Tian-Xiao He

Dimensionality Reducing Expansion of Multivariate Integration (eBook, PDF)

• Format: PDF

Produktbeschreibung

This book focuses primarily on a powerful tool: 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. This work will appeal to a broad audience of students and researchers in pure and applied mathematics, statistics, and physics.

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Produktdetails

• Verlag: Birkhäuser Boston
• Seitenzahl: 227
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9781461221005
• Artikelnr.: 44204142

Autorenporträt

First self-contained, comprehensive treatment of the method of dimensionality reducing expansion (DRE), a powerful technique for changing a higher dimensional integration to a lower dimensional one with or without remainder. DRE has broad connections to a number of areas: numerical integration, pdes and Green's function, harmonic analysis, numerical analysis and approximation theory. Exposition covers the history of the subject and includes up-to-date new results, related to many fields of current research such as boundary element methods for solving pdes and wavelet analysis. Examples, comprehensive bibliography and index included. Useful text or self-study resource for graduate/advanced undergaduate students and researchers in pure and applied mathematics, statistics, and physics.

Inhaltsangabe

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.