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Tensor Spaces and Numerical Tensor Calculus
Special numerical techniques are already needed to deal with n × n matrices for large n. Tensor data are of size n × n ×...× n=nd, where nd exceeds the computer memory by far. They appear for problems of high spatial dimensions. Since standard methods fail, a particular tensor calculus is needed to treat such problems. This monograph describes the methods by which tensors can be practically treated and shows how numerical operations can be performed. Applications include problems from quantum chemistry, approximation of multivariate functions, solution of partial differential equations, for example with stochastic coefficients, and more. In addition to containing corrections of the unavoidable misprints, this revised second edition includes new parts ranging from single additional statements to new subchapters. The book is mainly addressed to numerical mathematicians and researchers working with high-dimensional data. It also touches problems related to Geometric Algebra.
Hierarchical Matrices: Algorithms and Analysis
This self-contained monograph presents matrix algorithms and their analysis. The new technique enables not only the solution of linear systems but also the approximation of matrix functions, e.g., the matrix exponential. Other applications include the solution of matrix equations, e.g., the Lyapunov or Riccati equation. The required mathematical background can be found in the appendix. The numerical treatment of fully populated large-scale matrices is usually rather costly. However, the technique of hierarchical matrices makes it possible to store matrices and to perform matrix operations approximately with almost linear cost and a controllable degree of approximation error. For important classes of matrices, the computational cost increases only logarithmically with the approximation error. The operations provided include the matrix inversion and LU decomposition. Since large-scale linear algebra problems are standard in scientific computing, the subject of hierarchical matrices is of interest to scientists in computational mathematics, physics, chemistry and engineering.
Multi-Grid Methods and Applications
Multi-grid methods are the most efficient tools for solving elliptic boundary value problems. The reader finds here an elementary introduction to multi-grid algorithms as well as a comprehensive convergence analysis. One section describes special applications (convection-diffusion equations, singular perturbation problems, eigenvalue problems, etc.). The book also contains a complete presentation of the multi-grid method of the second kind, which has important applications to integral equations (e.g. the "panel method") and to numerous other problems. Readers with a practical interest in multi-grid methods will benefit from this book as well as readers with a more theoretical interest.
Integral Equations
The theory of integral equations has been an active research field for many years and is based on analysis, function theory, and functional analysis. On the other hand, integral equations are of practical interest because of the «boundary integral equation method», which transforms partial differential equations on a domain into integral equations over its boundary. This book grew out of a series of lectures given by the author at the Ruhr-Universitat Bochum and the Christian-Albrecht-Universitat zu Kiel to students of mathematics. The contents of the first six chapters correspond to an intensive lecture course of four hours per week for a semester. Readers of the book require background f...
Tensor Spaces and Numerical Tensor Calculus
Special numerical techniques are already needed to deal with nxn matrices for large n.Tensor data are of size nxnx...xn=n^d, where n^d exceeds the computer memory by far. They appear for problems of high spatial dimensions. Since standard methods fail, a particular tensor calculus is needed to treat such problems. The monograph describes the methods how tensors can be practically treated and how numerical operations can be performed. Applications are problems from quantum chemistry, approximation of multivariate functions, solution of pde, e.g., with stochastic coefficients, etc.
Direct and Large Eddy Simulation of Turbulence
This volume contains papers presented to a EUROMECH-Colloquium held in Munich, September 30 to October 2, 1985. The Colloquium is number 199 in a series of colloquia inaugurated by the European Mechanics Committee. The meeting was jointly organized by the 'Lehrstuhl fur Stromungsmechanik' at the 'Technische Universitat Munchen' and the 'Institut fur Physik der Atmosphare' of the 'Deutsche Forschungs- und Versuchsanstalt fur Luft- und Raumfahrt' (DFVLR) in Oberpfaffenhofen. 'Direct' and 'large eddy simulation' are terms which denote two closely con nected methods of turbulence research. In a 'direct simulation' (DS), turbu lent motion is simulated by numerically integrating the Navier-Stokes ...
Multigrid Methods IV
This volume contains a selection from the papers presented at the Fourth European Multigrid Conference, held in Amsterdam, July 6-9,1993. There were 78 registered participants from 14 different countries, and 56 presentations were given. The preceding conferences in this series were held in Cologne (1981, 1985) and in Bonn (1990). Also at the other side of the Atlantic special multigrid conferences are held regularly, at intervals of two years, always in Copper Mountain, Colorado, US. The Sixth Copper Mountain Conference on Multigrid Methods took place in April, 1993. Circumstances prevented us from putting a larger time interval between the Copper and Amsterdam meetings. The next European m...
Lectures on Advanced Computational Methods in Mechanics
This book contains four survey papers related to different topics in computational mechanics, in particular (1) novel discretization and solver techniques in mechanics and (2) inverse, control, and optimization problems in mechanics. These topics were considered in lectures, seminars, tutorials, and workshops at the Special Semester on Computational Mechanics held at the Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, in December 2005.
Sparse Grids and Applications
In the recent decade, there has been a growing interest in the numerical treatment of high-dimensional problems. It is well known that classical numerical discretization schemes fail in more than three or four dimensions due to the curse of dimensionality. The technique of sparse grids helps overcome this problem to some extent under suitable regularity assumptions. This discretization approach is obtained from a multi-scale basis by a tensor product construction and subsequent truncation of the resulting multiresolution series expansion. This volume of LNCSE is a collection of the papers from the proceedings of the workshop on sparse grids and its applications held in Bonn in May 2011. The selected articles present recent advances in the mathematical understanding and analysis of sparse grid discretization. Aspects arising from applications are given particular attention.
