$hp$-Finite Elements for Fractional Diffusion

Meidner, Dominik; Pfefferer, Johannes; Schürholz, Klemens; Vexler, Boris

doi:10.1137/17m1135517

Cited by 33 publications

(39 citation statements)

References 34 publications

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“…Let u d ∈ L 2 (Ω) and a, b ∈ R with a ≤ 0 ≤ b be given. We define the set of admissible controls Z ad by [19]. We may also consider the operator S acting on L 2 (Ω) with range in L 2 (Ω).…”

Section: Existence and Regularity Of Optimal Controlsmentioning

confidence: 99%

“…Exponential decay of the solution in the artificial dimension allows construction of different numerical methods, see e.g. [6,19,21]. In these publications, the problem is discretized by introducing a tensor product mesh of the domain C Y = Ω × (0, Y ), which is constructed by a conformal triangulation of Ω and a graded mesh in the artificial direction, see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…However, numerical experiments show that this convergence rate is not optimal in a specific range of fractional powers s. The cost of solving the problem is related to the number of elements in C Y , and not only to the number of elements in Ω, resulting in an increased computational complexity. This issue is first overcome in [19] by exploiting p-finite elements in the extended direction.…”

Section: Introductionmentioning

confidence: 99%

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A FEM for an optimal control problem subject to the fractional Laplace equation

Dohr

Kahle

Rogovs

et al. 2019

Calcolo

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We study the numerical approximation of linear-quadratic optimal control problems subject to the fractional Laplace equation with its spectral definition. We compute an approximation of the state equation using a discretization of the Balakrishnan formula that is based on a finite element discretization in space and a sinc quadrature approximation of the additionally involved integral. A tailored approach for the numerical solution of the resulting linear systems is proposed.Concerning the discretization of the optimal control problem we consider two schemes. The first one is the variational approach, where the control set is not discretized, and the second one is the fully discrete scheme where the control is discretized by piecewise constant functions. We derive finite element error estimates for both methods and illustrate our results by numerical experiments.

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Section: Existence and Regularity Of Optimal Controlsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A FEM for an optimal control problem subject to the fractional Laplace equation

Dohr

Kahle

Rogovs

et al. 2019

Calcolo

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“…This choice is motivated by the singular behavior of the solution towards the boundary Ω for a constant s. In that case anisotropically refined meshes are preferable as these can be used to compensate the singular effects [19,21]. In all our implementations we will choose a fixed constant s in (4.1).…”

Section: 1mentioning

confidence: 99%

Sobolev Spaces with Non-Muckenhoupt Weights, Fractional Elliptic Operators, and Applications

Antil¹,

Rautenberg²

2019

SIAM J. Math. Anal.

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We propose a new variational model in weighted Sobolev spaces with non-standard weights and applications to image processing. We show that these weights are, in general, not of Muckenhoupt type and therefore the classical analysis tools may not apply. For special cases of the weights, the resulting variational problem is known to be equivalent to the fractional Poisson problem. The trace space for the weighted Sobolev space is identified to be embedded in a weighted L 2 space. We propose a finite element scheme to solve the Euler-Lagrange equations, and for the image denoising application we propose an algorithm to identify the unknown weights. The approach is illustrated on several test problems and it yields better results when compared to the existing total variation techniques.

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“…The idea is to equivalently write (1) as a "local" PDE problem on C := Ω × (0, ∞), which can be solved using standard algorithms. Using this idea, finite element approaches have been developed in [31,33] by truncating the semiinfinite cylinder C to a finite cylinder C y + for y + > 0. Such a truncation is justified due to the exponential decay of solution in the extended dimension.…”

mentioning

confidence: 99%

Reduced Basis Methods for Fractional Laplace Equations via Extension

Antil¹,

Chen²,

Narayan³

2019

SIAM J. Sci. Comput.

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Fractional Laplace equations are becoming important tools for mathematical modeling and prediction. Recent years have shown much progress in developing accurate and robust algorithms to numerically solve such problems, yet most solvers for fractional problems are computationally expensive. Practitioners are often interested in choosing the fractional exponent of the mathematical model to match experimental and/or observational data; this requires the computational solution to the fractional equation for several values of the both exponent and other parameters that enter the model, which is a computationally expensive many-query problem. To address this difficulty, we present a model order reduction strategy for fractional Laplace problems utilizing the reduced basis method (RBM). Our RBM algorithm for this fractional partial differential equation (PDE) allows us to accomplish significant acceleration compared to a traditional PDE solver while maintaining accuracy. Our numerical results demonstrate this accuracy and efficiency of our RBM algorithm on fractional Laplace problems in two spatial dimensions. ). A. Narayan was partially supported by AFOSR FA9550-15-1-0467. arXiv:1808.00584v1 [math.NA] 1 Aug 2018 numerical solvers for (1) is an "extension" technique, and directly applies to more generic situations where we replace (−∆) s with L s where Lw = −div(A∇w) + cw when c is nonnegative and bounded on Ω, and A ∈ L ∞ (Ω) n×n is symmetric and uniformly positive definite. Extension techniques also apply to much more general operators [30].Fractional Laplace problems are useful in many contexts, for instance, image denoising [4,6,22], phase field models [3,4], electrical signal propagation in cardiac tissue where the occurrence of fractional Laplacian has been experimentally validated [11], diffusion of biological species [39]. In fact all heat kernels under fairly general assumptions are either equivalent to heat kernels of diffusion (exponential), or heat kernels for 2s-stable processes (polynomial) [24]. In particular, the fractional Laplacian is a special case of a 2s-stable process. We also refer to [19,35] for a general description of fractional heat kernels and their relation to stochastic processes.Several strategies exist for computing solutions to (1). We refer to [37] for the so called Stinga-Torrea extension which was originally proposed in R n in [13,32] and has come to be known as the Caffarelli-Silvestre extension. The idea is to equivalently write (1) as a "local" PDE problem on C := Ω × (0, ∞), which can be solved using standard algorithms. Using this idea, finite element approaches have been developed in [31,33] by truncating the semiinfinite cylinder C to a finite cylinder C y + for y + > 0. Such a truncation is justified due to the exponential decay of solution in the extended dimension. It is also possible to circumvent truncation and directly approximate solutions to the local problem on the unbounded domain C by using a spectral method in the extended direction [2].An excellent alternative to t...

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$hp$-Finite Elements for Fractional Diffusion

Cited by 33 publications

References 34 publications

A FEM for an optimal control problem subject to the fractional Laplace equation

A FEM for an optimal control problem subject to the fractional Laplace equation

Sobolev Spaces with Non-Muckenhoupt Weights, Fractional Elliptic Operators, and Applications

Reduced Basis Methods for Fractional Laplace Equations via Extension

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