Proper Generalized Decompositions

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The Proper Generalized Decomposition For Advanced Numerical Simulations

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Author : Francisco Chinesta
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-10-08

The Proper Generalized Decomposition For Advanced Numerical Simulations written by Francisco Chinesta and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-10-08 with Technology & Engineering categories.

Many problems in scientific computing are intractable with classical numerical techniques. These fail, for example, in the solution of high-dimensional models due to the exponential increase of the number of degrees of freedom. Recently, the authors of this book and their collaborators have developed a novel technique, called Proper Generalized Decomposition (PGD) that has proven to be a significant step forward. The PGD builds by means of a successive enrichment strategy a numerical approximation of the unknown fields in a separated form. Although first introduced and successfully demonstrated in the context of high-dimensional problems, the PGD allows for a completely new approach for addressing more standard problems in science and engineering. Indeed, many challenging problems can be efficiently cast into a multi-dimensional framework, thus opening entirely new solution strategies in the PGD framework. For instance, the material parameters and boundary conditions appearing in a particular mathematical model can be regarded as extra-coordinates of the problem in addition to the usual coordinates such as space and time. In the PGD framework, this enriched model is solved only once to yield a parametric solution that includes all particular solutions for specific values of the parameters. The PGD has now attracted the attention of a large number of research groups worldwide. The present text is the first available book describing the PGD. It provides a very readable and practical introduction that allows the reader to quickly grasp the main features of the method. Throughout the book, the PGD is applied to problems of increasing complexity, and the methodology is illustrated by means of carefully selected numerical examples. Moreover, the reader has free access to the Matlab© software used to generate these examples.

Proper Generalized Decompositions

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Author : Elías Cueto
language : en
Publisher: Springer
Release Date : 2016-03-01

Proper Generalized Decompositions written by Elías Cueto and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-03-01 with Science categories.

This book is intended to help researchers overcome the entrance barrier to Proper Generalized Decomposition (PGD), by providing a valuable tool to begin the programming task. Detailed Matlab Codes are included for every chapter in the book, in which the theory previously described is translated into practice. Examples include parametric problems, non-linear model order reduction and real-time simulation, among others. Proper Generalized Decomposition (PGD) is a method for numerical simulation in many fields of applied science and engineering. As a generalization of Proper Orthogonal Decomposition or Principal Component Analysis to an arbitrary number of dimensions, PGD is able to provide the analyst with very accurate solutions for problems defined in high dimensional spaces, parametric problems and even real-time simulation.

The Proper Generalized Decomposition For Advanced Numerical Simulations

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Author : Francisco Chinesta
language : en
Publisher:
Release Date : 2013-11-30

The Proper Generalized Decomposition For Advanced Numerical Simulations written by Francisco Chinesta and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-30 with categories.

Study Of The Proper Generalized Decomposition For Real Time Applications

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Author : Bernat Serra Zueras
language : en
Publisher:
Release Date : 2017

Study Of The Proper Generalized Decomposition For Real Time Applications written by Bernat Serra Zueras and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with categories.

Computational simulations are valuable tools for engineers because they facilitate design processes. Complex systems or problems are unreachable by the standard methods (like Finite Element Method, FEM) and Model Order Reduction (MOR) techniques are required. Among those, the Proper Orthogonal Decomposition has proved to be useful in many fields of science and engineering, and especially in solid mechanics. Alternatively, an a priori strategy MOR, the Proper Generalized Decomposition (PGD), is being developed to solve parametric problems in what, in certain areas, are considered "real-time" conditions. The purpose of this study is to introduce the PGD emergent technique and its potential. This approach has been put into context by qualitatively describing MOR techniques when applied in computational solid mechanics, like the also introduced FEM. The PGD key features and advantages, as well as drawbacks, have been introduced in comparison with the POD, a widely extended technique. Regarding the potential of the PGD, its adequacy in augmented reality in computational surgery has already been documented. In this study, the PGD has first been implemented to solve two 2D academic problems successfully (Poisson equation and linear elastics) with full mathematical formulation. Next, based on the linear elastic case study, a real-time application program has been contrived. In the latter, the problem's configuration has been updated, adding more complexity. Throughout this report, the good results obtained with the PGD have been demonstrated. Results show, first, the ability to solve simple cases using a PGD code applied in Matlab and, therefore, validates this approach. Second, it has been proved that it is useful to develop real-time applications, performing as expected. However, there are still open concepts and related issues to improve the methodology. The study has achieved to develop a simple and comfortable approach to the PGD. At the same time, it provides the readers with the basic tools to encourage them to make their first steps in the PGD methodology. With the hope that all this will allow the reader understand the importance and potential of this methodology.

Pgd Based Modeling Of Materials Structures And Processes

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Author : Francisco Chinesta
language : en
Publisher: Springer Science & Business
Release Date : 2014-04-23

Pgd Based Modeling Of Materials Structures And Processes written by Francisco Chinesta and has been published by Springer Science & Business this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-04-23 with Science categories.

This book focuses on the development of a new simulation paradigm allowing for the solution of models that up to now have never been resolved and which result in spectacular CPU time savings (in the order of millions) that, combined with supercomputing, could revolutionize future ICT (information and communication technologies) at the heart of science and technology. The authors have recently proposed a new paradigm for simulation-based engineering sciences called Proper Generalized Decomposition, PGD, which has proved a tremendous potential in many aspects of forming process simulation. In this book a review of the basics of the technique is made, together with different examples of application.

Snapshot Based Methods And Algorithms

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Author : Peter Benner
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2020-12-16

Snapshot Based Methods And Algorithms written by Peter Benner and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-12-16 with Mathematics categories.

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.

Reduced Order Models For The Biomechanics Of Living Organs

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Author : Francisco Chinesta
language : en
Publisher: Elsevier
Release Date : 2023-05-25

Reduced Order Models For The Biomechanics Of Living Organs written by Francisco Chinesta and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-05-25 with Technology & Engineering categories.

Reduced Order Models for the Biomechanics of Living Organs, a new volume in the Biomechanics of Living Organisms series, provides a comprehensive overview of the state-of-the-art in biomechanical computations using reduced order models, along with a deeper understanding of the associated reduction algorithms that will face students, researchers, clinicians and industrial partners in the future. The book gathers perspectives from key opinion scientists who describe and detail their approaches, methodologies and findings. It is the first to synthesize complementary advances in Biomechanical modelling of living organs using reduced order techniques in the design of medical devices and clinical interventions, including surgical procedures. This book provides an opportunity for students, researchers, clinicians and engineers to study the main topics related to biomechanics and reduced models in a single reference, with this volume summarizing all biomechanical aspects of each living organ in one comprehensive reference. - Introduces the fundamental aspects of reduced order models - Presents the main computational studies in the field of solid and fluid biomechanical modeling of living organs - Explores the use of reduced order models in the fields of biomechanical electrophysiology, tissue growth and prosthetic designs

Spectral And High Order Methods For Partial Differential Equations Icosahom 2018

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Author : Spencer J. Sherwin
language : en
Publisher: Springer Nature
Release Date : 2020-08-11

Spectral And High Order Methods For Partial Differential Equations Icosahom 2018 written by Spencer J. Sherwin and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-08-11 with Mathematics categories.

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.

Separated Representations And Pgd Based Model Reduction

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Author : Francisco Chinesta
language : en
Publisher: Springer
Release Date : 2014-09-02

Separated Representations And Pgd Based Model Reduction written by Francisco Chinesta and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-09-02 with Technology & Engineering categories.

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.

A Gentle Introduction To Data Learning And Model Order Reduction

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Author : Francisco Chinesta
language : en
Publisher: Springer Nature
Release Date : 2025-08-23

A Gentle Introduction To Data Learning And Model Order Reduction written by Francisco Chinesta and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-08-23 with Computers categories.

This open access book explores the latest advancements in simulation performance, driven by model order reduction, informed and augmented machine learning technologies and their combination into the so-called hybrid digital twins. It provides a comprehensive review of three key frameworks shaping modern engineering simulations: physics-based models, data-driven approaches, and hybrid techniques that integrate both. The book examines the limitations of traditional models, the role of data acquisition in uncovering underlying patterns, and how physics-informed and augmented learning techniques contribute to the development of digital twins. Organized into four sections—Around Data, Around Learning, Around Reduction, and Around Data Assimilation & Twinning—this book offers an essential resource for researchers, engineers, and students seeking to understand and apply cutting-edge simulation methodologies