A Galerkin method with two-dimensional Haar basis functions for the computation of the Karhunen–Loève expansion

We study the numerical approximation of a homogeneous Fredholm integral equation of second kind associated with the Karhunen–Loève expansion of Gaussian random fields. We develop, validate, and discuss an algorithm based on the Galerkin method with two-dimensional Haar wavelets as basis functions. T...

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Bibliographic Details
Published in:Computational & applied mathematics 2018-05, Vol.37 (2), p.1825-1846
Main Authors: Azevedo, J. S., Wisniewski, F., Oliveira, S. P.
Format: Article
Language:English
Subjects:
Algorithms
Applications of Mathematics
Applied physics
Approximation
Basis functions
Computational mathematics
Computational Mathematics and Numerical Analysis
Discrete Wavelet Transform
Galerkin method
Integral equations
Mathematical Applications in Computer Science
Mathematical Applications in the Physical Sciences
Mathematics
Mathematics and Statistics
Numerical methods
Shape functions
Sparsity
Wavelet transforms
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Summary:We study the numerical approximation of a homogeneous Fredholm integral equation of second kind associated with the Karhunen–Loève expansion of Gaussian random fields. We develop, validate, and discuss an algorithm based on the Galerkin method with two-dimensional Haar wavelets as basis functions. The shape functions are constructed from the orthogonal decomposition of tensor-product spaces of one-dimensional Haar functions, and a recursive algorithm is employed to compute the matrix of the discrete eigenvalue system without the explicit calculation of integrals, allowing the implementation of a fast and efficient algorithm that provides considerable reduction in CPU time, when compared with classical Galerkin methods. Numerical experiments confirm the convergence rate of the method and assess the approximation error and the sparsity of the eigenvalue system when the wavelet expansion is truncated. We illustrate the numerical solution of a diffusion problem with random input data with the present method. In this problem, accuracy was retained after dropping the coefficients below a threshold value that was numerically determined. A similar method with scaling functions rather than wavelet functions does not need a discrete wavelet transform and leads to eigenvalue systems with better conditioning but lower sparsity.
ISSN:0101-8205
2238-3603
1807-0302
DOI:10.1007/s40314-017-0422-4