scholarly journals A Reduced Basis Method For Fractional Diffusion Operators II

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tobias Danczul ◽  
Joachim Schöberl

Abstract We present a novel numerical scheme to approximate the solution map s ↦ u(s) := 𝓛−s f to fractional PDEs involving elliptic operators. Reinterpreting 𝓛−s as an interpolation operator allows us to write u(s) as an integral including solutions to a parametrized family of local PDEs. We propose a reduced basis strategy on top of a finite element method to approximate its integrand. Unlike prior works, we deduce the choice of snapshots for the reduced basis procedure analytically. The integral is interpreted in a spectral setting to evaluate the surrogate directly. Its computation boils down to a matrix approximation L of the operator whose inverse is projected to the s-independent reduced space, where explicit diagonalization is feasible. Exponential convergence rates are proven rigorously. A second algorithm is presented to avoid inversion of L. Instead, we directly project the matrix to the subspace, where its negative fractional power is evaluated. A numerical comparison with the predecessor highlights its competitive performance.

2020 ◽  
Vol 28 (3) ◽  
pp. 147-160
Author(s):  
Andrea Bonito ◽  
Diane Guignard ◽  
Ashley R. Zhang

AbstractWe consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.


2011 ◽  
Author(s):  
Jan Pomplun ◽  
Sven Burger ◽  
Lin Zschiedrich ◽  
Frank Schmidt

2011 ◽  
Author(s):  
Frank Schmidt ◽  
Jan Pomplun ◽  
Lin Zschiedrich ◽  
Sven Burger

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