  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.
