains did not show any signs of inflammation or neuronal death. Instead, the neuronal transcription factor FoxG1 was induced, and neuroproliferation leading to a 50% greater density of neurons in the cortical plate was observed. Bacterial infection of pregnant dams, followed by antibiotic treatment, which releases CW, yielded the same result. Neuroproliferation required TLR2 and was recapitulated in vitro with fetal neuronal precursor cells and TLR2/6, but not TLR2/1, ligands. The fetal neuroproliferative response correlated with abnormal cognitive behavior in CW-exposed pups following birth. Thus, the bacterial CW-TLR2 signaling axis affects fetal neurodevelopment and may underlie postnatal cognitive disorders.
{"title":"Bacterial Peptidoglycan Traverses the Placenta to Induce Fetal Neuroproliferation and Aberrant Postnatal Behavior","authors":"Jessica Humann, Beth Mann, Geli Gao, Philip Moresco, Joseph Ramahi, Lip Nam Loh, Arden Farr, Yunming Hu, Kelly Durick-Eder, Sophie A Fillon, Richard J Smeyne, Elaine I Tuomanen","doi":"10.1016/j.chom.2016.02.009","DOIUrl":"10.1016/j.chom.2016.02.009","url":null,"abstract":"<p><p>Maternal infection during pregnancy is associated with adverse outcomes for the fetus, including postnatal cognitive disorders. However, the underlying mechanisms are obscure. We find that bacterial cell wall peptidoglycan (CW), a universal PAMP for TLR2, traverses the murine placenta into the developing fetal brain. In contrast to adults, CW-exposed fetal brains did not show any signs of inflammation or neuronal death. Instead, the neuronal transcription factor FoxG1 was induced, and neuroproliferation leading to a 50% greater density of neurons in the cortical plate was observed. Bacterial infection of pregnant dams, followed by antibiotic treatment, which releases CW, yielded the same result. Neuroproliferation required TLR2 and was recapitulated in vitro with fetal neuronal precursor cells and TLR2/6, but not TLR2/1, ligands. The fetal neuroproliferative response correlated with abnormal cognitive behavior in CW-exposed pups following birth. Thus, the bacterial CW-TLR2 signaling axis affects fetal neurodevelopment and may underlie postnatal cognitive disorders.</p>","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"22 1","pages":"388-99"},"PeriodicalIF":30.3,"publicationDate":"2016-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4787272/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79452461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-04-27DOI: 10.1017/S0962492914000130
B. Fornberg, N. Flyer
Finite differences provided the first numerical approach that permitted large-scale simulations in many applications areas, such as geophysical fluid dynamics. As accuracy and integration time requirements gradually increased, the focus shifted from finite differences to a variety of different spectral methods. During the last few years, radial basis functions, in particular in their ‘local’ RBF-FD form, have taken the major step from being mostly a curiosity approach for small-scale PDE ‘toy problems’ to becoming a major contender also for very large simulations on advanced distributed memory computer systems. Being entirely mesh-free, RBF-FD discretizations are also particularly easy to implement, even when local refinements are needed. This article gives some background to this development, and highlights some recent results.
{"title":"Solving PDEs with radial basis functions *","authors":"B. Fornberg, N. Flyer","doi":"10.1017/S0962492914000130","DOIUrl":"https://doi.org/10.1017/S0962492914000130","url":null,"abstract":"Finite differences provided the first numerical approach that permitted large-scale simulations in many applications areas, such as geophysical fluid dynamics. As accuracy and integration time requirements gradually increased, the focus shifted from finite differences to a variety of different spectral methods. During the last few years, radial basis functions, in particular in their ‘local’ RBF-FD form, have taken the major step from being mostly a curiosity approach for small-scale PDE ‘toy problems’ to becoming a major contender also for very large simulations on advanced distributed memory computer systems. Being entirely mesh-free, RBF-FD discretizations are also particularly easy to implement, even when local refinements are needed. This article gives some background to this development, and highlights some recent results.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"24 1","pages":"215 - 258"},"PeriodicalIF":14.2,"publicationDate":"2015-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492914000130","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445993","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 : 2015-04-27DOI: 10.1017/S0962492915000021
A. Wathen
The computational solution of problems can be restricted by the availability of solution methods for linear(ized) systems of equations. In conjunction with iterative methods, preconditioning is often the vital component in enabling the solution of such systems when the dimension is large. We attempt a broad review of preconditioning methods.
{"title":"Preconditioning","authors":"A. Wathen","doi":"10.1017/S0962492915000021","DOIUrl":"https://doi.org/10.1017/S0962492915000021","url":null,"abstract":"The computational solution of problems can be restricted by the availability of solution methods for linear(ized) systems of equations. In conjunction with iterative methods, preconditioning is often the vital component in enabling the solution of such systems when the dimension is large. We attempt a broad review of preconditioning methods.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"24 1","pages":"329 - 376"},"PeriodicalIF":14.2,"publicationDate":"2015-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492915000021","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57446288","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 : 2015-04-27DOI: 10.1017/S0962492914000129
M. Floater
This paper surveys the construction, properties, and applications of generalized barycentric coordinates on polygons and polyhedra. Applications include: surface mesh parametrization in geometric modelling; image, curve, and surface deformation in computer graphics; and polygonal and polyhedral finite element methods.
{"title":"Generalized barycentric coordinates and applications *","authors":"M. Floater","doi":"10.1017/S0962492914000129","DOIUrl":"https://doi.org/10.1017/S0962492914000129","url":null,"abstract":"This paper surveys the construction, properties, and applications of generalized barycentric coordinates on polygons and polyhedra. Applications include: surface mesh parametrization in geometric modelling; image, curve, and surface deformation in computer graphics; and polygonal and polyhedral finite element methods.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"24 1","pages":"161 - 214"},"PeriodicalIF":14.2,"publicationDate":"2015-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492914000129","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445940","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 : 2015-02-24DOI: 10.1017/S0962492915000033
A. Cohen, R. DeVore
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.
{"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}
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