A reduced conjugate gradient basis method for fractional diffusion
arXiv:2305.18038 · doi:10.1137/23M1575913
Abstract
This work is on a fast and accurate reduced basis method for solving discretized fractional elliptic partial differential equations (PDEs) of the form by rational approximation. A direct computation of the action of such an approximation would require solving multiple (2030) large-scale sparse linear systems. Our method constructs the reduced basis using the first few directions obtained from the preconditioned conjugate gradient method applied to one of the linear systems. As shown in the theory and experiments, only a small number of directions (510) are needed to approximately solve all large-scale systems on the reduced basis subspace. This reduces the computational cost dramatically because: (1) We only use one of the large-scale problems to construct the basis; and (2) all large-scale problems restricted to the subspace have much smaller sizes. We test our algorithms for fractional PDEs on a 3d Euclidean domain, a 2d surface, and random combinatorial graphs. We also use a novel approach to construct the rational approximation for the fractional power function by the orthogonal greedy algorithm (OGA).
References in corpus (10)
- Nonlocal diffusion and applications
- The AAA algorithm for rational approximation
- Approximation and learning by greedy algorithms
- Numerical methods for nonlocal and fractional models
- Sharp Bounds on the Approximation Rates, Metric Entropy, and -widths of Shallow Neural Networks
- Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation
- Global superconvergence of the lowest order mixed finite element on mildly structured meshes
- Superconvergent recovery of Raviart--Thomas mixed finite elements on triangular grids
- Entropy-based convergence rates of greedy algorithms
- Nodal auxiliary space preconditioning for the surface de Rham complex