Integration and Cubature Methods A Geomathematically Oriented Course 1st Edition by Willi Freeden, Martin Gutting 1351764759 9781351764759

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ISBN-10 :  1351764759 

ISBN-13 :  9781351764759

Author:  Willi Freeden, Martin Gutting 

In industry and economics, the most common solutions of partial differential equations involving multivariate numerical integration over cuboids include techniques of iterated one-dimensional approximate integration. In geosciences, however, the integrals are extended over potato-like volumes (such as the ball, ellipsoid, geoid, or the Earth) and their boundary surfaces which require specific multi-variate approximate integration methods. Integration and Cubature Methods: A Geomathematically Oriented Course provides a basic foundation for students, researchers, and practitioners interested in precisely these areas, as well as breaking new ground in integration and cubature in geomathematics.

Integration and Cubature Methods: A Geomathematically Oriented Course 1st Table of contents:

I Preparatory 1D-Integration
1 Algebraic Polynomial Integration
1.1 Interpolatory Integration Rules
1.2 Peano’s Theorem
1.3 Error Truncation
2 Algebraic Spline Integration
2.1 Spline Integration Formulas
2.2 Spline Interpolation
2.3 Best Approximation and Spline Exact Formulas
3 Periodic Polynomial Integration
3.1 Integer Lattice and Periodic Polynomials
3.2 Lattice Functions
3.3 Euler Summation Formulas
3.4 Euler Summation Formulas for Periodic Functions
4 Periodic Spline Integration
4.1 Best Approximate Integration in Sobolev Spaces
4.2 Spline Lagrange Basis
4.3 Peano’s Theorem
4.4 Best Approximation and Spline Exact Formulas
4.5 Smoothing Splines for Erroneous Data Points
5 Trapezoidal Rules
5.1 Riemann Zeta Function and Lattice Function
5.2 Classical Trapezoidal Sums for Finite Intervals
5.3 Romberg Integration
5.4 Poisson Summation Based Integration
5.5 Trapezoidal Sums over Dilated Lattices
6 Adaptive Trapezoidal Rules
6.1 Lattice Functions for Helmholtz Operators
6.2 Adaptive Trapezoidal Sums over Finite Intervals
6.3 Adaptive Poisson Summation Formula over Infinite Intervals
6.4 Adaptive Trapezoidal Sums over Infinite Intervals
6.5 Discontinuous Integrals of Hardy–Landau Type
6.6 Periodic Polynomial Accuracy
7 Legendre Polynomial Reflected Integration
7.1 Legendre Polynomials
7.2 Legendre (Green’s) Functions
7.3 Integral Formulas
8 Gaussian Integration
8.1 Gaussian Quadrature Formulas
8.2 Adaptive Remainder Terms Involving Green’s Function
8.3 Convergence of Gaussian Quadrature
II Integration on 2D-Spheres
9 Remainder Terms Involving Beltrami Operators
9.1 Spherical Framework
9.2 Sphere Functions Involving Beltrami Operators
9.3 Best Approximate Integration by Splines
9.4 Integral Formulas under Boundary Conditions
9.5 Sphere Functions and Shannon Kernels
9.6 Peano’s Theorem Involving Beltrami Operators
10 Integration Rules with Polynomial Accuracy
10.1 Lagrangian Integration
10.2 Lebesgue Functions
10.3 Spherical Geometry and Polynomial Cubature Rules
10.4 Interpolatory Rules Based on Extremal Point Systems and Designs
10.5 Non-Existence of Spherical Gaussian Rules
11 Latitude-Longitude Cubature
11.1 Associated Legendre Functions
11.2 Legendre Spherical Harmonics
11.3 Latitude-Longitude Integration
12 Remainder Terms Involving Pseudodifferential Operators
12.1 Sobolev Spaces
12.2 Pseudodifferential Operators
12.3 Reproducing Kernels and Remainder Terms
12.4 Particular Types of Kernel Functions
12.5 Locally Supported Kernels
12.6 Zonal Function Exact Integration
13 Spline Exact Integration
13.1 Spline Interpolation
13.2 Peano’s Theorem in Terms of Pseudodifferential Operators
13.3 Best Approximations
13.4 Spline Exact Integration Formulas
14 Equidistributions and Discrepancy Methods
14.1 Equidistributions
14.2 Discrepancy Variants
14.3 Examples of Equidistributions on the Sphere
14.4 Sobolev Space Based Generalized Discrepancy
14.5 Statistics for Equidistributions
15 Multiscale Approximate Integration
15.1 Singular Integrals and Approximate Identities
15.2 Locally Supported Scaling Functions
15.3 Locally Supported Difference Wavelets
15.4 Integration for Large Equidistributed Data
15.5 Error Discussion
III Integration on 2D-Surfaces
16 Surface Integration
16.1 Transformation back to the Sphere
16.2 Use of Differential Geometric Means
16.3 Integral Formulas
16.4 Best Approximate Integration
16.5 Equidistributions
IV Integration over qD-Volumes
17 Lattices, Periodic Polynomials, and Integral Formulas
17.1 Lattices
17.2 Periodic Polynomials
17.3 Regular Regions and Green’s Integral Theorems
17.4 Fourier Transform in Euclidean Spaces
17.5 Periodization and Poisson Summation Formula
18 Euler Summation Based Integration
18.1 Euler Summation Formulas for Laplace Operators
18.2 Zeta Function and Euler Summation
18.3 Euler Lattice Point Cubature on Regular Regions
18.4 Romberg Extrapolation
19 Integration by Averaged Euler Summation
19.1 Lattice Ball Integral Means
19.2 Gauss–Weierstrass Integral Means
19.3 Averaged Cubature over 3D-Regular Regions
20 Adaptive Integration by Euler and Poisson Summation
20.1 Euler Summation Integration Involving Helmholtz Operators
20.2 Adaptive Cubature over Regular Regions
20.3 Poisson Summation under Adaptive Criteria
20.4 Adaptive Cubature over Euclidean Spaces
21 Lattice Spline Interpolation and Monospline Integration
21.1 Lattice Periodic Splines
21.2 Minimum Norm Interpolation
21.3 Periodic Sampling
21.4 Elliptic Operators, Monospline Integration, and Remainder Terms
22 Shannon Sampling and Paley–Wiener Integration
22.1 Gaussian Circle Problem and Hardy’s Conjecture
22.2 Higher-Dimensional Variants of the Circle Problem
22.3 Multivariate Shannon Sampling
22.4 Paley–Wiener Integration
22.5 Paley–Wiener Spline Interpolatory Integration

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