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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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.
|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.
Preface * Dimensionality Reducing Expansion of Multivariate Integration * Boundary Type Quadrature Formulas with Algebraic Precision * The Integration and DREs of Rapidly Oscillating Functions * Numerical Quadrature Formulas Associated with the Integration of Rapidly Oscillating Functions * DREs Over Complex Domains * Exact DREs Associated with Differential Equations * Bibliography * Index
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.
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.
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.
Inhaltsverzeichnis
Preface * Dimensionality Reducing Expansion of Multivariate Integration * Boundary Type Quadrature Formulas with Algebraic Precision * The Integration and DREs of Rapidly Oscillating Functions * Numerical Quadrature Formulas Associated with the Integration of Rapidly Oscillating Functions * DREs Over Complex Domains * Exact DREs Associated with Differential Equations * Bibliography * Index
Klappentext
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.
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.
