Proper Orthogonal Decomposition Methods for Partial Differential Equations Book

Proper Orthogonal Decomposition Methods for Partial Differential Equations


  • Author : Zhendong Luo
  • Publisher : Academic Press
  • Release Date : 2018-11-26
  • Genre: Mathematics
  • Pages : 278
  • ISBN 10 : 9780128167991

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Proper Orthogonal Decomposition Methods for Partial Differential Equations Excerpt :

Proper Orthogonal Decomposition Methods for Partial Differential Equations evaluates the potential applications of POD reduced-order numerical methods in increasing computational efficiency, decreasing calculating load and alleviating the accumulation of truncation error in the computational process. Introduces the foundations of finite-differences, finite-elements and finite-volume-elements. Models of time-dependent PDEs are presented, with detailed numerical procedures, implementation and error analysis. Output numerical data are plotted in graphics and compared using standard traditional methods. These models contain parabolic, hyperbolic and nonlinear systems of PDEs, suitable for the user to learn and adapt methods to their own R&D problems. Explains ways to reduce order for PDEs by means of the POD method so that reduced-order models have few unknowns Helps readers speed up computation and reduce computation load and memory requirements while numerically capturing system characteristics Enables readers to apply and adapt the methods to solve similar problems for PDEs of hyperbolic, parabolic and nonlinear types

Reduced Basis Methods for Partial Differential Equations Book

Reduced Basis Methods for Partial Differential Equations


  • Author : Alfio Quarteroni
  • Publisher : Springer
  • Release Date : 2015-08-19
  • Genre: Mathematics
  • Pages : 296
  • ISBN 10 : 9783319154312

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Reduced Basis Methods for Partial Differential Equations Excerpt :

This book provides a basic introduction to reduced basis (RB) methods for problems involving the repeated solution of partial differential equations (PDEs) arising from engineering and applied sciences, such as PDEs depending on several parameters and PDE-constrained optimization. The book presents a general mathematical formulation of RB methods, analyzes their fundamental theoretical properties, discusses the related algorithmic and implementation aspects, and highlights their built-in algebraic and geometric structures. More specifically, the authors discuss alternative strategies for constructing accurate RB spaces using greedy algorithms and proper orthogonal decomposition techniques, investigate their approximation properties and analyze offline-online decomposition strategies aimed at the reduction of computational complexity. Furthermore, they carry out both a priori and a posteriori error analysis. The whole mathematical presentation is made more stimulating by the use of representative examples of applicative interest in the context of both linear and nonlinear PDEs. Moreover, the inclusion of many pseudocodes allows the reader to easily implement the algorithms illustrated throughout the text. The book will be ideal for upper undergraduate students and, more generally, people interested in scientific computing. All these pseudocodes are in fact implemented in a MATLAB package that is freely available at https://github.com/redbkit

Snapshot Location in Proper Orthogonal Decomposition for Linear and Semi linear Parabolic Partial Differential Equations Book

Snapshot Location in Proper Orthogonal Decomposition for Linear and Semi linear Parabolic Partial Differential Equations


  • Author : Zhiheng Liu
  • Publisher : Unknown
  • Release Date : 2013
  • Genre: Mathematics
  • Pages : null
  • ISBN 10 : OCLC:905611871

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Snapshot Location in Proper Orthogonal Decomposition for Linear and Semi linear Parabolic Partial Differential Equations Excerpt :

It is well-known that the performance of POD and POD-DEIM methods depends on the selection of the snapshot locations. In this work, we consider the selections of the locations for POD and POD-DEIM snapshots for spatially semi-discretized linear or semi-linear parabolic PDEs. We present an approach that for a fixed number of snapshots the optimal locations may be selected such that the global discretization error is approximately the same in each associated sub-interval. The global discretization error is assessed by a hierarchical-type a posteriori error estimator developed from automatic time-stepping for systems of ODEs. We compare the global discretization error of this snapshot selection on error equilibration for the full order model (\textbf{FOM}) with that for the reduced order model (\textbf{ROM}) to study its impact. This contribution also shows that the equilibration of the global discretization error for the \textbf{FOM} is preserved by its corresponding POD and POD-DEIM-based \textbf{ROM}. The numerical examples illustrating the performance of this approach are provided.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations Book

Certified Reduced Basis Methods for Parametrized Partial Differential Equations


  • Author : Jan S Hesthaven
  • Publisher : Springer
  • Release Date : 2015-08-20
  • Genre: Mathematics
  • Pages : 131
  • ISBN 10 : 9783319224701

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Certified Reduced Basis Methods for Parametrized Partial Differential Equations Excerpt :

This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

Constrained Optimization and Optimal Control for Partial Differential Equations Book

Constrained Optimization and Optimal Control for Partial Differential Equations


  • Author : Günter Leugering
  • Publisher : Springer Science & Business Media
  • Release Date : 2012-01-03
  • Genre: Mathematics
  • Pages : 622
  • ISBN 10 : 9783034801331

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Constrained Optimization and Optimal Control for Partial Differential Equations Excerpt :

This special volume focuses on optimization and control of processes governed by partial differential equations. The contributors are mostly participants of the DFG-priority program 1253: Optimization with PDE-constraints which is active since 2006. The book is organized in sections which cover almost the entire spectrum of modern research in this emerging field. Indeed, even though the field of optimal control and optimization for PDE-constrained problems has undergone a dramatic increase of interest during the last four decades, a full theory for nonlinear problems is still lacking. The contributions of this volume, some of which have the character of survey articles, therefore, aim at creating and developing further new ideas for optimization, control and corresponding numerical simulations of systems of possibly coupled nonlinear partial differential equations. The research conducted within this unique network of groups in more than fifteen German universities focuses on novel methods of optimization, control and identification for problems in infinite-dimensional spaces, shape and topology problems, model reduction and adaptivity, discretization concepts and important applications. Besides the theoretical interest, the most prominent question is about the effectiveness of model-based numerical optimization methods for PDEs versus a black-box approach that uses existing codes, often heuristic-based, for optimization.

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 Book

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018


  • Author : Spencer J. Sherwin
  • Publisher : Springer Nature
  • Release Date : 2020-08-11
  • Genre: Mathematics
  • Pages : 658
  • ISBN 10 : 9783030396473

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Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 Excerpt :

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions.

Snapshot Based Methods and Algorithms Book

Snapshot Based Methods and Algorithms


  • Author : Peter Benner
  • Publisher : Walter de Gruyter GmbH & Co KG
  • Release Date : 2020-12-16
  • Genre: Mathematics
  • Pages : 356
  • ISBN 10 : 9783110671506

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Snapshot Based Methods and Algorithms Excerpt :

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science.

Numerical and Evolutionary Optimization 2018 Book

Numerical and Evolutionary Optimization 2018


  • Author : Adriana Lara
  • Publisher : MDPI
  • Release Date : 2019-11-19
  • Genre: Technology & Engineering
  • Pages : 230
  • ISBN 10 : 9783039218165

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Numerical and Evolutionary Optimization 2018 Excerpt :

This book was established after the 6th International Workshop on Numerical and Evolutionary Optimization (NEO), representing a collection of papers on the intersection of the two research areas covered at this workshop: numerical optimization and evolutionary search techniques. While focusing on the design of fast and reliable methods lying across these two paradigms, the resulting techniques are strongly applicable to a broad class of real-world problems, such as pattern recognition, routing, energy, lines of production, prediction, and modeling, among others. This volume is intended to serve as a useful reference for mathematicians, engineers, and computer scientists to explore current issues and solutions emerging from these mathematical and computational methods and their applications.

Incremental Proper Orthogonal Decomposition for PDE Simulation Data Book

Incremental Proper Orthogonal Decomposition for PDE Simulation Data


  • Author : Hiba Ghassan Fareed
  • Publisher : Unknown
  • Release Date : 2018
  • Genre: Uncategoriezed
  • Pages : 96
  • ISBN 10 : OCLC:1041858451

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Incremental Proper Orthogonal Decomposition for PDE Simulation Data Excerpt :

"We propose an incremental algorithm to compute the proper orthogonal decomposition (POD) of simulation data for a partial differential equation. Specifically, we modify an incremental matrix SVD algorithm of Brand to accommodate data arising from Galerkin-type simulation methods for time dependent PDEs. We introduce an incremental SVD algorithm with respect to a weighted inner product to compute the proper orthogonal decomposition (POD). The algorithm is applicable to data generated by many numerical methods for PDEs, including finite element and discontinuous Galerkin methods. We also modify the algorithm to initialize and incrementally update both the SVDand an error bound during the time stepping in a PDE solver without storing the simulation data. We show the algorithm produces the exact SVD of an approximate data matrix, and the operator norm error between the approximate and exact data matrices is bounded above by the computed error bound. This error bound also allows us to bound the error in the incrementally computed singular values and singular vectors. We demonstrate the effectiveness of the algorithm using finite element computations for a 1D Burgers' equation, a 1D FitzHugh-Nagumo PDE system, and a 2D Navier-Stokes problem"--Abstract, page iv.

Recent Trends in Computational Engineering   CE2014 Book

Recent Trends in Computational Engineering CE2014


  • Author : Miriam Mehl
  • Publisher : Springer
  • Release Date : 2015-10-12
  • Genre: Computers
  • Pages : 326
  • ISBN 10 : 9783319229973

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Recent Trends in Computational Engineering CE2014 Excerpt :

This book presents selected papers from the 3rd International Workshop on Computational Engineering held in Stuttgart from October 6 to 10, 2014, bringing together innovative contributions from related fields with computer science and mathematics as an important technical basis among others. The workshop discussed the state of the art and the further evolution of numerical techniques for simulation in engineering and science. We focus on current trends in numerical simulation in science and engineering, new requirements arising from rapidly increasing parallelism in computer architectures, and novel mathematical approaches. Accordingly, the chapters of the book particularly focus on parallel algorithms and performance optimization, coupled systems, and complex applications and optimization.

Lagrange Multiplier Approach to Variational Problems and Applications Book

Lagrange Multiplier Approach to Variational Problems and Applications


  • Author : Kazufumi Ito
  • Publisher : SIAM
  • Release Date : 2008-11-06
  • Genre: Mathematics
  • Pages : 354
  • ISBN 10 : 9780898716498

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Lagrange Multiplier Approach to Variational Problems and Applications Excerpt :

Analyses Lagrange multiplier theory and demonstrates its impact on the development of numerical algorithms for variational problems in function spaces.

Advances on Links Between Mathematics and Industry Book

Advances on Links Between Mathematics and Industry


  • Author : Peregrina Quintela Estévez
  • Publisher : Springer Nature
  • Release Date : 2021-03-01
  • Genre: Mathematics
  • Pages : 154
  • ISBN 10 : 9783030592233

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Advances on Links Between Mathematics and Industry Excerpt :

This book results from the talks presented at the First Conference on Transfer between Mathematics & Industry (CTMI 2019). Its goal is to promote and disseminate the mathematical tools for Statistics & Big Data, MSO (Modeling, Simulation and Optimization) and their industrial applications. In this volume, the reader will find innovative advances in the automotive, energy, railway, logistics, and materials sectors. In addition, Advances CTMI 2019 promotes the opening of new research lines aiming to provide suitable solutions for the industrial and societal challenges. Fostering effective interaction between Academia and Industry is our main purpose with this book. CTMI conferences are one of the main forums where significant advances in industrial mathematics are presented, bringing together outstanding leaders from business, science and Academia to promote the use of mathematics for an innovative industry.

Control and Optimization with PDE Constraints Book

Control and Optimization with PDE Constraints


  • Author : Kristian Bredies
  • Publisher : Springer Science & Business Media
  • Release Date : 2013-06-12
  • Genre: Mathematics
  • Pages : 215
  • ISBN 10 : 9783034806312

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Control and Optimization with PDE Constraints Excerpt :

Many mathematical models of physical, biological and social systems involve partial differential equations (PDEs). The desire to understand and influence these systems naturally leads to considering problems of control and optimization. This book presents important topics in the areas of control of PDEs and of PDE-constrained optimization, covering the full spectrum from analysis to numerical realization and applications. Leading scientists address current topics such as non-smooth optimization, Hamilton–Jacobi–Bellmann equations, issues in optimization and control of stochastic partial differential equations, reduced-order models and domain decomposition, discretization error estimates for optimal control problems, and control of quantum-dynamical systems. These contributions originate from the “International Workshop on Control and Optimization of PDEs” in Mariatrost in October 2011. This book is an excellent resource for students and researchers in control or optimization of differential equations. Readers interested in theory or in numerical algorithms will find this book equally useful.

Multiple Shooting and Time Domain Decomposition Methods Book

Multiple Shooting and Time Domain Decomposition Methods


  • Author : Thomas Carraro
  • Publisher : Springer
  • Release Date : 2015-10-26
  • Genre: Mathematics
  • Pages : 422
  • ISBN 10 : 9783319233215

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Multiple Shooting and Time Domain Decomposition Methods Excerpt :

This book offers a comprehensive collection of the most advanced numerical techniques for the efficient and effective solution of simulation and optimization problems governed by systems of time-dependent differential equations. The contributions present various approaches to time domain decomposition, focusing on multiple shooting and parareal algorithms. The range of topics covers theoretical analysis of the methods, as well as their algorithmic formulation and guidelines for practical implementation. Selected examples show that the discussed approaches are mandatory for the solution of challenging practical problems. The practicability and efficiency of the presented methods is illustrated by several case studies from fluid dynamics, data compression, image processing and computational biology, giving rise to possible new research topics. This volume, resulting from the workshop Multiple Shooting and Time Domain Decomposition Methods, held in Heidelberg in May 2013, will be of great interest to applied mathematicians, computer scientists and all scientists using mathematical methods.