  In this paper, we introduce a new reduced basis methodology for accelerating
the computation of large parameterized systems of high-fidelity integral
equations. Core to our methodology is the use of coarse-proxy models (i.e.,
lower resolution variants of the underlying high-fidelity equations) to
identify important samples in the parameter space from which a high quality
reduced basis is then constructed. Unlike the more traditional POD or greedy
methods for reduced basis construction, our methodology has the benefit of
being both easy to implement and embarrassingly parallel. We apply our
methodology to the under-served area of integral equations, where the density
of the underlying integral operators has traditionally made reduced basis
methods difficult to apply. To handle this difficulty, we introduce an operator
interpolation technique, based on random sub-sampling, that is aimed
specifically at integral operators. To demonstrate the effectiveness of our
techniques, we present two numerical case studies, based on the Radiative
Transport Equation and a boundary integral formation of the Laplace Equation
respectively, where our methodology provides a significant improvement in
performance over the underlying high-fidelity models for a wide range of error
tolerances. Moreover, we demonstrate that for these problems, as the
coarse-proxy selection threshold is made more aggressive, the approximation
error of our method decreases at an approximately linear rate.
