Alternatives for Generating a Reduced Basis to Solve the Hyperspectral Diffuse Optical Tomography Model

TL;DR

This paper compares alternative algorithms for generating reduced bases in the Reduced Basis Method to efficiently solve the parametric PDEs in hyperspectral diffuse optical tomography, showing potential improvements over the standard greedy approach.

Contribution

It introduces and evaluates Metropolis and gradient algorithms as new methods for selecting basis parameters in RBM, outperforming the traditional greedy algorithm in hyDOT applications.

Findings

Metropolis and gradient algorithms produce smaller relative errors.

These methods require less computational time.

They outperform the greedy algorithm in basis quality.

Abstract

The Reduced Basis Method (RBM) is a model reduction technique used to solve parametric PDEs that relies upon a basis set of solutions to the PDE at specific parameter values. To generate this reduced basis, the set of a small number of parameter values must be strategically chosen. We apply a Metropolis algorithm and a gradient algorithm to find the set of parameters and compare them to the standard greedy algorithm most commonly used in the RBM. We test our methods by using the RBM to solve a simplified version of the governing partial differential equation for hyperspectral diffuse optical tomography (hyDOT). The governing equation for hyDOT is an elliptic PDE parameterized by the wavelength of the laser source. For this one-dimensional problem, we find that both the Metropolis and gradient algorithms are potentially superior alternatives to the greedy algorithm in that they generate a reduced basis which produces solutions with a smaller relative error with respect to solutions found using the finite element method and in less time.