On rational Krylov and reduced basis methods for fractional diffusion

Danczul, Tobias; Hofreither, Clemens

doi:10.1515/jnma-2021-0032

Cited by 8 publications

(24 citation statements)

References 42 publications

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“…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.…”

Section: Introductionmentioning

confidence: 56%

“…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.…”

Section: Introductionmentioning

confidence: 74%

“…If the exact extremal eigenvalues of L are not available, one typically replaces them by some numerical approximations 0 < λ L ≤ λ min < λ max ≤ λ U to build Z thereupon. Thanks to (24), the performance of the surrogate u k+1 extracted from Q Z k+1 (L, b) deteriorates only logarithmically if the approximations of λ min and λ max become worse.…”

Section: Zolotarëv's Poles -Zmentioning

confidence: 99%

“…The idea is to replace f τ by a suitable rational function r τ such that r τ (L)b ≈ f τ (L)b. Although elementary, this concept can by found in a variety of modern approximation schemes for the stationary fractional diffusion problem [41,24] among which we discuss a few in the following. One class of methods is based on the prominent Caffarelli-Silvestre extension [21,68,20,22,18], whose susceptibility to standard finite element methods has evoked a large amount of research activity [55,4,8].…”

Section: Introductionmentioning

confidence: 99%

“…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, .…”

Section: Introductionmentioning

confidence: 99%

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

Danczul¹,

Hofreither²,

Schöberl³

2021

Preprint

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We present a unified framework to efficiently approximate solutions to fractional diffusion problems of stationary and parabolic type. After discretization, we can take the point of view that the solution is obtained by a matrix-vector product of the form f τ (L)b, where L is the discretization matrix of the spatial operator, b a prescribed vector, and f τ a parametric function, such as the fractional power or the Mittag-Leffler function. In the abstract framework of Stieltjes and complete Bernstein functions, to which the functions we are interested in belong to, we apply a rational Krylov method and prove uniform convergence when using poles based on Zolotarëv's minimal deviation problem. The latter are particularly suited for fractional diffusion as they allow for an efficient query of the map τ → f τ (L)b and do not degenerate as the fractional parameters approach zero.We also present a variety of both novel and existing pole selection strategies for which we develop a computable error certificate. Our numerical experiments comprise a detailed parameter study of space-time fractional diffusion problems and compare the performance of the poles with the ones predicted by our certificate.

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Section: Introductionmentioning

confidence: 56%

Section: Introductionmentioning

confidence: 74%

Section: Zolotarëv's Poles -Zmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Danczul¹,

Hofreither²,

Schöberl³

2021

Preprint

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A unified rational Krylov method for elliptic and parabolic fractional diffusion problems

Danczul

Hofreither

Schöberl

2023

Numerical Linear Algebra App

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SummaryWe present a unified framework to efficiently approximate solutions to fractional diffusion problems of stationary and parabolic type. After discretization, we can take the point of view that the solution is obtained by a matrix‐vector product of the form , where is the discretization matrix of the spatial operator, a prescribed vector, and a parametric function, such as a fractional power or the Mittag‐Leffler function. In the abstract framework of Stieltjes and complete Bernstein functions, to which the functions we are interested in belong to, we apply a rational Krylov method and prove uniform convergence when using poles based on Zolotarëv's minimal deviation problem. The latter are particularly suited for fractional diffusion as they allow for an efficient query of the map and do not degenerate as the fractional parameters approach zero. We also present a variety of both novel and existing pole selection strategies for which we develop a computable error certificate. Our numerical experiments comprise a detailed parameter study of space‐time fractional diffusion problems and compare the performance of the poles with the ones predicted by our certificate.

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Exponential convergence of hp FEM for spectral fractional diffusion in polygons

2022

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For the spectral fractional diffusion operator of order 2s, $$s \in (0,1)$$ s ∈ ( 0 , 1 ) , in bounded, curvilinear polygonal domains $$\varOmega \subset {{\mathbb {R}}}^2$$ Ω ⊂ R 2 we prove exponential convergence of two classes of hp discretizations under the assumption of analytic data (coefficients and source terms, without any boundary compatibility), in the natural fractional Sobolev norm $${\mathbb {H}}^s(\varOmega )$$ H s ( Ω ) . The first hp discretization is based on writing the solution as a co-normal derivative of a $$2+1$$ 2 + 1 -dimensional local, linear elliptic boundary value problem, to which an hp-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations in$$\varOmega $$ Ω . Leveraging results on robust exponential convergence of hp-FEM for second order, linear reaction diffusion boundary value problems in $$\varOmega $$ Ω , exponential convergence rates for solutions $$u\in {\mathbb {H}}^s(\varOmega )$$ u ∈ H s ( Ω ) of $$\mathcal {L}^s u = f$$ L s u = f follow. Key ingredient in this hp-FEM are boundary fitted meshes with geometric mesh refinement towards$$\partial \varOmega $$ ∂ Ω . The second discretization is based on exponentially convergent numerical sinc quadrature approximations of the Balakrishnan integral representation of $$\mathcal {L}^{-s}$$ L - s combined with hp-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations in$$\varOmega $$ Ω . The present analysis for either approach extends to (polygonal subsets $${\widetilde{\mathcal {M}}}$$ M ~ of) analytic, compact 2-manifolds $${\mathcal {M}}$$ M , parametrized by a global, analytic chart $$\chi $$ χ with polygonal Euclidean parameter domain $$\varOmega \subset {{\mathbb {R}}}^2$$ Ω ⊂ R 2 . Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogorov n-widths of solution sets for spectral fractional diffusion in curvilinear polygons and for analytic source terms are deduced.

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On rational Krylov and reduced basis methods for fractional diffusion

Cited by 8 publications

References 42 publications

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

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

A unified rational Krylov method for elliptic and parabolic fractional diffusion problems

Exponential convergence of hp FEM for spectral fractional diffusion in polygons

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