Abstract
We present a novel numerical scheme to approximate the solution map s ↦ u(s) := 𝓛–sf 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.
Acknowledgment
The authors acknowledge support from the Austrian Science Fund (FWF) through grant No. F65 and W1245.
References
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6 Appendix
Proof of Theorem 2.1
It suffices to show that
There holds
Let uk := 〈u, φk〉0 to deduce from Lemma 2.1
which proves (6.1) and concludes the proof.□
Proof of Theorem 2.2
One observes that for any F ∈ 𝒱0 we have
from which we conclude that
for all k ∈ ℕ implies that F = 0, it is also a basis. This proves the claim.□
Proof of Theorem 2.3
Due to
and Theorem 2.2, there holds
proving the first equality in (2.7). The second one follows by means of (2.2). Furthermore, one observes
confirming the first equality in (2.8). The latter is a consequence of (2.3).
The remainder follows as
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