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Numerical Homogenization by Localized Decomposition

Author: Axel Målqvist

Publisher: SIAM

Published: 2020-11-23

Total Pages: 120

ISBN-13: 1611976456

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This book presents the first survey of the Localized Orthogonal Decomposition (LOD) method, a pioneering approach for the numerical homogenization of partial differential equations with multiscale data beyond periodicity and scale separation. The authors provide a careful error analysis, including previously unpublished results, and a complete implementation of the method in MATLAB. They also reveal how the LOD method relates to classical homogenization and domain decomposition. Illustrated with numerical experiments that demonstrate the significance of the method, the book is enhanced by a survey of applications including eigenvalue problems and evolution problems. Numerical Homogenization by Localized Orthogonal Decomposition is appropriate for graduate students in applied mathematics, numerical analysis, and scientific computing. Researchers in the field of computational partial differential equations will find this self-contained book of interest, as will applied scientists and engineers interested in multiscale simulation.

Book Synopsis Numerical Homogenization by Localized Decomposition by : Axel Målqvist

Download or read book Numerical Homogenization by Localized Decomposition written by Axel Målqvist and published by SIAM. This book was released on 2020-11-23 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the first survey of the Localized Orthogonal Decomposition (LOD) method, a pioneering approach for the numerical homogenization of partial differential equations with multiscale data beyond periodicity and scale separation. The authors provide a careful error analysis, including previously unpublished results, and a complete implementation of the method in MATLAB. They also reveal how the LOD method relates to classical homogenization and domain decomposition. Illustrated with numerical experiments that demonstrate the significance of the method, the book is enhanced by a survey of applications including eigenvalue problems and evolution problems. Numerical Homogenization by Localized Orthogonal Decomposition is appropriate for graduate students in applied mathematics, numerical analysis, and scientific computing. Researchers in the field of computational partial differential equations will find this self-contained book of interest, as will applied scientists and engineers interested in multiscale simulation.

Domain Decomposition Methods in Science and Engineering XXIII

Author: Chang-Ock Lee

Publisher: Springer

Published: 2017-03-15

Total Pages: 419

ISBN-13: 3319523899

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This book is a collection of papers presented at the 23rd International Conference on Domain Decomposition Methods in Science and Engineering, held on Jeju Island, Korea on July 6-10, 2015. Domain decomposition methods solve boundary value problems by splitting them into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. Domain decomposition methods have considerable potential for a parallelization of the finite element methods, and serve a basis for distributed, parallel computations.

Book Synopsis Domain Decomposition Methods in Science and Engineering XXIII by : Chang-Ock Lee

Download or read book Domain Decomposition Methods in Science and Engineering XXIII written by Chang-Ock Lee and published by Springer. This book was released on 2017-03-15 with total page 419 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a collection of papers presented at the 23rd International Conference on Domain Decomposition Methods in Science and Engineering, held on Jeju Island, Korea on July 6-10, 2015. Domain decomposition methods solve boundary value problems by splitting them into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. Domain decomposition methods have considerable potential for a parallelization of the finite element methods, and serve a basis for distributed, parallel computations.

Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization

Author: Houman Owhadi

Publisher: Cambridge University Press

Published: 2019-10-24

Total Pages: 491

ISBN-13: 1108484360

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Presents interplays between numerical approximation and statistical inference as a pathway to simple solutions to fundamental problems.

Book Synopsis Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization by : Houman Owhadi

Download or read book Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization written by Houman Owhadi and published by Cambridge University Press. This book was released on 2019-10-24 with total page 491 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents interplays between numerical approximation and statistical inference as a pathway to simple solutions to fundamental problems.

Domain Decomposition Methods in Science and Engineering XXV

Author: Ronald Haynes

Publisher: Springer Nature

Published: 2020-10-24

Total Pages: 508

ISBN-13: 3030567508

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These are the proceedings of the 25th International Conference on Domain Decomposition Methods in Science and Engineering, which was held in St. John's, Newfoundland, Canada in July 2018. Domain decomposition methods are iterative methods for solving the often very large systems of equations that arise when engineering problems are discretized, frequently using finite elements or other modern techniques. These methods are specifically designed to make effective use of massively parallel, high-performance computing systems. The book presents both theoretical and computational advances in this domain, reflecting the state of art in 2018.

Book Synopsis Domain Decomposition Methods in Science and Engineering XXV by : Ronald Haynes

Download or read book Domain Decomposition Methods in Science and Engineering XXV written by Ronald Haynes and published by Springer Nature. This book was released on 2020-10-24 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt: These are the proceedings of the 25th International Conference on Domain Decomposition Methods in Science and Engineering, which was held in St. John's, Newfoundland, Canada in July 2018. Domain decomposition methods are iterative methods for solving the often very large systems of equations that arise when engineering problems are discretized, frequently using finite elements or other modern techniques. These methods are specifically designed to make effective use of massively parallel, high-performance computing systems. The book presents both theoretical and computational advances in this domain, reflecting the state of art in 2018.

Homogenization Theory for Multiscale Problems

Author: Xavier Blanc

Publisher: Springer Nature

Published: 2023-04-29

Total Pages: 469

ISBN-13: 3031218337

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The book provides a pedagogic and comprehensive introduction to homogenization theory with a special focus on problems set for non-periodic media. The presentation encompasses both deterministic and probabilistic settings. It also mixes the most abstract aspects with some more practical aspects regarding the numerical approaches necessary to simulate such multiscale problems. Based on lecture courses of the authors, the book is suitable for graduate students of mathematics and engineering.

Book Synopsis Homogenization Theory for Multiscale Problems by : Xavier Blanc

Download or read book Homogenization Theory for Multiscale Problems written by Xavier Blanc and published by Springer Nature. This book was released on 2023-04-29 with total page 469 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book provides a pedagogic and comprehensive introduction to homogenization theory with a special focus on problems set for non-periodic media. The presentation encompasses both deterministic and probabilistic settings. It also mixes the most abstract aspects with some more practical aspects regarding the numerical approaches necessary to simulate such multiscale problems. Based on lecture courses of the authors, the book is suitable for graduate students of mathematics and engineering.

Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations

Author: Tarek Mathew

Publisher: Springer Science & Business Media

Published: 2008-06-25

Total Pages: 775

ISBN-13: 354077209X

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Domain decomposition methods are divide and conquer computational methods for the parallel solution of partial differential equations of elliptic or parabolic type. The methodology includes iterative algorithms, and techniques for non-matching grid discretizations and heterogeneous approximations. This book serves as a matrix oriented introduction to domain decomposition methodology. A wide range of topics are discussed include hybrid formulations, Schwarz, and many more.

Book Synopsis Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations by : Tarek Mathew

Download or read book Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations written by Tarek Mathew and published by Springer Science & Business Media. This book was released on 2008-06-25 with total page 775 pages. Available in PDF, EPUB and Kindle. Book excerpt: Domain decomposition methods are divide and conquer computational methods for the parallel solution of partial differential equations of elliptic or parabolic type. The methodology includes iterative algorithms, and techniques for non-matching grid discretizations and heterogeneous approximations. This book serves as a matrix oriented introduction to domain decomposition methodology. A wide range of topics are discussed include hybrid formulations, Schwarz, and many more.

Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations

Author: Gabriel R. Barrenechea

Publisher: Springer

Published: 2016-10-03

Total Pages: 443

ISBN-13: 3319416405

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This volume contains contributed survey papers from the main speakers at the LMS/EPSRC Symposium “Building bridges: connections and challenges in modern approaches to numerical partial differential equations”. This meeting took place in July 8-16, 2014, and its main purpose was to gather specialists in emerging areas of numerical PDEs, and explore the connections between the different approaches. The type of contributions ranges from the theoretical foundations of these new techniques, to the applications of them, to new general frameworks and unified approaches that can cover one, or more than one, of these emerging techniques.

Book Synopsis Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations by : Gabriel R. Barrenechea

Download or read book Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations written by Gabriel R. Barrenechea and published by Springer. This book was released on 2016-10-03 with total page 443 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains contributed survey papers from the main speakers at the LMS/EPSRC Symposium “Building bridges: connections and challenges in modern approaches to numerical partial differential equations”. This meeting took place in July 8-16, 2014, and its main purpose was to gather specialists in emerging areas of numerical PDEs, and explore the connections between the different approaches. The type of contributions ranges from the theoretical foundations of these new techniques, to the applications of them, to new general frameworks and unified approaches that can cover one, or more than one, of these emerging techniques.

75 Years of Mathematics of Computation

Author: Susanne C. Brenner

Publisher: American Mathematical Soc.

Published: 2020-07-29

Total Pages: 364

ISBN-13: 1470451638

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The year 2018 marked the 75th anniversary of the founding of Mathematics of Computation, one of the four primary research journals published by the American Mathematical Society and the oldest research journal devoted to computational mathematics. To celebrate this milestone, the symposium “Celebrating 75 Years of Mathematics of Computation” was held from November 1–3, 2018, at the Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, Rhode Island. The sixteen papers in this volume, written by the symposium speakers and editors of the journal, include both survey articles and new contributions. On the discrete side, there are four papers covering topics in computational number theory and computational algebra. On the continuous side, there are twelve papers covering topics in machine learning, high dimensional approximations, nonlocal and fractional elliptic problems, gradient flows, hyperbolic conservation laws, Maxwell's equations, Stokes's equations, a posteriori error estimation, and iterative methods. Together they provide a snapshot of significant achievements in the past quarter century in computational mathematics and also in important current trends.

Book Synopsis 75 Years of Mathematics of Computation by : Susanne C. Brenner

Download or read book 75 Years of Mathematics of Computation written by Susanne C. Brenner and published by American Mathematical Soc.. This book was released on 2020-07-29 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt: The year 2018 marked the 75th anniversary of the founding of Mathematics of Computation, one of the four primary research journals published by the American Mathematical Society and the oldest research journal devoted to computational mathematics. To celebrate this milestone, the symposium “Celebrating 75 Years of Mathematics of Computation” was held from November 1–3, 2018, at the Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, Rhode Island. The sixteen papers in this volume, written by the symposium speakers and editors of the journal, include both survey articles and new contributions. On the discrete side, there are four papers covering topics in computational number theory and computational algebra. On the continuous side, there are twelve papers covering topics in machine learning, high dimensional approximations, nonlocal and fractional elliptic problems, gradient flows, hyperbolic conservation laws, Maxwell's equations, Stokes's equations, a posteriori error estimation, and iterative methods. Together they provide a snapshot of significant achievements in the past quarter century in computational mathematics and also in important current trends.

Error Norm Estimation in the Conjugate Gradient Algorithm

Author: Gérard Meurant

Publisher: SIAM

Published: 2024-01-30

Total Pages: 138

ISBN-13: 161197786X

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The conjugate gradient (CG) algorithm is almost always the iterative method of choice for solving linear systems with symmetric positive definite matrices. This book describes and analyzes techniques based on Gauss quadrature rules to cheaply compute bounds on norms of the error. The techniques can be used to derive reliable stopping criteria. How to compute estimates of the smallest and largest eigenvalues during CG iterations is also shown. The algorithms are illustrated by many numerical experiments, and they can be easily incorporated into existing CG codes. The book is intended for those in academia and industry who use the conjugate gradient algorithm, including the many branches of science and engineering in which symmetric linear systems have to be solved.

Book Synopsis Error Norm Estimation in the Conjugate Gradient Algorithm by : Gérard Meurant

Download or read book Error Norm Estimation in the Conjugate Gradient Algorithm written by Gérard Meurant and published by SIAM. This book was released on 2024-01-30 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: The conjugate gradient (CG) algorithm is almost always the iterative method of choice for solving linear systems with symmetric positive definite matrices. This book describes and analyzes techniques based on Gauss quadrature rules to cheaply compute bounds on norms of the error. The techniques can be used to derive reliable stopping criteria. How to compute estimates of the smallest and largest eigenvalues during CG iterations is also shown. The algorithms are illustrated by many numerical experiments, and they can be easily incorporated into existing CG codes. The book is intended for those in academia and industry who use the conjugate gradient algorithm, including the many branches of science and engineering in which symmetric linear systems have to be solved.

Multiscale Model Reduction

Author: Eric Chung

Publisher: Springer Nature

Published: 2023-06-07

Total Pages: 499

ISBN-13: 3031204093

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This monograph is devoted to the study of multiscale model reduction methods from the point of view of multiscale finite element methods. Multiscale numerical methods have become popular tools for modeling processes with multiple scales. These methods allow reducing the degrees of freedom based on local offline computations. Moreover, these methods allow deriving rigorous macroscopic equations for multiscale problems without scale separation and high contrast. Multiscale methods are also used to design efficient solvers. This book offers a combination of analytical and numerical methods designed for solving multiscale problems. The book mostly focuses on methods that are based on multiscale finite element methods. Both applications and theoretical developments in this field are presented. The book is suitable for graduate students and researchers, who are interested in this topic.

Book Synopsis Multiscale Model Reduction by : Eric Chung

Download or read book Multiscale Model Reduction written by Eric Chung and published by Springer Nature. This book was released on 2023-06-07 with total page 499 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is devoted to the study of multiscale model reduction methods from the point of view of multiscale finite element methods. Multiscale numerical methods have become popular tools for modeling processes with multiple scales. These methods allow reducing the degrees of freedom based on local offline computations. Moreover, these methods allow deriving rigorous macroscopic equations for multiscale problems without scale separation and high contrast. Multiscale methods are also used to design efficient solvers. This book offers a combination of analytical and numerical methods designed for solving multiscale problems. The book mostly focuses on methods that are based on multiscale finite element methods. Both applications and theoretical developments in this field are presented. The book is suitable for graduate students and researchers, who are interested in this topic.