# Proper Orthogonal Decomposition Methods for Partial Differential Equations

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## Proper Orthogonal Decomposition Methods for Partial Differential Equations

Author | : Zhendong Luo,Goong Chen |

Publsiher | : Academic Press |

Total Pages | : 278 |

Release | : 2018-11-26 |

ISBN 10 | : 0128167998 |

ISBN 13 | : 9780128167991 |

Language | : EN, FR, DE, ES & NL |

**Proper Orthogonal Decomposition Methods for Partial Differential Equations Book Review:**

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

Author | : Alfio Quarteroni,Andrea Manzoni,Federico Negri |

Publsiher | : Springer |

Total Pages | : 296 |

Release | : 2015-08-19 |

ISBN 10 | : 3319154311 |

ISBN 13 | : 9783319154312 |

Language | : EN, FR, DE, ES & NL |

**Reduced Basis Methods for Partial Differential Equations Book Review:**

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

## Incremental Proper Orthogonal Decomposition for PDE Simulation Data

Author | : Hiba Ghassan Fareed |

Publsiher | : Unknown |

Total Pages | : 96 |

Release | : 2018 |

ISBN 10 | : 1928374650XXX |

ISBN 13 | : OCLC:1041858451 |

Language | : EN, FR, DE, ES & NL |

**Incremental Proper Orthogonal Decomposition for PDE Simulation Data Book Review:**

"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.

## Separated Representations and PGD Based Model Reduction

Author | : Francisco Chinesta,Pierre Ladevèze |

Publsiher | : Springer |

Total Pages | : 227 |

Release | : 2014-09-02 |

ISBN 10 | : 3709117941 |

ISBN 13 | : 9783709117941 |

Language | : EN, FR, DE, ES & NL |

**Separated Representations and PGD Based Model Reduction Book Review:**

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.

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

Author | : Zhiheng Liu |

Publsiher | : Unknown |

Total Pages | : 135 |

Release | : 2013 |

ISBN 10 | : 1928374650XXX |

ISBN 13 | : OCLC:905611871 |

Language | : EN, FR, DE, ES & NL |

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

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

Author | : Jan S Hesthaven,Gianluigi Rozza,Benjamin Stamm |

Publsiher | : Springer |

Total Pages | : 131 |

Release | : 2015-08-20 |

ISBN 10 | : 3319224700 |

ISBN 13 | : 9783319224701 |

Language | : EN, FR, DE, ES & NL |

**Certified Reduced Basis Methods for Parametrized Partial Differential Equations Book Review:**

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.

## Model Reduction of Parametrized Systems

Author | : Peter Benner,Mario Ohlberger,Anthony Patera,Gianluigi Rozza,Karsten Urban |

Publsiher | : Springer |

Total Pages | : 504 |

Release | : 2017-09-05 |

ISBN 10 | : 3319587862 |

ISBN 13 | : 9783319587868 |

Language | : EN, FR, DE, ES & NL |

**Model Reduction of Parametrized Systems Book Review:**

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).

## Constrained Optimization and Optimal Control for Partial Differential Equations

Author | : Günter Leugering,Sebastian Engell,Andreas Griewank,Michael Hinze,Rolf Rannacher,Volker Schulz,Michael Ulbrich,Stefan Ulbrich |

Publsiher | : Springer Science & Business Media |

Total Pages | : 624 |

Release | : 2012-01-03 |

ISBN 10 | : 3034801335 |

ISBN 13 | : 9783034801331 |

Language | : EN, FR, DE, ES & NL |

**Constrained Optimization and Optimal Control for Partial Differential Equations Book Review:**

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

Author | : S. S. Ravindran |

Publsiher | : Unknown |

Total Pages | : 30 |

Release | : 1999 |

ISBN 10 | : 1928374650XXX |

ISBN 13 | : NASA:31769000632292 |

Language | : EN, FR, DE, ES & NL |

**Proper Orthogonal Decomposition in Optimal Control of Fluids Book Review:**

## Multiscale Wavelet Methods for Partial Differential Equations

Author | : Wolfgang Dahmen,Andrew Kurdila,Peter Oswald |

Publsiher | : Elsevier |

Total Pages | : 570 |

Release | : 1997-08-13 |

ISBN 10 | : 9780080537146 |

ISBN 13 | : 0080537146 |

Language | : EN, FR, DE, ES & NL |

**Multiscale Wavelet Methods for Partial Differential Equations Book Review:**

This latest volume in the Wavelets Analysis and Its Applications Series provides significant and up-to-date insights into recent developments in the field of wavelet constructions in connection with partial differential equations. Specialists in numerical applications and engineers in a variety of fields will find Multiscale Wavelet for Partial Differential Equations to be a valuable resource. Covers important areas of computational mechanics such as elasticity and computational fluid dynamics Includes a clear study of turbulence modeling Contains recent research on multiresolution analyses with operator-adapted wavelet discretizations Presents well-documented numerical experiments connected with the development of algorithms, useful in specific applications

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

Author | : Spencer J. Sherwin |

Publsiher | : Springer Nature |

Total Pages | : 658 |

Release | : 2020 |

ISBN 10 | : 3030396479 |

ISBN 13 | : 9783030396473 |

Language | : EN, FR, DE, ES & NL |

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

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

Author | : Adriana Lara,Marcela Quiroz,Efrén Mezura-Montes,Oliver Schütze |

Publsiher | : MDPI |

Total Pages | : 230 |

Release | : 2019-11-19 |

ISBN 10 | : 3039218166 |

ISBN 13 | : 9783039218165 |

Language | : EN, FR, DE, ES & NL |

**Numerical and Evolutionary Optimization 2018 Book Review:**

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.

## Proper Orthogonal Decomposition in Optimal Control of Fluids

Author | : National Aeronautics and Space Adm Nasa |

Publsiher | : Independently Published |

Total Pages | : 32 |

Release | : 2018-09-16 |

ISBN 10 | : 9781723748813 |

ISBN 13 | : 1723748811 |

Language | : EN, FR, DE, ES & NL |

**Proper Orthogonal Decomposition in Optimal Control of Fluids Book Review:**

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

## Recent Trends in Computational Engineering CE2014

Author | : Miriam Mehl,Manfred Bischoff,Michael Schäfer |

Publsiher | : Springer |

Total Pages | : 326 |

Release | : 2015-10-12 |

ISBN 10 | : 3319229974 |

ISBN 13 | : 9783319229973 |

Language | : EN, FR, DE, ES & NL |

**Recent Trends in Computational Engineering CE2014 Book Review:**

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.

## Control and Optimization with PDE Constraints

Author | : Kristian Bredies,Christian Clason,Karl Kunisch,Gregory Winckel |

Publsiher | : Springer Science & Business Media |

Total Pages | : 215 |

Release | : 2013-06-12 |

ISBN 10 | : 3034806310 |

ISBN 13 | : 9783034806312 |

Language | : EN, FR, DE, ES & NL |

**Control and Optimization with PDE Constraints Book Review:**

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.

## Trust region Proper Orthogonal Decomposition for Flow Control

Author | : E. Arian,Institute for Computer Applications in Science and Engineering |

Publsiher | : Unknown |

Total Pages | : 18 |

Release | : 2000 |

ISBN 10 | : 1928374650XXX |

ISBN 13 | : NASA:31769000711625 |

Language | : EN, FR, DE, ES & NL |

**Trust region Proper Orthogonal Decomposition for Flow Control Book Review:**

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.

## Advances on Links Between Mathematics and Industry

Author | : Peregrina Quintela Estévez |

Publsiher | : Springer Nature |

Total Pages | : 135 |

Release | : 2021 |

ISBN 10 | : 3030592235 |

ISBN 13 | : 9783030592233 |

Language | : EN, FR, DE, ES & NL |

**Advances on Links Between Mathematics and Industry Book Review:**

## Control and Estimation of Distributed Parameter Systems

Author | : Wolfgang Desch,Gertrud Desch,Franz Kappel,Karl Kunisch |

Publsiher | : Springer Science & Business Media |

Total Pages | : 269 |

Release | : 2003 |

ISBN 10 | : 9783764370046 |

ISBN 13 | : 3764370041 |

Language | : EN, FR, DE, ES & NL |

**Control and Estimation of Distributed Parameter Systems Book Review:**

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

## Optimal Control of Partial Differential Equations

Author | : Fredi Tröltzsch |

Publsiher | : American Mathematical Soc. |

Total Pages | : 399 |

Release | : 2010 |

ISBN 10 | : 0821849042 |

ISBN 13 | : 9780821849040 |

Language | : EN, FR, DE, ES & NL |

**Optimal Control of Partial Differential Equations Book Review:**

"Optimal control theory is concerned with finding control functions that minimize cost functions for systems described by differential equations. The methods have found widespread applications in aeronautics, mechanical engineering, the life sciences, and many other disciplines. This book focuses on optimal control problems where the state equation is an elliptic or parabolic partial differential equation. Included are topics such as the existence of optimal solutions, necessary optimality conditions and adjoint equations, second-order sufficient conditions, and main principles of selected numerical techniques. It also contains a survey on the Karush-Kuhn-Tucker theory of nonlinear programming in Banach spaces. The exposition begins with control problems with linear equations, quadratic cost functions and control constraints. To make the book self-contained, basic facts on weak solutions of elliptic and parabolic equations are introduced. Principles of functional analysis are introduced and explained as they are needed. Many simple examples illustrate the theory and its hidden difficulties. This start to the book makes it fairly self-contained and suitable for advanced undergraduates or beginning graduate students. Advanced control problems for nonlinear partial differential equations are also discussed. As prerequisites, results on boundedness and continuity of solutions to semilinear elliptic and parabolic equations are addressed. These topics are not yet readily available in books on PDEs, making the exposition also interesting for researchers. Alongside the main theme of the analysis of problems of optimal control, Tr'oltzsch also discusses numerical techniques. The exposition is confined to brief introductions into the basic ideas in order to give the reader an impression of how the theory can be realized numerically. After reading this book, the reader will be familiar with the main principles of the numerical analysis of PDE-constrained optimization."--Publisher's description.

## Lagrange Multiplier Approach to Variational Problems and Applications

Author | : Kazufumi Ito,Karl Kunisch |

Publsiher | : SIAM |

Total Pages | : 341 |

Release | : 2008-11-06 |

ISBN 10 | : 0898716497 |

ISBN 13 | : 9780898716498 |

Language | : EN, FR, DE, ES & NL |

**Lagrange Multiplier Approach to Variational Problems and Applications Book Review:**

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