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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

Incremental Proper Orthogonal Decomposition for PDE Simulation Data

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

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Book Description :

Abstract: 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. The algorithm is applicable to data generated by many numerical methods for PDEs, including finite element and discontinuous Galerkin methods. The algorithm initializes and efficiently updates the dominant POD eigenvalues and modes during the time stepping in a PDE solver without storing the simulation data. We prove that the algorithm without truncation updates the POD exactly. We demonstrate the effectiveness of the algorithm using finite element computations for a 1D Burgers' equation and a 2D Navier–Stokes problem.

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

Proper Orthogonal Decomposition in Optimal Control of Fluids

Proper Orthogonal Decomposition in Optimal Control of Fluids Book
Author : National Aeronautics and Space Administration (NASA)
Publisher : Createspace Independent Publishing Platform
Release : 2018-06-03
ISBN : 9781720667094
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.

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.

Real Time PDE Constrained Optimization

Real Time PDE Constrained Optimization Book
Author : Lorenz T. Biegler,Omar Ghattas,Matthias Heinkenschloss,David Keyes,Bart van Bloemen Waanders
Publisher : SIAM
Release : 2007-07-12
ISBN : 0898716217
Language : En, Es, Fr & De

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Book Description :

“…a timely contribution to a field of growing importance. This carefully edited book presents a rich collection of chapters ranging from mathematical methodology to emerging applications. I recommend it to students as a rigorous and comprehensive presentation of simulation-based optimization and to researchers as an overview of recent advances and challenges in the field.” — Jorge Nocedal, Professor, Northwestern University.Many engineering and scientific problems in design, control, and parameter estimation can be formulated as optimization problems that are governed by partial differential equations (PDEs). The complexities of the PDEs—and the requirement for rapid solution—pose significant difficulties. A particularly challenging class of PDE-constrained optimization problems is characterized by the need for real-time solution, i.e., in time scales that are sufficiently rapid to support simulation-based decision making. Real-Time PDE-Constrained Optimization, the first book devoted to real-time optimization for systems governed by PDEs, focuses on new formulations, methods, and algorithms needed to facilitate real-time, PDE-constrained optimization. In addition to presenting state-of-the-art algorithms and formulations, the text illustrates these algorithms with a diverse set of applications that includes problems in the areas of aerodynamics, biology, fluid dynamics, medicine, chemical processes, homeland security, and structural dynamics. Despite difficulties, there is a pressing need to capitalize on continuing advances in computing power to develop optimization methods that will replace simple rule-based decision making with optimized decisions based on complex PDE simulations. Audience The book is aimed at readers who have expertise in simulation and are interested in incorporating optimization into their simulations, who have expertise in numerical optimization and are interested in adapting optimization methods to the class of infinite-dimensional simulation problems, or who have worked in “offline” optimization contexts and are interested in moving to “online” optimization.Contents Preface; Part I: Concepts and Properties of Real-Time, Online Strategies. Chapter 1: Constrained Optimal Feedback Control of Systems Governed by Large Differential Algebraic Equations; Chapter 2: A Stabilizing Real-Time Implementation of Nonlinear Model Predictive Control; Chapter 3: Numerical Feedback Controller Design for PDE Systems Using Model Reduction: Techniques and Case Studies; Chapter 4: Least-Squares Finite Element Method for Optimization and Control Problems; Part II: Fast PDE-Constrained Optimization Solvers. Chapter 5: Space-Time Multigrid Methods for Solving Unsteady Optimal Control Problems; Chapter 6: A Time-Parallel Implicit Methodology for the Near-Real-Time Solution of Systems of Linear Oscillators; Chapter 7: Generalized SQP Methods with “Parareal” Time-Domain Decomposition for Time-Dependent PDE-Constrained Optimization; Chapter 8: Simultaneous Pseudo-Timestepping for State-Constrained Optimization Problems in Aerodynamics; Chapter 9: Digital Filter Stepsize Control in DASPK and Its Effect on Control Optimization Performance; Part III: Reduced Order Modeling. Chapter 10: Certified Rapid Solution of Partial Differential Equations for Real-Time Parameter Estimation and Optimization; Chapter 11: Model Reduction for Large-Scale Applications in Computational Fluid Dynamics; Chapter 12: Suboptimal Feedback Control of Flow Separation by POD Model Reduction; Part IV: Applications. Chapter 13: A Combined Shape-Newton and Topology Optimization Technique in Real-Time Image Segmentation; Chapter 14: COFIR: Coarse and Fine Image Registration; Chapter 15: Real-Time, Large Scale Optimization of Water Network Systems Using a Sub-domain Approach; Index.

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).

Proceedings

Proceedings Book
Author : Anonim
Publisher : Unknown
Release : 2002
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download Proceedings book written by , available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

SIAM Journal on Scientific Computing

SIAM Journal on Scientific Computing Book
Author : Anonim
Publisher : Unknown
Release : 2009
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download SIAM Journal on Scientific Computing book written by , available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Discussiones Mathematicae

Discussiones Mathematicae Book
Author : Anonim
Publisher : Unknown
Release : 2006
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download Discussiones Mathematicae book written by , available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

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.

Grazer mathematische Berichte

Grazer mathematische Berichte Book
Author : Anonim
Publisher : Unknown
Release : 1999
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Publications du Laboratoire Jacques Louis Lions

Publications du Laboratoire Jacques Louis Lions Book
Author : Anonim
Publisher : Unknown
Release : 2004
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

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Mathematical Reviews

Mathematical Reviews Book
Author : Anonim
Publisher : Unknown
Release : 2008
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

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Proper Orthogonal Decomposition and Tomographic Analysis of Combustion Systems for Control Applications

Proper Orthogonal Decomposition and Tomographic Analysis of Combustion Systems for Control Applications Book
Author : Jennifer Leslie Edwards
Publisher : Unknown
Release : 2007
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Control of combustion systems is of considerable importance to the improvement of system performance and is currently an active field of research. An understanding of combustion system dynamics is crucial to the development of effective control systems. Combustion dynamics and control combine the different aspects of combustion research such as theoretical analysis of the governing equations and phenomena, computational simulation, modeling, and measurement using advanced sensors.

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.

Nonlinear Dynamics and Controls

Nonlinear Dynamics and Controls Book
Author : Anil K. Bajaj,Navaratnam Sri Namachchivaya,Matthew A. Franchek
Publisher : Amer Society of Mechanical
Release : 1996
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Proceedings of the November 1996 symposium. Contains 35 papers related to theoretical, experimental, and computational aspects of dynamics, vibration, and control of nonlinear mechanical and structural systems. The papers are divided according to the symposium's seven technical sessions: analytical

Proceedings of the ASME Applied Mechanics Division

Proceedings of the ASME Applied Mechanics Division Book
Author : Anonim
Publisher : Unknown
Release : 2003
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

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