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Adaptive Wavelet Frame Domain Decomposition Methods For Elliptic Operator Equations
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Book Synopsis Adaptive Wavelet Frame Domain Decomposition Methods for Elliptic Operator Equations by : Manuel Werner
Download or read book Adaptive Wavelet Frame Domain Decomposition Methods for Elliptic Operator Equations written by Manuel Werner and published by Logos Verlag Berlin. This book was released on 2009 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work, new adaptive numerical wavelet algorithms for the solution of elliptic operator equations posed in a bounded domain or on a closed manifold are developed. To circumvent the complicated construction of a wavelet Riesz basis for the solution space, we work with the weaker concept of wavelet frames. Using an overlapping domain decomposition technique, suitable frames can easily be constructed and implemented. In a first step, we show that classical results on the convergence rates of best N-term approximations of the solution with respect to wavelet Riesz bases essentially carry over to the considered class of wavelet frames. We then develop an adaptive method based on a steepest descent iteration for the frame coordinate representation of the elliptic equation, and, most importantly, we develop algorithms based on multiplicative and additive Schwarz overlapping domain decomposition methods. We prove that our adaptive schemes are of asymptotically optimal complexity, in the sense that they realize the same convergence rate as the sequence of best N-term frame approximations of the solution. Moreover, using special numerical quadrature rules for the computation of the frame representation of the elliptic operator, the overall computational cost stays proportional to the number of wavelets selected by the algorithms. The results of a series of numerical tests for non-trivial one- and two-dimensional Poisson and biharmonic model problems confirm our theoretical findings and particularly demonstrate the efficiency of the domain decomposition approach. A comparison with a standard adaptive finite element solver shows that our multiplicative Schwarz method potentially generates significantly sparser approximations. In addition, a parallel implementation of the new adaptive additive Schwarz wavelet solver is developed and tested.
Book Synopsis Adaptive wavelet frame methods for nonlinear elliptic problems by : Jens Kappei
Download or read book Adaptive wavelet frame methods for nonlinear elliptic problems written by Jens Kappei and published by Logos Verlag Berlin GmbH. This book was released on 2012-02-06 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Wavelet Methods for Elliptic Partial Differential Equations by : Karsten Urban
Download or read book Wavelet Methods for Elliptic Partial Differential Equations written by Karsten Urban and published by Numerical Mathematics and Scie. This book was released on 2009 with total page 509 pages. Available in PDF, EPUB and Kindle. Book excerpt: Wavelet methods are by now a well-known tool in image processing (jpeg2000). These functions have been used successfully in other areas, however. Elliptic Partial Differential Equations which model several processes in, for example, science and engineering, is one such field. This book, based on the author's course, 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.
Author : Publisher :Springer Nature ISBN 13 :3031743709 Total Pages :444 pages Book Rating :4.0/5 (317 download)
Download or read book written by and published by Springer Nature. This book was released on with total page 444 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Numerical Methods for Nonlinear Elliptic Differential Equations by : Klaus Böhmer
Download or read book Numerical Methods for Nonlinear Elliptic Differential Equations written by Klaus Böhmer and published by Oxford University Press. This book was released on 2010-10-07 with total page 775 pages. Available in PDF, EPUB and Kindle. Book excerpt: Boehmer systmatically handles the different numerical methods for nonlinear elliptic problems.
Book Synopsis Handbook of Geomathematics by : Willi Freeden
Download or read book Handbook of Geomathematics written by Willi Freeden and published by Springer Science & Business Media. This book was released on 2010-08-13 with total page 1371 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Besov Regularity of Stochastic Partial Differential Equations on Bounded Lipschitz Domains by : Petru A. Cioica
Download or read book Besov Regularity of Stochastic Partial Differential Equations on Bounded Lipschitz Domains written by Petru A. Cioica and published by Logos Verlag Berlin GmbH. This book was released on 2015-03-01 with total page 166 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Matrix Preconditioning Techniques and Applications by : Ke Chen
Download or read book Matrix Preconditioning Techniques and Applications written by Ke Chen and published by Cambridge University Press. This book was released on 2005-07-14 with total page 616 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive introduction to preconditioning techniques, now an essential part of successful and efficient iterative solutions of matrices.
Book Synopsis Multiscale and Adaptivity: Modeling, Numerics and Applications by : Silvia Bertoluzza
Download or read book Multiscale and Adaptivity: Modeling, Numerics and Applications written by Silvia Bertoluzza and published by Springer Science & Business Media. This book was released on 2012-01-07 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Mathematics of Surfaces XIII by : Edwin R. Hancock
Download or read book Mathematics of Surfaces XIII written by Edwin R. Hancock and published by Springer Science & Business Media. This book was released on 2009-08-06 with total page 418 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Journal of the Society for Industrial and Applied Mathematics. Series B: Numerical Analysis by : Society for Industrial and Applied Mathematics
Download or read book Journal of the Society for Industrial and Applied Mathematics. Series B: Numerical Analysis written by Society for Industrial and Applied Mathematics and published by . This book was released on 2005 with total page 944 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Multiscale Wavelet Methods for Partial Differential Equations by : Wolfgang Dahmen
Download or read book Multiscale Wavelet Methods for Partial Differential Equations written by Wolfgang Dahmen and published by Elsevier. This book was released on 1997-08-13 with total page 587 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis Perspectives in Partial Differential Equations, Harmonic Analysis and Applications by : Dorina Mitrea
Download or read book Perspectives in Partial Differential Equations, Harmonic Analysis and Applications written by Dorina Mitrea and published by American Mathematical Soc.. This book was released on 2008 with total page 446 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains a collection of papers contributed on the occasion of Mazya's 70th birthday by a distinguished group of experts of international stature in the fields of harmonic analysis, partial differential equations, function theory, and spectral analysis, reflecting the state of the art in these areas.
Book Synopsis Singular Phenomena and Scaling in Mathematical Models by : Michael Griebel
Download or read book Singular Phenomena and Scaling in Mathematical Models written by Michael Griebel and published by Springer Science & Business Media. This book was released on 2013-11-18 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book integrates theoretical analysis, numerical simulation and modeling approaches for the treatment of singular phenomena. The projects covered focus on actual applied problems, and develop qualitatively new and mathematically challenging methods for various problems from the natural sciences. Ranging from stochastic and geometric analysis over nonlinear analysis and modelling to numerical analysis and scientific computation, the book is divided into the three sections: A) Scaling limits of diffusion processes and singular spaces, B) Multiple scales in mathematical models of materials science and biology and C) Numerics for multiscale models and singular phenomena. Each section addresses the key aspects of multiple scales and model hierarchies, singularities and degeneracies, and scaling laws and self-similarity.
Download or read book Mathematical Reviews written by and published by . This book was released on 2008 with total page 836 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization by : Houman Owhadi
Download or read book Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization written by Houman Owhadi and published by Cambridge University Press. This book was released on 2019-10-24 with total page 491 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents interplays between numerical approximation and statistical inference as a pathway to simple solutions to fundamental problems.
Book Synopsis Applied Stochastic Differential Equations by : Simo Särkkä
Download or read book Applied Stochastic Differential Equations written by Simo Särkkä and published by Cambridge University Press. This book was released on 2019-05-02 with total page 327 pages. Available in PDF, EPUB and Kindle. Book excerpt: With this hands-on introduction readers will learn what SDEs are all about and how they should use them in practice.
