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：本文致力于发展降阶模型(ROMs)，用于对受多个参数（如扩散系数或分数阶指数）控制的分数阶椭圆偏微分方程(PDEs)进行反复模拟。这些问题广泛存在于高斯过程模拟和地球物理电磁学等应用中。该方法利用加藤积分公式将解表示为涉及参数化椭圆PDE解的积分形式，并采用空间有限元离散与分数阶部分的sinc求积法进行数值离散。ROM的离线阶段通过使用移位线性系统求解器MPGMRES-Sh以及随机化方法压缩快照矩阵来加速。我们的方法兼具计算高效性与内存节约性。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展示了该方法的优势。