Pub Date : 2017-11-14DOI: 10.1017/S0962492917000101
Nawaf Bou-Rabee, J. Sanz-Serna
This paper surveys in detail the relations between numerical integration and the Hamiltonian (or hybrid) Monte Carlo method (HMC). Since the computational cost of HMC mainly lies in the numerical integrations, these should be performed as efficiently as possible. However, HMC requires methods that have the geometric properties of being volume-preserving and reversible, and this limits the number of integrators that may be used. On the other hand, these geometric properties have important quantitative implications for the integration error, which in turn have an impact on the acceptance rate of the proposal. While at present the velocity Verlet algorithm is the method of choice for good reasons, we argue that Verlet can be improved upon. We also discuss in detail the behaviour of HMC as the dimensionality of the target distribution increases.
{"title":"Geometric integrators and the Hamiltonian Monte Carlo method","authors":"Nawaf Bou-Rabee, J. Sanz-Serna","doi":"10.1017/S0962492917000101","DOIUrl":"https://doi.org/10.1017/S0962492917000101","url":null,"abstract":"This paper surveys in detail the relations between numerical integration and the Hamiltonian (or hybrid) Monte Carlo method (HMC). Since the computational cost of HMC mainly lies in the numerical integrations, these should be performed as efficiently as possible. However, HMC requires methods that have the geometric properties of being volume-preserving and reversible, and this limits the number of integrators that may be used. On the other hand, these geometric properties have important quantitative implications for the integration error, which in turn have an impact on the acceptance rate of the proposal. While at present the velocity Verlet algorithm is the method of choice for good reasons, we argue that Verlet can be improved upon. We also discuss in detail the behaviour of HMC as the dimensionality of the target distribution increases.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"27 1","pages":"113 - 206"},"PeriodicalIF":14.2,"publicationDate":"2017-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492917000101","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49018849","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 : 2017-05-05DOI: 10.1017/s096249291700006x
Onur Özyeşil, Vladislav Voroninski, Ronen Basri, Amit Singer
The structure from motion (SfM) problem in computer vision is to recover the three-dimensional (3D) structure of a stationary scene from a set of projective measurements, represented as a collection of two-dimensional (2D) images, via estimation of motion of the cameras corresponding to these images. In essence, SfM involves the three main stages of (i) extracting features in images (e.g. points of interest, lines,etc.) and matching these features between images, (ii) camera motion estimation (e.g. using relative pairwise camera positions estimated from the extracted features), and (iii) recovery of the 3D structure using the estimated motion and features (e.g. by minimizing the so-calledreprojection error). This survey mainly focuses on relatively recent developments in the literature pertaining to stages (ii) and (iii). More specifically, after touching upon the early factorization-based techniques for motion and structure estimation, we provide a detailed account of some of the recent cameralocationestimation methods in the literature, followed by discussion of notable techniques for 3D structure recovery. We also cover the basics of thesimultaneous localization and mapping(SLAM) problem, which can be viewed as a specific case of the SfM problem. Further, our survey includes a review of the fundamentals of feature extraction and matching (i.e. stage (i) above), various recent methods for handling ambiguities in 3D scenes, SfM techniques involving relatively uncommon camera models and image features, and popular sources of data and SfM software.
{"title":"A survey of structure from motion.","authors":"Onur Özyeşil, Vladislav Voroninski, Ronen Basri, Amit Singer","doi":"10.1017/s096249291700006x","DOIUrl":"https://doi.org/10.1017/s096249291700006x","url":null,"abstract":"The structure from motion (SfM) problem in computer vision is to recover the three-dimensional (3D) structure of a stationary scene from a set of projective measurements, represented as a collection of two-dimensional (2D) images, via estimation of motion of the cameras corresponding to these images. In essence, SfM involves the three main stages of (i) extracting features in images (<jats:italic>e.g.</jats:italic> points of interest, lines,<jats:italic>etc.</jats:italic>) and matching these features between images, (ii) camera motion estimation (<jats:italic>e.g.</jats:italic> using relative pairwise camera positions estimated from the extracted features), and (iii) recovery of the 3D structure using the estimated motion and features (<jats:italic>e.g.</jats:italic> by minimizing the so-called<jats:italic>reprojection error</jats:italic>). This survey mainly focuses on relatively recent developments in the literature pertaining to stages (ii) and (iii). More specifically, after touching upon the early factorization-based techniques for motion and structure estimation, we provide a detailed account of some of the recent camera<jats:italic>location</jats:italic>estimation methods in the literature, followed by discussion of notable techniques for 3D structure recovery. We also cover the basics of the<jats:italic>simultaneous localization and mapping</jats:italic>(SLAM) problem, which can be viewed as a specific case of the SfM problem. Further, our survey includes a review of the fundamentals of feature extraction and matching (<jats:italic>i.e.</jats:italic> stage (i) above), various recent methods for handling ambiguities in 3D scenes, SfM techniques involving relatively uncommon camera models and image features, and popular sources of data and SfM software.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"229 1","pages":""},"PeriodicalIF":14.2,"publicationDate":"2017-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138530597","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 : 2017-05-01DOI: 10.1017/S0962492917000058
R. Kannan, S. Vempala
This survey provides an introduction to the use of randomization in the design of fast algorithms for numerical linear algebra. These algorithms typically examine only a subset of the input to solve basic problems approximately, including matrix multiplication, regression and low-rank approximation. The survey describes the key ideas and gives complete proofs of the main results in the field. A central unifying idea is sampling the columns (or rows) of a matrix according to their squared lengths.
{"title":"Randomized algorithms in numerical linear algebra","authors":"R. Kannan, S. Vempala","doi":"10.1017/S0962492917000058","DOIUrl":"https://doi.org/10.1017/S0962492917000058","url":null,"abstract":"This survey provides an introduction to the use of randomization in the design of fast algorithms for numerical linear algebra. These algorithms typically examine only a subset of the input to solve basic problems approximately, including matrix multiplication, regression and low-rank approximation. The survey describes the key ideas and gives complete proofs of the main results in the field. A central unifying idea is sampling the columns (or rows) of a matrix according to their squared lengths.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"26 1","pages":"95 - 135"},"PeriodicalIF":14.2,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492917000058","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45036614","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 : 2017-05-01DOI: 10.1017/S0962492917000046
A. Quarteroni, A. Manzoni, C. Vergara
Mathematical and numerical modelling of the cardiovascular system is a research topic that has attracted remarkable interest from the mathematical community because of its intrinsic mathematical difficulty and the increasing impact of cardiovascular diseases worldwide. In this review article we will address the two principal components of the cardiovascular system: arterial circulation and heart function. We will systematically describe all aspects of the problem, ranging from data imaging acquisition, stating the basic physical principles, analysing the associated mathematical models that comprise PDE and ODE systems, proposing sound and efficient numerical methods for their approximation, and simulating both benchmark problems and clinically inspired problems. Mathematical modelling itself imposes tremendous challenges, due to the amazing complexity of the cardiocirculatory system, the multiscale nature of the physiological processes involved, and the need to devise computational methods that are stable, reliable and efficient. Critical issues involve filtering the data, identifying the parameters of mathematical models, devising optimal treatments and accounting for uncertainties. For this reason, we will devote the last part of the paper to control and inverse problems, including parameter estimation, uncertainty quantification and the development of reduced-order models that are of paramount importance when solving problems with high complexity, which would otherwise be out of reach.
{"title":"The cardiovascular system: Mathematical modelling, numerical algorithms and clinical applications *","authors":"A. Quarteroni, A. Manzoni, C. Vergara","doi":"10.1017/S0962492917000046","DOIUrl":"https://doi.org/10.1017/S0962492917000046","url":null,"abstract":"Mathematical and numerical modelling of the cardiovascular system is a research topic that has attracted remarkable interest from the mathematical community because of its intrinsic mathematical difficulty and the increasing impact of cardiovascular diseases worldwide. In this review article we will address the two principal components of the cardiovascular system: arterial circulation and heart function. We will systematically describe all aspects of the problem, ranging from data imaging acquisition, stating the basic physical principles, analysing the associated mathematical models that comprise PDE and ODE systems, proposing sound and efficient numerical methods for their approximation, and simulating both benchmark problems and clinically inspired problems. Mathematical modelling itself imposes tremendous challenges, due to the amazing complexity of the cardiocirculatory system, the multiscale nature of the physiological processes involved, and the need to devise computational methods that are stable, reliable and efficient. Critical issues involve filtering the data, identifying the parameters of mathematical models, devising optimal treatments and accounting for uncertainties. For this reason, we will devote the last part of the paper to control and inverse problems, including parameter estimation, uncertainty quantification and the development of reduced-order models that are of paramount importance when solving problems with high complexity, which would otherwise be out of reach.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"26 1","pages":"365 - 590"},"PeriodicalIF":14.2,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492917000046","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43213446","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 : 2017-02-21DOI: 10.1017/S0962492917000034
Stefan Güttel, F. Tisseur
Nonlinear eigenvalue problems arise in a variety of science and engineering applications, and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton’s method, contour integration and sampling via rational interpolation are reviewed. Problems of selecting the appropriate parameters for each of the solver classes are discussed and illustrated with numerical examples. This survey also contains numerous MATLAB code snippets that can be used for interactive exploration of the discussed methods.
{"title":"The nonlinear eigenvalue problem *","authors":"Stefan Güttel, F. Tisseur","doi":"10.1017/S0962492917000034","DOIUrl":"https://doi.org/10.1017/S0962492917000034","url":null,"abstract":"Nonlinear eigenvalue problems arise in a variety of science and engineering applications, and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton’s method, contour integration and sampling via rational interpolation are reviewed. Problems of selecting the appropriate parameters for each of the solver classes are discussed and illustrated with numerical examples. This survey also contains numerous MATLAB code snippets that can be used for interactive exploration of the discussed methods.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"26 1","pages":"1 - 94"},"PeriodicalIF":14.2,"publicationDate":"2017-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492917000034","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42951565","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 : 2016-11-07DOI: 10.1017/S0962492917000083
Jinchao Xu, L. Zikatanov
This paper provides an overview of AMG methods for solving large-scale systems of equations, such as those from discretizations of partial differential equations. AMG is often understood as the acronym of ‘algebraic multigrid’, but it can also be understood as ‘abstract multigrid’. Indeed, we demonstrate in this paper how and why an algebraic multigrid method can be better understood at a more abstract level. In the literature, there are many different algebraic multigrid methods that have been developed from different perspectives. In this paper we try to develop a unified framework and theory that can be used to derive and analyse different algebraic multigrid methods in a coherent manner. Given a smoother $R$ for a matrix $A$ , such as Gauss–Seidel or Jacobi, we prove that the optimal coarse space of dimension $n_{c}$ is the span of the eigenvectors corresponding to the first $n_{c}$ eigenvectors $bar{R}A$ (with $bar{R}=R+R^{T}-R^{T}AR$ ). We also prove that this optimal coarse space can be obtained via a constrained trace-minimization problem for a matrix associated with $bar{R}A$ , and demonstrate that coarse spaces of most existing AMG methods can be viewed as approximate solutions of this trace-minimization problem. Furthermore, we provide a general approach to the construction of quasi-optimal coarse spaces, and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, classical AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG and spectral AMGe.
本文概述了求解大型方程组的AMG方法，如偏微分方程的离散化。AMG通常被理解为“代数多重网格”的缩写，但它也可以被理解为“抽象多重网格”。实际上，我们在本文中演示了如何以及为什么代数多网格方法可以在更抽象的层面上更好地理解。在文献中，有许多不同的代数多重网格方法已经从不同的角度发展。在本文中，我们试图发展一个统一的框架和理论，可以用来推导和分析不同的代数多网格方法在一个连贯的方式。给定矩阵a $的一个更光滑的$R$，如高斯-塞德尔或雅可比，我们证明了维数$n_{c}$的最优粗空间是第一个$n_{c}$特征向量$bar{R} a $所对应的特征向量张成的空间(其中$bar{R}=R+R^{T}-R^{T}AR$)。我们还证明了该最优粗空间可以通过与$bar{R} a $相关的矩阵的约束迹最小化问题得到，并证明了大多数现有AMG方法的粗空间可以视为该迹最小化问题的近似解。在此基础上，给出了拟最优粗糙空间构造的一般方法，并证明了在适当的假设下，所得到的二阶AMG方法对于问题的大小、系数变化和各向异性是一致收敛的。我们的理论适用于大多数现有的多网格方法，包括标准几何多网格法、经典多网格法、能量最小化多网格法、非光滑和光滑聚集多网格法以及频谱多网格法。
{"title":"Algebraic multigrid methods *","authors":"Jinchao Xu, L. Zikatanov","doi":"10.1017/S0962492917000083","DOIUrl":"https://doi.org/10.1017/S0962492917000083","url":null,"abstract":"This paper provides an overview of AMG methods for solving large-scale systems of equations, such as those from discretizations of partial differential equations. AMG is often understood as the acronym of ‘algebraic multigrid’, but it can also be understood as ‘abstract multigrid’. Indeed, we demonstrate in this paper how and why an algebraic multigrid method can be better understood at a more abstract level. In the literature, there are many different algebraic multigrid methods that have been developed from different perspectives. In this paper we try to develop a unified framework and theory that can be used to derive and analyse different algebraic multigrid methods in a coherent manner. Given a smoother $R$ for a matrix $A$ , such as Gauss–Seidel or Jacobi, we prove that the optimal coarse space of dimension $n_{c}$ is the span of the eigenvectors corresponding to the first $n_{c}$ eigenvectors $bar{R}A$ (with $bar{R}=R+R^{T}-R^{T}AR$ ). We also prove that this optimal coarse space can be obtained via a constrained trace-minimization problem for a matrix associated with $bar{R}A$ , and demonstrate that coarse spaces of most existing AMG methods can be viewed as approximate solutions of this trace-minimization problem. Furthermore, we provide a general approach to the construction of quasi-optimal coarse spaces, and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, classical AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG and spectral AMGe.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"26 1","pages":"591 - 721"},"PeriodicalIF":14.2,"publicationDate":"2016-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492917000083","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57446874","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 : 2016-10-25DOI: 10.1017/S0962492917000071
M. Neilan, A. Salgado, Wujun Zhang
We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and non-convex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental result in this area which states that stable, consistent and monotone schemes converge as the discretization parameter tends to zero. We review methodologies to construct finite difference, finite element and semi-Lagrangian schemes that satisfy these criteria, and, in addition, discuss some rather novel tools that have paved the way to derive rates of convergence within this framework.
{"title":"Numerical analysis of strongly nonlinear PDEs *","authors":"M. Neilan, A. Salgado, Wujun Zhang","doi":"10.1017/S0962492917000071","DOIUrl":"https://doi.org/10.1017/S0962492917000071","url":null,"abstract":"We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and non-convex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental result in this area which states that stable, consistent and monotone schemes converge as the discretization parameter tends to zero. We review methodologies to construct finite difference, finite element and semi-Lagrangian schemes that satisfy these criteria, and, in addition, discuss some rather novel tools that have paved the way to derive rates of convergence within this framework.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"22 1","pages":"137 - 303"},"PeriodicalIF":14.2,"publicationDate":"2016-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492917000071","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57446832","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 : 2016-05-01DOI: 10.1017/S0962492916000027
F. Cucker
In recent decades, condition numbers have joined forces with probabilistic analysis to give rise to a form of condition-based analysis of algorithms. In this paper we survey how this analysis is done via a number of examples. We precede this catalogue of examples with short primers on both condition numbers and probabilistic analyses.
{"title":"Probabilistic analyses of condition numbers*","authors":"F. Cucker","doi":"10.1017/S0962492916000027","DOIUrl":"https://doi.org/10.1017/S0962492916000027","url":null,"abstract":"In recent decades, condition numbers have joined forces with probabilistic analysis to give rise to a form of condition-based analysis of algorithms. In this paper we survey how this analysis is done via a number of examples. We precede this catalogue of examples with short primers on both condition numbers and probabilistic analyses.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"25 1","pages":"321 - 382"},"PeriodicalIF":14.2,"publicationDate":"2016-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492916000027","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57446144","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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