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This work is on a fast and accurate reduced basis method for solvingdiscretized fractional elliptic partial differential equations (PDEs) of theform $\mathcal{A}^su=f$ by rational approximation. A direct computation of theaction of such an approximation would require solving multiple (20$\sim$30)large-scale sparse linear systems. Our method constructs the reduced basisusing the first few directions obtained from the preconditioned conjugategradient method applied to one of the linear systems. As shown in the theoryand experiments, only a small number of directions (5$\sim$10) are needed toapproximately solve all large-scale systems on the reduced basis subspace. Thisreduces the computational cost dramatically because: (1) We only use one of thelarge-scale problems to construct the basis; and (2) all large-scale problemsrestricted to the subspace have much smaller sizes. We test our algorithms forfractional PDEs on a 3d Euclidean domain, a 2d surface, and randomcombinatorial graphs. We also use a novel approach to construct the rationalapproximation for the fractional power function by the orthogonal greedyalgorithm (OGA).

A reduced conjugate gradient basis method for fractional diffusion