Adaptive Wavelet Frame Methods For Nonlinear Elliptic Problems
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Adaptive Wavelet Frame Methods For Nonlinear Elliptic Problems
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Author : Jens Kappei
language : en
Publisher: Logos Verlag Berlin GmbH
Release Date : 2012-02-06
Adaptive Wavelet Frame Methods For Nonlinear Elliptic Problems written by Jens Kappei and has been published by Logos Verlag Berlin GmbH this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-02-06 with Mathematics categories.
Over the last ten years, adaptive wavelet methods have turned out to be a powerful tool in the numerical treatment of operator equations given on a bounded domain or closed manifold. In this work, we consider semi-nonlinear operator equations, including an elliptic linear operator as well as a nonlinear monotone one. Since the classical approach to construct a wavelet Riesz basis for the solution space is still afflicted with some notable problems, we use the weaker concept of wavelet frames to design an adaptive algorithm for the numerical solution of problems of this type. Choosing an appropriate overlapping decomposition of the given domain, a suitable frame system can be constructed easily. Applying it to the given continuous problem yields a discrete, bi-infinite nonlinear system of equations, which is shown to be solvable by a damped Richardson iteration method. We then successively introduce all building blocks for the numerical implementation of the iteration method. Here, we concentrate on the evaluation of the discrete nonlinearity, where we show that the previously developed auxiliary of tree-structured index sets can be generalized to the wavelet frame setting in a proper way. This allows an effective numerical treatment of the nonlinearity by so-called aggregated trees. Choosing the error tolerances appropriately, we show that our adaptive scheme is asymptotically optimal with respect to aggregated tree-structured index sets, i.e., it realizes the same convergence rate as the sequence of best N-term frame approximations of the solution respecting aggregated trees. Moreover, under the assumption of a sufficiently precise numerical quadrature method, the computational cost of our algorithm stays the same order as the number of wavelets used by it. The theoretical results are widely confirmed by one- and two-dimensional test problems over non-trivial bounded domains.
Numerical Methods For Nonlinear Elliptic Differential Equations
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Author : Klaus Böhmer
language : en
Publisher: Oxford University Press
Release Date : 2010-10-07
Numerical Methods For Nonlinear Elliptic Differential Equations written by Klaus Böhmer and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-10-07 with Computers categories.
Boehmer systmatically handles the different numerical methods for nonlinear elliptic problems.
Adaptive Wavelet Methods For Variational Formulations Of Nonlinear Elliptic Pdes On Tensor Product Domains
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Author : Roland Pabel
language : en
Publisher: Logos Verlag Berlin GmbH
Release Date : 2015-09-30
Adaptive Wavelet Methods For Variational Formulations Of Nonlinear Elliptic Pdes On Tensor Product Domains written by Roland Pabel and has been published by Logos Verlag Berlin GmbH this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-09-30 with Mathematics categories.
This thesis is concerned with the numerical solution of boundary value problems (BVPs) governed by nonlinear elliptic partial differential equations (PDEs). To iteratively solve such BVPs, it is of primal importance to develop efficient schemes that guarantee convergence of the numerically approximated PDE solutions towards the exact solution. The new adaptive wavelet theory guarantees convergence of adaptive schemes with fixed approximation rates. Furthermore, optimal, i.e., linear, complexity estimates of such adaptive solution methods have been established. These achievements are possible since wavelets allow for a completely new perspective to attack BVPs: namely, to represent PDEs in their original infinite dimensional realm. Wavelets in this context represent function bases with special analytical properties, e.g., the wavelets considered herein are piecewise polynomials, have compact support and norm equivalences between certain function spaces and the $ell_2$ sequence spaces of expansion coefficients exist. This theoretical framework is implemented in the course of this thesis in a truly dimensionally unrestricted adaptive wavelet program code, which allows one to harness the proven theoretical results for the first time when numerically solving the above mentioned BVPs. Numerical studies of 2D and 3D PDEs and BVPs demonstrate the feasibility and performance of the developed schemes. The BVPs are solved using an adaptive Uzawa algorithm, which requires repeated solution of nonlinear PDE sub-problems. This thesis presents for the first time a numerically competitive implementation of a new theoretical paradigm to solve nonlinear elliptic PDEs in arbitrary space dimensions with a complete convergence and complexity theory.
Multiscale Nonlinear And Adaptive Approximation
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Author : Ronald DeVore
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-09-16
Multiscale Nonlinear And Adaptive Approximation written by Ronald DeVore 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 2009-09-16 with Mathematics categories.
The book of invited articles offers a collection of high-quality papers in selected and highly topical areas of Applied and Numerical Mathematics and Approximation Theory which have some connection to Wolfgang Dahmen's scientific work. On the occasion of his 60th birthday, leading experts have contributed survey and research papers in the areas of Nonlinear Approximation Theory, Numerical Analysis of Partial Differential and Integral Equations, Computer-Aided Geometric Design, and Learning Theory. The main focus and common theme of all the articles in this volume is the mathematics building the foundation for most efficient numerical algorithms for simulating complex phenomena.
Extraction Of Quantifiable Information From Complex Systems
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Author : Stephan Dahlke
language : en
Publisher: Springer
Release Date : 2014-11-13
Extraction Of Quantifiable Information From Complex Systems written by Stephan Dahlke and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-11-13 with Mathematics categories.
In April 2007, the Deutsche Forschungsgemeinschaft (DFG) approved the Priority Program 1324 “Mathematical Methods for Extracting Quantifiable Information from Complex Systems.” This volume presents a comprehensive overview of the most important results obtained over the course of the program. Mathematical models of complex systems provide the foundation for further technological developments in science, engineering and computational finance. Motivated by the trend toward steadily increasing computer power, ever more realistic models have been developed in recent years. These models have also become increasingly complex, and their numerical treatment poses serious challenges. Recent developments in mathematics suggest that, in the long run, much more powerful numerical solution strategies could be derived if the interconnections between the different fields of research were systematically exploited at a conceptual level. Accordingly, a deeper understanding of the mathematical foundations as well as the development of new and efficient numerical algorithms were among the main goals of this Priority Program. The treatment of high-dimensional systems is clearly one of the most challenging tasks in applied mathematics today. Since the problem of high-dimensionality appears in many fields of application, the above-mentioned synergy and cross-fertilization effects were expected to make a great impact. To be truly successful, the following issues had to be kept in mind: theoretical research and practical applications had to be developed hand in hand; moreover, it has proven necessary to combine different fields of mathematics, such as numerical analysis and computational stochastics. To keep the whole program sufficiently focused, we concentrated on specific but related fields of application that share common characteristics and as such, they allowed us to use closely related approaches.
Mathematics Of Surfaces Xiii
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Author : Ralph R. Martin
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-08-06
Mathematics Of Surfaces Xiii written by Ralph R. Martin 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 2009-08-06 with Computers categories.
This book constitutes the refereed proceedings of the 13th IMA International Conference on the Mathematics of Surfaces held in York, UK in September 2009. The papers in the present volume include seven invited papers, as well as 16 submitted papers. The topics covered include subdivision schemes and their continuity, polar patchworks, compressive algorithms for PDEs, surface invariant functions, swept volume parameterization, Willmore flow, computational conformal geometry, heat kernel embeddings, and self-organizing maps on manifolds, mesh and manifold construction, editing, flattening, morphing and interrogation, dissection of planar shapes, symmetry processing, morphable models, computation of isophotes, point membership classification and vertex blends. Surface types considered encompass polygon meshes as well as parametric and implicit surfaces.
Wavelet Methods For Elliptic Partial Differential Equations
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Author : Karsten Urban
language : en
Publisher: OUP Oxford
Release Date : 2008-11-27
Wavelet Methods For Elliptic Partial Differential Equations written by Karsten Urban and has been published by OUP Oxford this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-11-27 with Mathematics categories.
The origins of wavelets go back to the beginning of the last century and wavelet methods are by now a well-known tool in image processing (jpeg2000). These functions have, however, been used successfully in other areas, such as elliptic partial differential equations, which can be used to model many processes in science and engineering. This book, based on the author's course and accessible to those with basic knowledge of analysis and numerical mathematics, gives an introduction to wavelet methods in general and then describes their application for the numerical solution of elliptic partial differential equations. Recently developed adaptive methods are also covered and each scheme is complemented with numerical results, exercises, and corresponding software tools.
Multiscale And Adaptivity Modeling Numerics And Applications
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Author : Silvia Bertoluzza
language : en
Publisher: Springer
Release Date : 2012-01-06
Multiscale And Adaptivity Modeling Numerics And Applications written by Silvia Bertoluzza and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-01-06 with Mathematics categories.
This book is a collection of lecture notes for the CIME course on "Multiscale and Adaptivity: Modeling, Numerics and Applications," held in Cetraro (Italy), in July 2009. Complex systems arise in several physical, chemical, and biological processes, in which length and time scales may span several orders of magnitude. Traditionally, scientists have focused on methods that are particularly applicable in only one regime, and knowledge of the system on one scale has been transferred to another scale only indirectly. Even with modern computer power, the complexity of such systems precludes their being treated directly with traditional tools, and new mathematical and computational instruments have had to be developed to tackle such problems. The outstanding and internationally renowned lecturers, coming from different areas of Applied Mathematics, have themselves contributed in an essential way to the development of the theory and techniques that constituted the subjects of the courses.
Besov Regularity Of Stochastic Partial Differential Equations On Bounded Lipschitz Domains
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Author : Petru A. Cioica
language : en
Publisher: Logos Verlag Berlin GmbH
Release Date : 2015-03-01
Besov Regularity Of Stochastic Partial Differential Equations On Bounded Lipschitz Domains written by Petru A. Cioica and has been published by Logos Verlag Berlin GmbH this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-03-01 with Mathematics categories.
Stochastic partial differential equations (SPDEs, for short) are the mathematical models of choice for space time evolutions corrupted by noise. Although in many settings it is known that the resulting SPDEs have a unique solution, in general, this solution is not given explicitly. Thus, in order to make those mathematical models ready to use for real life applications, appropriate numerical algorithms are needed. To increase efficiency, it would be tempting to design suitable adaptive schemes based, e.g., on wavelets. However, it is not a priori clear whether such adaptive strategies can outperform well-established uniform alternatives. Their theoretical justification requires a rigorous regularity analysis in so-called non-linear approximation scales of Besov spaces. In this thesis the regularity of (semi-)linear second order SPDEs of Itô type on general bounded Lipschitz domains is analysed. The non-linear approximation scales of Besov spaces are used to measure the regularity with respect to the space variable, the time regularity being measured first in terms of integrability and afterwards in terms of Hölder norms. In particular, it is shown that in specific situations the spatial Besov regularity of the solution in the non-linear approximation scales is generically higher than its corresponding classical Sobolev regularity. This indicates that it is worth developing spatially adaptive wavelet methods for solving SPDEs instead of using uniform alternatives.
Handbook Of Geomathematics
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Author : Willi Freeden
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-08-13
Handbook Of Geomathematics written by Willi Freeden 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 2010-08-13 with Mathematics categories.
During the last three decades geosciences and geo-engineering were influenced by two essential scenarios: First, the technological progress has changed completely the observational and measurement techniques. Modern high speed computers and satellite based techniques are entering more and more all geodisciplines. Second, there is a growing public concern about the future of our planet, its climate, its environment, and about an expected shortage of natural resources. Obviously, both aspects, viz. efficient strategies of protection against threats of a changing Earth and the exceptional situation of getting terrestrial, airborne as well as spaceborne data of better and better quality explain the strong need of new mathematical structures, tools, and methods. Mathematics concerned with geoscientific problems, i.e., Geomathematics, is becoming increasingly important. The ‘Handbook Geomathematics’ as a central reference work in this area comprises the following scientific fields: (I) observational and measurement key technologies (II) modelling of the system Earth (geosphere, cryosphere, hydrosphere, atmosphere, biosphere) (III) analytic, algebraic, and operator-theoretic methods (IV) statistical and stochastic methods (V) computational and numerical analysis methods (VI) historical background and future perspectives.