Acta Numerica最新文献_第7页

Randomized algorithms in numerical linear algebra 数值线性代数中的随机化算法

IF 14.2 1区数学

Acta Numerica Pub Date : 2017-05-01 DOI: 10.1017/S0962492917000058

R. Kannan, S. Vempala

引用次数: 52

The cardiovascular system: Mathematical modelling, numerical algorithms and clinical applications * 心血管系统:数学建模、数值算法和临床应用*

IF 14.2 1区数学

Acta Numerica Pub Date : 2017-05-01 DOI: 10.1017/S0962492917000046

A. Quarteroni, A. Manzoni, C. Vergara

{"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}

引用次数: 148

The nonlinear eigenvalue problem * 非线性特征值问题*

IF 14.2 1区数学

Acta Numerica Pub Date : 2017-02-21 DOI: 10.1017/S0962492917000034

Stefan Güttel, F. Tisseur

引用次数: 7

Algebraic multigrid methods * 代数多重网格方法*

IF 14.2 1区数学

Acta Numerica Pub Date : 2016-11-07 DOI: 10.1017/S0962492917000083

Jinchao Xu, L. Zikatanov

{"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}

引用次数: 168

Numerical analysis of strongly nonlinear PDEs * 强非线性偏微分方程的数值分析

IF 14.2 1区数学

Acta Numerica Pub Date : 2016-10-25 DOI: 10.1017/S0962492917000071

M. Neilan, A. Salgado, Wujun Zhang

引用次数: 66

Probabilistic analyses of condition numbers* 条件数的概率分析*

IF 14.2 1区数学

Acta Numerica Pub Date : 2016-05-01 DOI: 10.1017/S0962492916000027

F. Cucker

引用次数: 8

On the computation of measure-valued solutions 论测度值解的计算

IF 14.2 1区数学

Acta Numerica Pub Date : 2016-05-01 DOI: 10.1017/S0962492916000088

U. S. Fjordholm, Siddhartha Mishra, E. Tadmor

{"title":"On the computation of measure-valued solutions","authors":"U. S. Fjordholm, Siddhartha Mishra, E. Tadmor","doi":"10.1017/S0962492916000088","DOIUrl":"https://doi.org/10.1017/S0962492916000088","url":null,"abstract":"A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multi-dimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures – parametrized probability measures which can describe the limits of such oscillatory sequences – offer the more general paradigm of measure-valued solutions for these problems. We present viable numerical algorithms to compute approximate measure-valued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. We also discuss the use of these algorithms, and their extensions, in uncertainty quantification and contexts other than fluid dynamics, such as non-convex variational problems in materials science.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"25 1","pages":"567 - 679"},"PeriodicalIF":14.2,"publicationDate":"2016-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492916000088","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57446733","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}

引用次数: 81

An introduction to continuous optimization for imaging 介绍成像的连续优化

IF 14.2 1区数学

Acta Numerica Pub Date : 2016-05-01 DOI: 10.1017/S096249291600009X

A. Chambolle, T. Pock

引用次数: 416

Partial differential equations and stochastic methods in molecular dynamics* 分子动力学中的偏微分方程和随机方法*

IF 14.2 1区数学

Acta Numerica Pub Date : 2016-05-01 DOI: 10.1017/S0962492916000039

T. Lelièvre, G. Stoltz

引用次数: 173

Linear algebra software for large-scale accelerated multicore computing* 线性代数软件大规模加速多核计算*

IF 14.2 1区数学

Acta Numerica Pub Date : 2016-05-01 DOI: 10.1017/S0962492916000015

A. Abdelfattah, H. Anzt, J. Dongarra, M. Gates, A. Haidar, J. Kurzak, P. Luszczek, S. Tomov, I. Yamazaki, A. YarKhan

{"title":"Linear algebra software for large-scale accelerated multicore computing*","authors":"A. Abdelfattah, H. Anzt, J. Dongarra, M. Gates, A. Haidar, J. Kurzak, P. Luszczek, S. Tomov, I. Yamazaki, A. YarKhan","doi":"10.1017/S0962492916000015","DOIUrl":"https://doi.org/10.1017/S0962492916000015","url":null,"abstract":"Many crucial scientific computing applications, ranging from national security to medical advances, rely on high-performance linear algebra algorithms and technologies, underscoring their importance and broad impact. Here we present the state-of-the-art design and implementation practices for the acceleration of the predominant linear algebra algorithms on large-scale accelerated multicore systems. Examples are given with fundamental dense linear algebra algorithms – from the LU, QR, Cholesky, and LDLT factorizations needed for solving linear systems of equations, to eigenvalue and singular value decomposition (SVD) problems. The implementations presented are readily available via the open-source PLASMA and MAGMA libraries, which represent the next generation modernization of the popular LAPACK library for accelerated multicore systems. To generate the extreme level of parallelism needed for the efficient use of these systems, algorithms of interest are redesigned and then split into well-chosen computational tasks. The task execution is scheduled over the computational components of a hybrid system of multicore CPUs with GPU accelerators and/or Xeon Phi coprocessors, using either static scheduling or light-weight runtime systems. The use of light-weight runtime systems keeps scheduling overheads low, similar to static scheduling, while enabling the expression of parallelism through sequential-like code. This simplifies the development effort and allows exploration of the unique strengths of the various hardware components. Finally, we emphasize the development of innovative linear algebra algorithms using three technologies – mixed precision arithmetic, batched operations, and asynchronous iterations – that are currently of high interest for accelerated multicore systems.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"25 1","pages":"1 - 160"},"PeriodicalIF":14.2,"publicationDate":"2016-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492916000015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57446084","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}

引用次数: 14