Download or read book Proper Orthogonal Decomposition Methods for Partial Differential Equations written by Zhendong Luo and published by Academic Press. This book was released on 2018-11-26 with total page 278 pages. Available in PDF, EPUB and Kindle. 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

Download or read book Proper Orthogonal Decomposition in Optimal Control of Fluids written by S. S. Ravindran and published by . This book was released on 1999 with total page 34 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Reduced Basis Methods for Partial Differential Equations written by Alfio Quarteroni and published by Springer. This book was released on 2015-08-19 with total page 305 pages. Available in PDF, EPUB and Kindle. 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

Download or read book Trust region Proper Orthogonal Decomposition for Flow Control written by E. Arian and published by . This book was released on 2000 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Proper Orthogonal Decomposition written by Sarah Katherine Locke and published by . This book was released on 2021 with total page 95 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Proper orthogonal decomposition (POD) projection errors and error bounds for POD reduced order models of partial differential equations have been studied by many. In this research we obtain new results regarding POD data approximation theory and present a new difference quotient (DQ) approach for computing the POD modes of the data. First, we improve on earlier results concerning POD projection errors by extending to a more general framework that allows for non-orthogonal POD projections and seminorms. We obtain new exact error formulas and convergence results for POD data approximation errors, and also prove new pointwise convergence results and error bounds for POD projections. We consider both the discrete and continuous cases of POD within this generalized framework. We also apply our results to several example problems, and show how the new results improve on previous work. Next, we consider the relationship between POD, difference quotients (DQs), and pointwise ROM error bounds. It is known that including DQs is necessary in order to prove optimal pointwise in time error bounds for POD reduced order models of the heat equation. We introduce a new approach to including DQs in the POD procedure to further investigate the role DQs play in POD numerical analysis. Instead of computing the POD modes using all of the snapshot data and DQs, we only use the first snapshot along with all of the DQs and special POD weights. We show that this approach retains all of the numerical analysis benefits of the standard POD DQ approach, while using a POD data set that has half the number of snapshots as the standard POD DQ approach, i.e., the new approach is more computationally efficient. We illustrate our theoretical results with numerical experiments"--Abstract, page iii.

Download or read book Snapshot Location in Proper Orthogonal Decomposition for Linear and Semi linear Parabolic Partial Differential Equations written by Zhiheng Liu and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. 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.

Download or read book Certified Reduced Basis Methods for Parametrized Partial Differential Equations written by Jan S Hesthaven and published by Springer. This book was released on 2015-08-20 with total page 139 pages. Available in PDF, EPUB and Kindle. 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.

Download or read book Model Order Reduction Theory Research Aspects and Applications written by Wilhelmus H. Schilders and published by Springer Science & Business Media. This book was released on 2008-08-27 with total page 471 pages. Available in PDF, EPUB and Kindle. Book excerpt: The idea for this book originated during the workshop “Model order reduction, coupled problems and optimization” held at the Lorentz Center in Leiden from S- tember 19–23, 2005. During one of the discussion sessions, it became clear that a book describing the state of the art in model order reduction, starting from the very basics and containing an overview of all relevant techniques, would be of great use for students, young researchers starting in the ?eld, and experienced researchers. The observation that most of the theory on model order reduction is scattered over many good papers, making it dif?cult to ?nd a good starting point, was supported by most of the participants. Moreover, most of the speakers at the workshop were willing to contribute to the book that is now in front of you. The goal of this book, as de?ned during the discussion sessions at the workshop, is three-fold: ?rst, it should describe the basics of model order reduction. Second, both general and more specialized model order reduction techniques for linear and nonlinear systems should be covered, including the use of several related numerical techniques. Third, the use of model order reduction techniques in practical appli- tions and current research aspects should be discussed. We have organized the book according to these goals. In Part I, the rationale behind model order reduction is explained, and an overview of the most common methods is described.

Download or read book Numerical Homogenization by Localized Decomposition written by Axel Målqvist and published by SIAM. This book was released on 2020-11-23 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the first survey of the Localized Orthogonal Decomposition (LOD) method, a pioneering approach for the numerical homogenization of partial differential equations with multiscale data beyond periodicity and scale separation. The authors provide a careful error analysis, including previously unpublished results, and a complete implementation of the method in MATLAB. They also reveal how the LOD method relates to classical homogenization and domain decomposition. Illustrated with numerical experiments that demonstrate the significance of the method, the book is enhanced by a survey of applications including eigenvalue problems and evolution problems. Numerical Homogenization by Localized Orthogonal Decomposition is appropriate for graduate students in applied mathematics, numerical analysis, and scientific computing. Researchers in the field of computational partial differential equations will find this self-contained book of interest, as will applied scientists and engineers interested in multiscale simulation.

Download or read book Optimal and Suboptimal Control of Partial Differential Equations written by Stefan Volkwein and published by . This book was released on 2001 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Data Driven Science and Engineering written by Steven L. Brunton and published by Cambridge University Press. This book was released on 2022-05-05 with total page 615 pages. Available in PDF, EPUB and Kindle. Book excerpt: A textbook covering data-science and machine learning methods for modelling and control in engineering and science, with Python and MATLAB®.

Download or read book Proper Orthogonal Decomposition in Optimal Control of Fluids written by National Aeronautics and Space Administration (NASA) and published by Createspace Independent Publishing Platform. This book was released on 2018-06-03 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Download or read book Separated Representations and PGD Based Model Reduction written by Francisco Chinesta and published by Springer. This book was released on 2014-09-02 with total page 234 pages. Available in PDF, EPUB and Kindle. 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.

Download or read book Third International Symposium on Domain Decomposition Methods for Partial Differential Equations written by Tony F. Chan and published by SIAM. This book was released on 1990-01-01 with total page 518 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Model Reduction and Approximation written by Peter Benner and published by SIAM. This book was released on 2017-07-06 with total page 421 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Domain Decomposition Methods in Optimal Control of Partial Differential Equations written by John E. Lagnese and published by Birkhäuser. This book was released on 2012-12-06 with total page 454 pages. Available in PDF, EPUB and Kindle. Book excerpt: While domain decomposition methods have a long history dating back well over one hundred years, it is only during the last decade that they have become a major tool in numerical analysis of partial differential equations. This monograph emphasizes domain decomposition methods in the context of so-called virtual optimal control problems and treats optimal control problems for partial differential equations and their decompositions using an all-at-once approach.

Download or read book Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations written by David E. Keyes and published by SIAM. This book was released on 1992-01-01 with total page 642 pages. Available in PDF, EPUB and Kindle. Book excerpt: Papers presented at the May 1991 symposium reflect continuing interest in the role of domain decomposition in the effective utilization of parallel systems; applications in fluid mechanics, structures, biology, and design optimization; and maturation of analysis of elliptic equations, with theoretic