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, geophysical electromagnetics. The approach uses the Kato integral formula to express the solution as an integral involving the solution of a parameterized 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.

本文感兴趣的是开发降阶模型（ROM），用于重复模拟分数椭圆偏微分方程（PDE），以获得控制这些模型的多个参数值（例如，扩散系数或分数指数）。这些问题出现在许多应用中，包括模拟高斯过程、地球物理电磁学。该方法使用加藤积分公式将解表示为涉及参数化椭圆偏微分方程解的积分，该积分使用空间中的有限元和正弦求积来离散化小数部分。使用移位线性系统 MPGMRES-Sh 的求解器并使用随机方法压缩快照矩阵来加速 ROM 的离线阶段。我们的方法具有计算效率和内存效率。对一系列模型问题的数值实验（包括高斯过程的应用）显示了我们方法的优点。