A reduced basis method for fractional diffusion operators II

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.

A Unified Rational Krylov Method for Elliptic and Parabolic Fractional Diffusion Problems

“…In view of [24], the presented results can be seen as improvement and extension of [25,26]. Our analytical findings show that the results of [49] admit a natural generalization to complete Bernstein functions.…”

“…over the spectral interval Σ of L. Minimizing r Ξ L∞(Σ) leads to Zolotarëv's well-known minimal deviation problem whose analytical solution provides a τ -independent selection of real poles and allows for an efficient querying of the solution map τ → f τ (L)b. Initiated by the work of [25,26,24], which proves pointwise convergence in the parameter s ∈ (−1, 1) for the special case f τ (λ) = λ s , we show exponential convergence rates which are uniform in τ . A computational inconvenience of Zolotarëv's poles is the fact that they are not nested.…”

“…Another class for approximating f τ (L)b are the rational Krylov methods (RKM) [54,2,53,11], to which, in view of [24], a variety of reduced basis methods [28,17,25,26] belong to. The RKM extracts a low-dimensional surrogate u k+1 ≈ f τ (L)b from a search space of the form Q Ξ k+1 (L, b) := span{q k (L) −1 b, .…”

An exponentially convergent discretization for space-time fractional parabolic equations using $hp$-FEM

“…See [BBN + 18] for a summary of discretization methods for the elliptic fractional diffusion problem. Recently another approach for discretizing both the fractional Laplace and heat equation based on a reduced basis method [DS19,DS20]. Further methods that are based on discretizing first and computing fractional powers of the resulting stiffness matrix include [HHT08, GHK05, HLM + 18, HLM + 20]; see also the discussion in [Hof20].…”

An a posteriori error estimator for the spectral fractional power of the Laplacian

“…in [3,66,77]. Among the methods addressing the above numerical issues, we can cite: methods to efficiently solve eigenvalue problems [40], multigrid methods for performing efficient dense matrixvector products [6,7], hybrid finite element-spectral schemes [8], Dirichlet-to-Neumann maps (such as the Caffarelli-Silvestre extension) [13,36,44,57,75,79], semigroups methods [46,45,78], rational approximation methods [1,65,67], Dunford-Taylor integrals [21,26,25,28,23,31,60,67] (which can be considered as particular examples of rational approximation methods) and reduced basis methods [47,48,51].…”