  This paper is interested in developing reduced order models (ROMs) for
repeated simulation of fractional elliptic partial differential equations
(PDEs) for multiple values of the parameters (e.g., diffusion coefficients or
fractional exponent) governing these models. These problems arise in many
applications including simulating Gaussian processes, and geophysical
electromagnetics. The approach uses the Kato integral formula to express the
solution as an integral involving the solution of a parametrized elliptic PDE,
which is discretized using finite elements in space and sinc quadrature for the
fractional part. The offline stage of the ROM is accelerated using a solver for
shifted linear systems, MPGMRES-Sh, and using a randomized approach for
compressing the snapshot matrix. Our approach is both computational and memory
efficient. Numerical experiments on a range of model problems, including an
application to Gaussian processes, show the benefits of our approach.
