Pub Date : 2016-05-01DOI: 10.1017/S0962492916000088
U. S. Fjordholm, Siddhartha Mishra, E. Tadmor
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.
{"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}
Pub Date : 2016-05-01DOI: 10.1017/S096249291600009X
A. Chambolle, T. Pock
A large number of imaging problems reduce to the optimization of a cost function, with typical structural properties. The aim of this paper is to describe the state of the art in continuous optimization methods for such problems, and present the most successful approaches and their interconnections. We place particular emphasis on optimal first-order schemes that can deal with typical non-smooth and large-scale objective functions used in imaging problems. We illustrate and compare the different algorithms using classical non-smooth problems in imaging, such as denoising and deblurring. Moreover, we present applications of the algorithms to more advanced problems, such as magnetic resonance imaging, multilabel image segmentation, optical flow estimation, stereo matching, and classification.
{"title":"An introduction to continuous optimization for imaging","authors":"A. Chambolle, T. Pock","doi":"10.1017/S096249291600009X","DOIUrl":"https://doi.org/10.1017/S096249291600009X","url":null,"abstract":"A large number of imaging problems reduce to the optimization of a cost function, with typical structural properties. The aim of this paper is to describe the state of the art in continuous optimization methods for such problems, and present the most successful approaches and their interconnections. We place particular emphasis on optimal first-order schemes that can deal with typical non-smooth and large-scale objective functions used in imaging problems. We illustrate and compare the different algorithms using classical non-smooth problems in imaging, such as denoising and deblurring. Moreover, we present applications of the algorithms to more advanced problems, such as magnetic resonance imaging, multilabel image segmentation, optical flow estimation, stereo matching, and classification.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"25 1","pages":"161 - 319"},"PeriodicalIF":14.2,"publicationDate":"2016-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S096249291600009X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57446771","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/S0962492916000039
T. Lelièvre, G. Stoltz
The objective of molecular dynamics computations is to infer macroscopic properties of matter from atomistic models via averages with respect to probability measures dictated by the principles of statistical physics. Obtaining accurate results requires efficient sampling of atomistic configurations, which are typically generated using very long trajectories of stochastic differential equations in high dimensions, such as Langevin dynamics and its overdamped limit. Depending on the quantities of interest at the macroscopic level, one may also be interested in dynamical properties computed from averages over paths of these dynamics. This review describes how techniques from the analysis of partial differential equations can be used to devise good algorithms and to quantify their efficiency and accuracy. In particular, a crucial role is played by the study of the long-time behaviour of the solution to the Fokker–Planck equation associated with the stochastic dynamics.
{"title":"Partial differential equations and stochastic methods in molecular dynamics*","authors":"T. Lelièvre, G. Stoltz","doi":"10.1017/S0962492916000039","DOIUrl":"https://doi.org/10.1017/S0962492916000039","url":null,"abstract":"The objective of molecular dynamics computations is to infer macroscopic properties of matter from atomistic models via averages with respect to probability measures dictated by the principles of statistical physics. Obtaining accurate results requires efficient sampling of atomistic configurations, which are typically generated using very long trajectories of stochastic differential equations in high dimensions, such as Langevin dynamics and its overdamped limit. Depending on the quantities of interest at the macroscopic level, one may also be interested in dynamical properties computed from averages over paths of these dynamics. This review describes how techniques from the analysis of partial differential equations can be used to devise good algorithms and to quantify their efficiency and accuracy. In particular, a crucial role is played by the study of the long-time behaviour of the solution to the Fokker–Planck equation associated with the stochastic dynamics.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"13 1","pages":"681 - 880"},"PeriodicalIF":14.2,"publicationDate":"2016-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492916000039","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57446940","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/S0962492916000015
A. Abdelfattah, H. Anzt, J. Dongarra, M. Gates, A. Haidar, J. Kurzak, P. Luszczek, S. Tomov, I. Yamazaki, A. YarKhan
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.
{"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}
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。
{"title":"A survey of direct methods for sparse linear systems","authors":"T. Davis, S. Rajamanickam, Wissam M. Sid-Lakhdar","doi":"10.1017/S0962492916000076","DOIUrl":"https://doi.org/10.1017/S0962492916000076","url":null,"abstract":"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.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"25 1","pages":"383 - 566"},"PeriodicalIF":14.2,"publicationDate":"2016-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492916000076","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57446691","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 : 2016-03-09DOI: 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}
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