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" Dimensionality Reducing Expansion of Multivariate Integration "
by Tian-Xiao He.
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BL
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573647
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b402866
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| Main Entry
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He, Tian-Xiao.
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Dimensionality Reducing Expansion of Multivariate Integration\ by Tian-Xiao He.
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| Publication Statement
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Boston, MA :: Birkhäuser Boston,, 2001.
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| ISBN
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9781461221005
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: 9781461274148
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| Contents
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1 Dimensionality Reducing Expansion of Multivariate Integration -- 1.1 Darboux formulas and their special forms -- 1.2 Generalized integration by parts rule -- 1.3 DREs with algebraic precision -- 1.4 Minimum estimation of remainders in DREs with algebraic precision -- 2 Boundary Type Quadrature Formulas with Algebraic Precision -- 2.1 Construction of BTQFs using DREs -- 2.2 BTQFs with homogeneous precision -- 2.3 Numerical integration associated with wavelet functions -- 2.4 Some applications of DREs and BTQFs -- 2.5 BTQFs over axially symmetric regions -- 3 The Integration and DREs of Rapidly Oscillating Functions -- 3.1 DREs for approximating a double integral -- 3.2 Basic lemma -- 3.3 DREs with large parameters -- 3.4 Basic expansion theorem for integrals with large parameters -- 3.5 Asymptotic expansion formulas for oscillatory integrals with singular factors -- 4 Numerical Quadrature Formulas Associated with the Integration of Rapidly Oscillating Functions -- 4.1 Numerical quadrature formulas of double integrals -- 4.2 Numerical integration of oscillatory integrals -- 4.3 Numerical quadrature of strongly oscillatory integrals with compound precision -- 4.4 Fast numerical computations of oscillatory integrals -- 4.5 DRE construction and numerical integration using measure theory -- 4.6 Error analysis of numerical integration -- 5 DREs Over Complex Domains -- 5.1 DREs of the double integrals of analytic functions -- 5.2 Construction of quadrature formulas using DREs -- 5.3 Integral regions suitable for DREs -- 5.4 Additional topics -- 6 Exact DREs Associated With Differential Equations -- 6.1 DREs and ordinary differential equations -- 6.2 DREs and partial differential equations -- 6.3 Applications of DREs in the construction of BTQFs -- 6.4 Applications of DREs in the boundary element method.
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| Abstract
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Multivariate integration has been a fundamental subject in mathematics, with broad connections to a number of areas: numerical analysis, approximation theory, partial differential equations, integral equations, harmonic analysis, etc. In this work the exposition focuses primarily on a powerful tool that has become especially important in our computerized age, namely, dimensionality reducing expansion (DRE). The method of DRE is a technique for changing a higher dimensional integration to a lower dimensional one with or without remainder. To date, there is no comprehensive treatment of this subject in monograph or textbook form. Key features of this self-contained monograph include: * fine exposition covering the history of the subject * up-to-date new results, related to many fields of current research such as boundary element methods for solving PDEs and wavelet analysis * presentation of DRE techniques using a broad array of examples * good balance between theory and application * coverage of such related topics as boundary type quadratures and asymptotic expansions of oscillatory integrals * excellent and comprehensive bibliography and index This work will appeal to a broad audience of students and researchers in pure and applied mathematics, statistics, and physics, and can be used in a graduate/advanced undergraduate course or as a standard reference text.
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Mathematics.
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Differential equations, Partial.
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Computer science-- Mathematics.
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Economics-- Statistics.
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SpringerLink (Online service)
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