A Reduced Basis Method For Fractional Diffusion Operators II
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We present a novel numerical scheme to approximate the solution map s u(s) := L^-sf to partial differential equations involving fractional elliptic operators. Reinterpreting L^-s as interpolation operator allows us to derive an integral representation of u(s) which includes solutions to parametrized reaction-diffusion problems. 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. Avoiding further discretization, 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 a low-dimensional space, where explicit diagonalization is feasible. The universal character of the underlying s-independent reduced space allows the approximation of (u(s))_s∈(0,1) in its entirety. We prove exponential convergence rates and confirm the analysis with a variety of numerical examples. Further improvements are proposed in the second part of this investigation to avoid inversion of L. Instead, we directly project the matrix to the reduced space, where its negative fractional power is evaluated. A numerical comparison with the predecessor highlights its competitive performance.
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