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Indispensable for students, invaluable for researchers, this comprehensive treatment of contemporary quasi-Monte Carlo methods, digital nets and sequences, and discrepancy theory starts from scratch with detailed explanations of the basic concepts and then advances to current methods used in research. As deterministic versions of the Monte Carlo method, quasi-Monte Carlo rules have increased in popularity, with many fruitful applications in mathematical practice. These rules require nodes with good uniform distribution properties, and digital nets and sequences in the sense of Niederreiter are known to be excellent candidates. Besides the classical theory, the book contains chapters on reproducing kernel Hilbert spaces and weighted integration, duality theory for digital nets, polynomial lattice rules, the newest constructions by Niederreiter and Xing and many more. The authors present an accessible introduction to the subject based mainly on material taught in undergraduate courses with numerous examples, exercises and illustrations.
Monte Carlo method. --- Nets (Mathematics) --- Sequences (Mathematics) --- Numerical integration. --- Digital filters (Mathematics) --- Data smoothing filters --- Filters, Digital (Mathematics) --- Linear digital filters (Mathematics) --- Linear filters (Mathematics) --- Numerical filters --- Smoothing filters (Mathematics) --- Digital electronics --- Filters (Mathematics) --- Fourier transformations --- Functional analysis --- Numerical analysis --- Numerical calculations --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Mathematical sequences --- Numerical sequences --- Algebra --- Mathematics --- Moore-Smith convergence --- Net equations --- Net methods (Mathematics) --- Convergence --- Set theory --- Topology --- Artificial sampling --- Model sampling --- Monte Carlo simulation --- Monte Carlo simulation method --- Stochastic sampling --- Games of chance (Mathematics) --- Mathematical models --- Stochastic processes
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Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches. The second edition is substantially revised and enlarged, with many improvements in the presentation and additions concerning in particular non-canonical Hamiltonian systems, highly oscillatory mechanical systems, and the dynamics of multistep methods.
Numerical analysis --- analyse (wiskunde) --- Computer. Automation --- Biomathematics. Biometry. Biostatistics --- algoritmen --- biomathematica --- Mathematical physics --- numerieke analyse --- wiskunde --- Mathematical analysis --- fysica --- 519.63 --- 519.63 Numerical methods for solution of partial differential equations --- Numerical methods for solution of partial differential equations --- 517.91 --- Hamiltonian systems --- Numerical integration --- 519.62 --- 681.3*G17 --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Differentiable dynamical systems --- 517.91 Differential equations --- Differential equations --- 681.3*G17 Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- 519.62 Numerical methods for solution of ordinary differential equations --- Numerical methods for solution of ordinary differential equations --- Numerical solutions --- Numerical solutions. --- Numerical solutions&delete& --- Differential equations - Numerical solutions
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