# 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 | : 0128167998 |

Category | : Mathematics |

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

**Proper Orthogonal Decomposition Methods for Partial Differential Equations Book 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

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

Publsiher | : Springer |

Total Pages | : 296 |

Release | : 2015-08-19 |

ISBN | : 3319154311 |

Category | : Mathematics |

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

**Reduced Basis Methods for Partial Differential Equations Book 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

## 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 | : 3319224700 |

Category | : Mathematics |

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

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

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

Publsiher | : Springer Science & Business Media |

Total Pages | : 622 |

Release | : 2012-01-03 |

ISBN | : 3034801335 |

Category | : Mathematics |

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

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

Author | : Spencer J. Sherwin,David Moxey,Joaquim Peiró,Peter E. Vincent,Christoph Schwab |

Publsiher | : Springer Nature |

Total Pages | : 658 |

Release | : 2020-08-11 |

ISBN | : 3030396479 |

Category | : Mathematics |

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

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

## Proper Orthogonal Decomposition in Optimal Control of Fluids

Author | : S. S. Ravindran |

Publsiher | : Unknown |

Total Pages | : 30 |

Release | : 1999 |

ISBN | : 1928374650XXX |

Category | : Fluid dynamics |

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

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

## 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 | : 3039218166 |

Category | : Technology & Engineering |

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

**Numerical and Evolutionary Optimization 2018 Book 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.

## Snapshot Based Methods and Algorithms

Author | : Peter Benner,et al. |

Publsiher | : Walter de Gruyter GmbH & Co KG |

Total Pages | : 356 |

Release | : 2020-12-16 |

ISBN | : 3110671506 |

Category | : Mathematics |

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

**Snapshot Based Methods and Algorithms Book 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.

## 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 | : 1928374650XXX |

Category | : Mathematics |

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

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

## Recent Trends in Computational Engineering CE2014

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

Publsiher | : Springer |

Total Pages | : 326 |

Release | : 2015-10-12 |

ISBN | : 3319229974 |

Category | : Computers |

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

**Recent Trends in Computational Engineering CE2014 Book 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

Author | : Kazufumi Ito,Karl Kunisch |

Publsiher | : SIAM |

Total Pages | : 341 |

Release | : 2008-11-06 |

ISBN | : 0898716497 |

Category | : Mathematics |

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

**Lagrange Multiplier Approach to Variational Problems and Applications Book 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

Author | : Peregrina Quintela Estévez,Bartomeu Coll,Rosa M. Crujeiras,José Durany,Laureano Escudero |

Publsiher | : Springer Nature |

Total Pages | : 154 |

Release | : 2021-03-01 |

ISBN | : 3030592235 |

Category | : Mathematics |

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

**Advances on Links Between Mathematics and Industry Book 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

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

Publsiher | : Springer Science & Business Media |

Total Pages | : 215 |

Release | : 2013-06-12 |

ISBN | : 3034806310 |

Category | : Mathematics |

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

**Control and Optimization with PDE Constraints Book 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.

## Separated Representations and PGD Based Model Reduction

Author | : Francisco Chinesta,Pierre Ladevèze |

Publsiher | : Springer |

Total Pages | : 227 |

Release | : 2014-09-02 |

ISBN | : 3709117941 |

Category | : Technology & Engineering |

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

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

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.

## Multiple Shooting and Time Domain Decomposition Methods

Author | : Thomas Carraro,Michael Geiger,Stefan Körkel,Rolf Rannacher |

Publsiher | : Springer |

Total Pages | : 422 |

Release | : 2015-10-26 |

ISBN | : 3319233211 |

Category | : Mathematics |

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

**Multiple Shooting and Time Domain Decomposition Methods Book 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.

## Reduced Order Modeling ROM for Simulation and Optimization

Author | : Winfried Keiper,Anja Milde,Stefan Volkwein |

Publsiher | : Springer |

Total Pages | : 179 |

Release | : 2018-04-11 |

ISBN | : 3319753193 |

Category | : Mathematics |

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

**Reduced Order Modeling ROM for Simulation and Optimization Book Excerpt:**

This edited monograph collects research contributions and addresses the advancement of efficient numerical procedures in the area of model order reduction (MOR) for simulation, optimization and control. The topical scope includes, but is not limited to, new out-of-the-box algorithmic solutions for scientific computing, e.g. reduced basis methods for industrial problems and MOR approaches for electrochemical processes. The target audience comprises research experts and practitioners in the field of simulation, optimization and control, but the book may also be beneficial for graduate students alike.

## Optimal Control of Partial Differential Equations

Author | : Fredi Tröltzsch |

Publsiher | : American Mathematical Soc. |

Total Pages | : 399 |

Release | : 2010 |

ISBN | : 0821849042 |

Category | : Mathematics |

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

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

"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.

## Real Time PDE Constrained Optimization

Author | : Lorenz T. Biegler,Omar Ghattas,Matthias Heinkenschloss,David Keyes,Bart van Bloemen Waanders |

Publsiher | : SIAM |

Total Pages | : 312 |

Release | : 2007-07-12 |

ISBN | : 0898716217 |

Category | : Mathematics |

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

**Real Time PDE Constrained Optimization Book Excerpt:**

“…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.

## System Modeling and Optimization

Author | : Adam Korytowski,Maciej Szymkat,Kazimierz Malanowski,Wojciech Mitkowski |

Publsiher | : Springer Science & Business Media |

Total Pages | : 507 |

Release | : 2009-10-15 |

ISBN | : 3642048013 |

Category | : Technology & Engineering |

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

**System Modeling and Optimization Book Excerpt:**

rd This book constitutes a collection of extended versions of papers presented at the 23 IFIP TC7 Conference on System Modeling and Optimization, which was held in C- cow, Poland, on July 23–27, 2007. It contains 7 plenary and 22 contributed articles, the latter selected via a peer reviewing process. Most of the papers are concerned with optimization and optimal control. Some of them deal with practical issues, e. g. , p- formance-based design for seismic risk reduction, or evolutionary optimization in structural engineering. Many contributions concern optimization of infini- dimensional systems, ranging from a general overview of the variational analysis, through optimization and sensitivity analysis of PDE systems, to optimal control of neutral systems. A significant group of papers is devoted to shape analysis and opti- zation. Sufficient optimality conditions for ODE problems, and stochastic control methods applied to mathematical finance, are also investigated. The remaining papers are on mathematical programming, modeling, and information technology. The conference was the 23rd event in the series of such meetings biennially org- ized under the auspices of the Seventh Technical Committee “Systems Modeling and Optimization” of the International Federation for Information Processing (IFIP TC7).

## Cardiovascular Mechanics

Author | : Michel F. Labrosse |

Publsiher | : CRC Press |

Total Pages | : 386 |

Release | : 2018-09-13 |

ISBN | : 1315280272 |

Category | : Medical |

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

**Cardiovascular Mechanics Book Excerpt:**

The objective of this book is to illustrate in specific detail how cardiovascular mechanics stands as a common pillar supporting such different clinical successes as drugs for high blood pressure, prosthetic heart valves and coronary artery bypass grafting, among others. This information is conveyed through a comprehensive treatment of the overarching principles and theories that are behind mechanobiological processes, aortic and arterial mechanics, atherosclerosis, blood and microcirculation, hear valve mechanics, as well as medical devices and drugs. Examines all major theoretical and practical aspects of mechanical forces related to the cardiovascular system. Discusses a unique coverage of mechanical changes related to an aging cardiovascular system. Provides an overview of experimental methods in cardiovascular mechanics. Written by world-class researchers from Canada, the US and EU. Extensive references are provided at the end of each chapter to enhance further study. Michel R. Labrosse is the founder of the Cardiovascular Mechanics Laboratory at the University of Ottawa, where he is a full professor within the Department of Mechanical Engineering. He has been an active researcher in academia along with being heavily associated with the University of Ottawa Heart Institute. He has authored or co-authored over 90 refereed communications, and supervised or co-supervised over 40 graduate students and post-docs.