## All Research

#### Title of Oral/Poster Presentation

Multi-resolution Analysis Using Wavelet Basis Conditioned on Homogenization

Article

#### College

College of Science

Joseph Koebbe

#### Presentation Type

Oral Presentation

#### Abstract

Wavelets and homogenization methods have been used in the development of algorithms for the approximate solution of differential equations. In this work, we propose a homogenization wavelets reconstruction algorithm for computing the solution of elliptic partial differential equations. The proposed algorithm is based on the characterization of homogenization methods in multi-resolution analysis. We employ orthogonal decomposition of the problem into two scale problems at various scale levels using wavelets multi-resolution analysis. To illustrate the proposed methodology, we restrict the problem to that of one dimensional case. The problem of flow in a porous medium with conductivity/permeability depending on the spatial variable provides an example of one dimensional situation. It is well known that if an average value of the conductivity/permeability is computed, it must be equal to the harmonic average of the function representing the fine scale parameter values. Our fast transform algorithm also preserves the harmonic average of the conductivity/permeability.

Room 421

#### Start Date

4-12-2018 12:00 PM

#### End Date

4-12-2018 1:15 PM

#### Share

COinS

Apr 12th, 12:00 PM Apr 12th, 1:15 PM

Multi-resolution Analysis Using Wavelet Basis Conditioned on Homogenization

Room 421

Wavelets and homogenization methods have been used in the development of algorithms for the approximate solution of differential equations. In this work, we propose a homogenization wavelets reconstruction algorithm for computing the solution of elliptic partial differential equations. The proposed algorithm is based on the characterization of homogenization methods in multi-resolution analysis. We employ orthogonal decomposition of the problem into two scale problems at various scale levels using wavelets multi-resolution analysis. To illustrate the proposed methodology, we restrict the problem to that of one dimensional case. The problem of flow in a porous medium with conductivity/permeability depending on the spatial variable provides an example of one dimensional situation. It is well known that if an average value of the conductivity/permeability is computed, it must be equal to the harmonic average of the function representing the fine scale parameter values. Our fast transform algorithm also preserves the harmonic average of the conductivity/permeability.