Numerical Analysis

The third edition of the book introduces the fundamentals of the finite element method through simple examples and an applications-oriented approach using the latest computational tools. Using the transport equation for heat transfer as the foundation for the governing equations,... text demonstrates the versatility of the method of weighted residuals for a wide range of applications including structural analysis and fluid flow. It introduces the boundary element method and meshless, or mesh-free, methods through two additonal chapters. User-friendly computer codes written in MATLAB, MAPLE and FORTRAN are listed.
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This award-winning best-seller addresses the fundamentals of computer science. The authors discuss the design of algorithms, their efficiency and correctness, quantum algorithms, concurrency, large systems and artificial intelligence.

Algorithms are usually classified based on their convergence order - linear, quadratic, etc - and their convergence rate, defined as a bound as the number of iterates goes to infinity. The rate is calculated with respect to the number of steps (iterates) an algorithm takes to con...verge, with no thought on how much effort is applied at each iterate. This volume demonstrates that the use of partial information, refined at each new iteration of the algorithm, can increase the efficiency of the algorithm, while guaranteeing convergence to the optimal solution, provided that the rate of refinement is carefully chosen. Based on approximate algorithms for problem solving, the authors present a new theory, which serves as a general framework for the problem, and consider general and particularly Markov Decision processes. A more suitable definition of the convergence rate with respect to the computational effort is derived for the proposed class of approximate iterative algorithms. In addition, an optimal refinement strategy is derived, which maximizes the rate of convergence with respect to the computation effort of this class of algorithms. Hence, a class of algorithms is derived that makes optimal use of the computational resources available. With a given, fixed amount of computation time, the proposed class of algorithms gets closer to the solution than any other algorithm, including the classical (exact) algorithm. Many numerical examples are included for illustration. This volume is intended for mathematicians, engineers and computer scientists, who work on learning processes in numerical analysis and are involved with optimization, optimal control, decision analysis and machine learning.
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In this volume, experts describe how methods of analysis and modelling are used to answer questions associated with geomechanics, and give an insight into the future directions of those methods.

Studying the relationship between the geometry, arithmetic and spectra of fractals has been a subject of significant interest in contemporary mathematics. This book contributes to the literature on the subject in several different and new ways. In particular, the authors provide ...a rigorous and detailed study of the spectral operator, a map that sends the geometry of fractal strings onto their spectrum. To that effect, they use and develop methods from fractal geometry, functional analysis, complex analysis, operator theory, partial differential equations, analytic number theory and mathematical physics.Originally, M L Lapidus and M van Frankenhuijsen 'heuristically' introduced the spectral operator in their development of the theory of fractal strings and their complex dimensions, specifically in their reinterpretation of the earlier work of M L Lapidus and H Maier on inverse spectral problems for fractal strings and the Riemann hypothesis.One of the main themes of the book is to provide a rigorous framework within which the corresponding question 'Can one hear the shape of a fractal string?' or, equivalently, 'Can one obtain information about the geometry of a fractal string, given its spectrum?' can be further reformulated in terms of the invertibility or the quasi-invertibility of the spectral operator.The infinitesimal shift of the real line is first precisely defined as a differentiation operator on a family of suitably weighted Hilbert spaces of functions on the real line and indexed by a dimensional parameter c. Then, the spectral operator is defined via the functional calculus as a function of the infinitesimal shift. In this manner, it is viewed as a natural 'quantum' analog of the Riemann zeta function. More precisely, within this framework, the spectral operator is defined as the composite map of the Riemann zeta function with the infinitesimal shift, viewed as an unbounded normal operator acting on the above Hilbert space.It is shown that the quasi-inver
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