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

## Abstract

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.

- Authors:

- CETEC-UFRB, Centro (Brazil)
- UFPR, Departamento de Matemática (Brazil)

- Publication Date:

- OSTI Identifier:
- 22769330

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 2; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; ALGORITHMS; APPROXIMATIONS; COMPARATIVE EVALUATIONS; CONVERGENCE; EIGENVALUES; INTEGRAL EQUATIONS; NUMERICAL SOLUTION; RANDOMNESS; TWO-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Azevedo, J. S., E-mail: juarez@ufrb.edu.br, Wisniewski, F., E-mail: felipewisniewski@yahoo.com.br, and Oliveira, S. P., E-mail: saulopo@ufpr.br.
```*A Galerkin method with two-dimensional Haar basis functions for the computation of the Karhunen–Loève expansion*. United States: N. p., 2018.
Web. doi:10.1007/S40314-017-0422-4.

```
Azevedo, J. S., E-mail: juarez@ufrb.edu.br, Wisniewski, F., E-mail: felipewisniewski@yahoo.com.br, & Oliveira, S. P., E-mail: saulopo@ufpr.br.
```*A Galerkin method with two-dimensional Haar basis functions for the computation of the Karhunen–Loève expansion*. United States. doi:10.1007/S40314-017-0422-4.

```
Azevedo, J. S., E-mail: juarez@ufrb.edu.br, Wisniewski, F., E-mail: felipewisniewski@yahoo.com.br, and Oliveira, S. P., E-mail: saulopo@ufpr.br. Tue .
"A Galerkin method with two-dimensional Haar basis functions for the computation of the Karhunen–Loève expansion". United States. doi:10.1007/S40314-017-0422-4.
```

```
@article{osti_22769330,
```

title = {A Galerkin method with two-dimensional Haar basis functions for the computation of the Karhunen–Loève expansion},

author = {Azevedo, J. S., E-mail: juarez@ufrb.edu.br and Wisniewski, F., E-mail: felipewisniewski@yahoo.com.br and Oliveira, S. P., E-mail: saulopo@ufpr.br},

abstractNote = {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.},

doi = {10.1007/S40314-017-0422-4},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 2,

volume = 37,

place = {United States},

year = {2018},

month = {5}

}