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Proper Orthogonal Decomposition Methods For Partial Differential Equations

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Proper Orthogonal Decomposition Methods for Partial Differential Equations

Proper Orthogonal Decomposition Methods for Partial Differential Equations Book
Author : Zhendong Luo,Goong Chen
Publisher : Academic Press
Release : 2018-11-26
ISBN : 0128167998
Language : En, Es, Fr & De

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Book Description :

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

Reduced Basis Methods for Partial Differential Equations Book
Author : Alfio Quarteroni,Andrea Manzoni,Federico Negri
Publisher : Springer
Release : 2015-08-19
ISBN : 3319154311
Language : En, Es, Fr & De

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Book Description :

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

Separated Representations and PGD Based Model Reduction

Separated Representations and PGD Based Model Reduction Book
Author : Francisco Chinesta,Pierre Ladevèze
Publisher : Springer
Release : 2014-09-02
ISBN : 3709117941
Language : En, Es, Fr & De

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Book Description :

The papers in this volume start with a description of the construction of reduced models through a review of Proper Orthogonal Decomposition (POD) and reduced basis models, including their mathematical foundations and some challenging applications, then followed by a description of a new generation of simulation strategies based on the use of separated representations (space-parameters, space-time, space-time-parameters, space-space,...), which have led to what is known as Proper Generalized Decomposition (PGD) techniques. The models can be enriched by treating parameters as additional coordinates, leading to fast and inexpensive online calculations based on richer offline parametric solutions. Separated representations are analyzed in detail in the course, from their mathematical foundations to their most spectacular applications. It is also shown how such an approximation could evolve into a new paradigm in computational science, enabling one to circumvent various computational issues in a vast array of applications in engineering science.

Incremental Proper Orthogonal Decomposition for PDE Simulation Data

Incremental Proper Orthogonal Decomposition for PDE Simulation Data Book
Author : Hiba Ghassan Fareed
Publisher : Unknown
Release : 2018
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

"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.

Model Reduction of Parametrized Systems

Model Reduction of Parametrized Systems Book
Author : Peter Benner,Mario Ohlberger,Anthony Patera,Gianluigi Rozza,Karsten Urban
Publisher : Springer
Release : 2017-09-05
ISBN : 3319587862
Language : En, Es, Fr & De

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Book Description :

The special volume offers a global guide to new concepts and approaches concerning the following topics: reduced basis methods, proper orthogonal decomposition, proper generalized decomposition, approximation theory related to model reduction, learning theory and compressed sensing, stochastic and high-dimensional problems, system-theoretic methods, nonlinear model reduction, reduction of coupled problems/multiphysics, optimization and optimal control, state estimation and control, reduced order models and domain decomposition methods, Krylov-subspace and interpolatory methods, and applications to real industrial and complex problems. The book represents the state of the art in the development of reduced order methods. It contains contributions from internationally respected experts, guaranteeing a wide range of expertise and topics. Further, it reflects an important effor t, carried out over the last 12 years, to build a growing research community in this field. Though not a textbook, some of the chapters can be used as reference materials or lecture notes for classes and tutorials (doctoral schools, master classes).

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

Snapshot Location in Proper Orthogonal Decomposition for Linear and Semi linear Parabolic Partial Differential Equations Book
Author : Zhiheng Liu
Publisher : Unknown
Release : 2013
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

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.

Proper Orthogonal Decomposition in Optimal Control of Fluids

Proper Orthogonal Decomposition in Optimal Control of Fluids Book
Author : National Aeronautics and Space Adm Nasa
Publisher : Independently Published
Release : 2018-09-16
ISBN : 9781723748813
Language : En, Es, Fr & De

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Book Description :

In this article, we present a reduced order modeling approach suitable for active control of fluid dynamical systems based on proper orthogonal decomposition (POD). The rationale behind the reduced order modeling is that numerical simulation of Navier-Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. We examine the possibility of obtaining reduced order models that reduce computational complexity associated with the Navier-Stokes equations while capturing the essential dynamics by using the POD. The POD allows extraction of certain optimal set of basis functions, perhaps few, from a computational or experimental data-base through an eigenvalue analysis. The solution is then obtained as a linear combination of these optimal set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations. We here use it in active control of fluid flows governed by the Navier-Stokes equations. We show that the resulting reduced order model can be very efficient for the computations of optimization and control problems in unsteady flows. Finally, implementational issues and numerical experiments are presented for simulations and optimal control of fluid flow through channels.Ravindran, S. S.Langley Research CenterNAVIER-STOKES EQUATION; TURBULENT FLOW; OPTIMAL CONTROL; UNSTEADY FLOW; ACTIVE CONTROL; FLUID FLOW; GALERKIN METHOD; PARTIAL DIFFERENTIAL EQUATIONS; ORTHOGONAL FUNCTIONS; DYNAMICAL SYSTEMS; EIGENVALUES; SIMULATION; MODELS

Model Reduction and Approximation

Model Reduction and Approximation Book
Author : Peter Benner,Albert Cohen,Mario Ohlberger,Karen Willcox
Publisher : SIAM
Release : 2017-07-06
ISBN : 161197481X
Language : En, Es, Fr & De

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Book Description :

Many physical, chemical, biomedical, and technical processes can be described by partial differential equations or dynamical systems. In spite of increasing computational capacities, many problems are of such high complexity that they are solvable only with severe simplifications, and the design of efficient numerical schemes remains a central research challenge. This book presents a tutorial introduction to recent developments in mathematical methods for model reduction and approximation of complex systems. Model Reduction and Approximation: Theory and Algorithms contains three parts that cover (I) sampling-based methods, such as the reduced basis method and proper orthogonal decomposition, (II) approximation of high-dimensional problems by low-rank tensor techniques, and (III) system-theoretic methods, such as balanced truncation, interpolatory methods, and the Loewner framework. It is tutorial in nature, giving an accessible introduction to state-of-the-art model reduction and approximation methods. It also covers a wide range of methods drawn from typically distinct communities (sampling based, tensor based, system-theoretic).?? This book is intended for researchers interested in model reduction and approximation, particularly graduate students and young researchers.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Certified Reduced Basis Methods for Parametrized Partial Differential Equations Book
Author : Jan S Hesthaven,Gianluigi Rozza,Benjamin Stamm
Publisher : Springer
Release : 2015-08-20
ISBN : 3319224700
Language : En, Es, Fr & De

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Book Description :

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

Constrained Optimization and Optimal Control for Partial Differential Equations Book
Author : Günter Leugering,Sebastian Engell,Andreas Griewank,Michael Hinze,Rolf Rannacher,Volker Schulz,Michael Ulbrich,Stefan Ulbrich
Publisher : Springer Science & Business Media
Release : 2012-01-03
ISBN : 3034801335
Language : En, Es, Fr & De

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Book Description :

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.

Proper Orthogonal Decomposition in Optimal Control of Fluids

Proper Orthogonal Decomposition in Optimal Control of Fluids Book
Author : S. S. Ravindran
Publisher : Unknown
Release : 1999
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download Proper Orthogonal Decomposition in Optimal Control of Fluids book written by S. S. Ravindran, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Optimal and Suboptimal Control of Partial Differential Equations

Optimal and Suboptimal Control of Partial Differential Equations Book
Author : Stefan Volkwein
Publisher : Unknown
Release : 2001
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download Optimal and Suboptimal Control of Partial Differential Equations book written by Stefan Volkwein, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Trust region Proper Orthogonal Decomposition for Flow Control

Trust region Proper Orthogonal Decomposition for Flow Control Book
Author : E. Arian,Institute for Computer Applications in Science and Engineering
Publisher : Unknown
Release : 2000
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

The proper orthogonal decomposition (POD) is a model reduction technique for the simulation of physical processes governed by partial differential equations, e.g., fluid flows. It can also be used to develop reduced order control models. Fundamental is the computation of POD basis functions that represent the influence of the control action on the system in order to get a suitable control model. We present an approach where suitable reduced order models are derived successively and give global convergence results.

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 Book
Author : Spencer J. Sherwin
Publisher : Springer Nature
Release : 2020
ISBN : 3030396479
Language : En, Es, Fr & De

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Book Description :

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.

Numerical and Evolutionary Optimization 2018

Numerical and Evolutionary Optimization 2018 Book
Author : Adriana Lara,Marcela Quiroz,Efrén Mezura-Montes,Oliver Schütze
Publisher : MDPI
Release : 2019-11-19
ISBN : 3039218166
Language : En, Es, Fr & De

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Book Description :

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.

Control and Optimization with PDE Constraints

Control and Optimization with PDE Constraints Book
Author : Kristian Bredies,Christian Clason,Karl Kunisch,Gregory Winckel
Publisher : Springer Science & Business Media
Release : 2013-06-12
ISBN : 3034806310
Language : En, Es, Fr & De

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Book Description :

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.

Advances on Links Between Mathematics and Industry

Advances on Links Between Mathematics and Industry Book
Author : Peregrina Quintela Estévez
Publisher : Springer Nature
Release : 2021-09-20
ISBN : 3030592235
Language : En, Es, Fr & De

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Book Description :

Download Advances on Links Between Mathematics and Industry book written by Peregrina Quintela Estévez, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Control and Estimation of Distributed Parameter Systems

Control and Estimation of Distributed Parameter Systems Book
Author : Wolfgang Desch,Gertrud Desch,Franz Kappel,Karl Kunisch
Publisher : Springer Science & Business Media
Release : 2003
ISBN : 9783764370046
Language : En, Es, Fr & De

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Book Description :

Consisting of 16 refereed original contributions, this volume presents a diversified collection of recent results in control of distributed parameter systems, including applications in fluid mechanics, partial differential equations, perturbation theory and shape optimization. Advanced graduate students and researchers will find the book an excellent guide to the forefront of control and estimation of distributed pa

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016 Book
Author : Marco L. Bittencourt,Ney A. Dumont,Jan S. Hesthaven
Publisher : Springer
Release : 2017-11-07
ISBN : 3319658700
Language : En, Es, Fr & De

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Book Description :

This book features a selection of high-quality papers chosen from the best presentations at the International Conference on Spectral and High-Order Methods (2016), 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.