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.

翻译：本文旨在开发适用于分数阶椭圆型偏微分方程（PDE）在多个参数（如扩散系数或分数阶指数）取值下重复模拟的降阶模型（ROM）。此类问题广泛存在于包括高斯过程模拟和地球物理电磁学在内的诸多应用中。该方法利用Kato积分公式将解表示为涉及参数化椭圆型PDE解的积分形式，并通过空间有限元离散和分数阶部分的sinc求积实现数值离散。ROM的离线阶段采用移位线性系统求解器MPGMRES-Sh加速，并通过随机化方法压缩快照矩阵。本方法兼具计算效率与内存效率。在包括高斯过程应用在内的一系列模型问题上的数值实验证明了该方法的优势。