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Constructing numerical schemes which are both adaptive and suitable for parallel architectures is very challenging. The challenge lies in the need to maintain a balanced load across the processing elements using a method that is both efficient and scalable. Here we propose a method which is adaptive, load balanced, absolutely efficient and scalable offering significant speedup over lower order adaptive schemes. The ability of wavelets to accurately and efficiently represent functions with localized features has spawned intensive research into applying wavelets for the solution of partial differential equations with the promise of significantly reducing the necessary computational effort and memory requirements. Traditionally, this effort has been centered around using wavelets as an orthogonal and complete basis, spanning a space in which to seek approximate solutions satisfying the equation in a Galerkin sense. Besides from the well known difficulties associated with such an approach for non-linear problems, one is also faced with the problem of dealing with non-trivial boundary conditions in an accurate and stable manner. Such restrictions on the applicability of wavelet based methods for the solution of problems of more general interest have, in recent years, induced significant interest into grid-based collocation wavelet methods, with various different approaches being taken. The formulation and implementation of multi-dimensional pure wavelet collocation methods, however, remains a challenging task and many issues require attention. In the present work we take a somewhat different approach to arrive at a grid based method utilizing the unique properties of wavelets. Rather than using the wavelets as a basis, we utilize the ability of wavelets to not only detect the existence of high-frequency information but also to supply information about the spatial location of such strongly inhomogeneous regions.
The three-volume set LNCS 3514-3516 constitutes the refereed proceedings of the 5th International Conference on Computational Science, ICCS 2005, held in Atlanta, GA, USA in May 2005.The 464 papers presented were carefully reviewed and selected from a total of 834 submissions for the main conference and its 21 topical workshops. The papers span the whole range of computational science, ranging from numerical methods, algorithms, and computational kernels to programming environments, grids, networking, and tools. These fundamental contributions dealing with computer science methodologies and techniques are complemented by papers discussing computational applications and needs in virtually all scientific disciplines applying advanced computational methods and tools to achieve new discoveries with greater accuracy and speed.
A common feature of multiphase flows is that a dispersed or discontinuous phase is being carried by a continuous phase, for example water drops in gas flow, solid particles in water flow, or gas bubbles in liquid flow. The overall behavior of the flow is shaped largely by the interaction between the discontinuous elements--drops, particles, bubbles
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.
This book provides comprehensive information on the conceptual basis of wavelet theory and it applications. Maintaining an essential balance between mathematical rigour and the practical applications of wavelet theory, the book is closely linked to the wavelet MATLAB toolbox, which is accompanied, wherever applicable, by relevant MATLAB codes. The book is divided into four parts, the first of which is devoted to the mathematical foundations. The second part offers a basic introduction to wavelets. The third part discusses wavelet-based numerical methods for differential equations, while the last part highlights applications of wavelets in other fields. The book is ideally suited as a text for undergraduate and graduate students of mathematics and engineering.