Fractional operators with inhomogeneous boundary conditions: analysis, control, and discretization

Antil, Harbir; Pfefferer, Johannes; Rogovs, Sergejs

doi:10.4310/cms.2018.v16.n5.a11

Cited by 57 publications

(85 citation statements)

References 30 publications

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“…As it is customary in the PDE theory, we have stated this definition for smooth functions, however by standard density arguments it immediately extends to Sobolev spaces, we refer to [3] for details. In addition, we emphasize that when u = 0 on the boundary Γ, the definition above is nothing but the standard spectral fractional Laplacian (−∆ 0 ) s with zero boundary conditions.…”

Section: 3] and Xmentioning

confidence: 99%

Fractional Operators Applied to Geophysical Electromagnetics

Weiss¹,

Waanders

Antil

2019

Geophysical Journal International

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A growing body of applied mathematics literature in recent years has focussed on the application of fractional calculus to problems of anomalous transport. In these analyses, the anomalous transport (of charge, tracers, fluid, etc.) is presumed attributable to long-range correlations of material properties within an inherently complex, and in some cases self-similar, conducting medium. Rather than considering an exquisitely discretized (and computationally intractable) representation of the medium, the complex and spatially correlated heterogeneity is represented through reformulation of the PDE governing the relevant transport physics such that its coefficients are, instead, smooth but paired with fractional-order space derivatives. Here we apply these concepts to the scalar Helmholtz equation and its use in electromagnetic interrogation of Earth's interior through the magnetotelluric method. We outline a practical algorithm for solving the Helmholtz equation using spectral methods coupled with finite element discretization. Execution of this algorithm for the magnetotelluric problem reveals several interesting features observable in field data: long-range correlation of the predicted electromagnetic fields; a power-law relationship between the squared impedance amplitude and squared wavenumber whose slope is a function of the fractional exponent within the governing Helmholtz equation; and, a non-constant apparent resistivity spectrum whose variability arises solely from the fractional exponent. In geologic settings characterized by selfsimilarity (e.g. fracture systems; thick and richly-textured sedimentary sequences, etc.) we posit that diagnostics are useful for geologic characterization of features far below the typical resolution limit of electromagnetic methods in geophysics.

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Section: 3] and Xmentioning

confidence: 99%

Fractional Operators Applied to Geophysical Electromagnetics

Weiss¹,

Waanders

Antil

2019

Geophysical Journal International

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“…In the connection of well-posedness and regularity results, we refer to [1,2] for the case of the fractional negative Laplacian with zero Dirichlet boundary conditions; general operators other than the negative Laplacian have apparently only been studied in [24,[33][34][35]. As of now, aspects of optimal control have been scarcely dealt with even for simpler linear evolutionary systems involving fractional operators; for such systems, some identification problems were addressed in the recent contributions [36,45], while for optimal control problems for such cases we refer to [6] (for the stationary (elliptic) case, see also [3][4][5][7][8][9]). However, to the authors' best knowledge, the present paper appears to be the first contribution that addresses optimal control problems for Cahn-Hilliard systems with general fractional order operators and potentials of double obstacle type.…”

Section: Introductionmentioning

confidence: 99%

Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials

Colli¹,

Gilardi²,

Sprekels³

2021

Discrete &Amp; Continuous Dynamical Systems - S

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In the recent paper "Well-posedness and regularity for a generalized fractional Cahn-Hilliard system", the same authors derived general well-posedness and regularity results for a rather general system of evolutionary operator equations having the structure of a Cahn-Hilliard system. The operators appearing in the system equations were fractional versions in the spectral sense of general linear operators A and B having compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. The associated double-well potentials driving the phase separation process modeled by the Cahn-Hilliard system could be of a very general type that includes standard physically meaningful cases such as polynomial, logarithmic, and double obstacle nonlinearities. In the subsequent paper "Optimal distributed control of a generalized fractional Cahn-Hilliard system" (Appl. Math. Optim. (2018), https://doi.org/10.1007/s00245-018-9540-7) by the same authors, an analysis of distributed optimal control problems was performed for such evolutionary systems, where only the differentiable case of certain polynomial and logarithmic doublewell potentials could be admitted. Results concerning existence of optimizers and Colli -Gilardi -Sprekels first-order necessary optimality conditions were derived, where more restrictive conditions on the operators A and B had to be assumed in order to be able to show differentiability properties for the associated control-to-state operator. In the present paper, we complement these results by studying a distributed control problem for such evolutionary systems in the case of nondifferentiable nonlinearities of double obstacle type. For such nonlinearities, it is well known that the standard constraint qualifications cannot be applied to construct appropriate Lagrange multipliers. To overcome this difficulty, we follow here the so-called "deep quench" method. This technique, in which the nondifferentiable double obstacle nonlinearity is approximated by differentiable logarithmic nonlinearities, was first developed by P. Colli, M.H. Farshbaf-Shaker and J. Sprekels in the paper "A deep quench approach to the optimal control of an Allen-Cahn equation with dynamic boundary conditions and double obstacles" (Appl. Math. Optim. 71 (2015), pp. 1-24) and has proved to be a powerful tool in a number of optimal control problems with double obstacle potentials in the framework of systems of Cahn-Hilliard type. We first give a general convergence analysis of the deep quench approximation that includes an error estimate and then demonstrate that its use leads in the double obstacle case to appropriate first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint state system.

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“…Regularity issues have been treated for example by Seeley in [57,58,59] and by several other authors in more recent works (e.g., [40,21,9,69]). While the analysis of boundary behaviour for the solution to such problems has been investigated in [11], non-homogeneous boundary conditions for these nonlocal operators have been considered in [1,5,26,48].…”

Section: Introductionmentioning

confidence: 99%

Numerical approximations for fractional elliptic equationsviathe method of semigroups

2020

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We provide a novel approach to the numerical solution of the family of nonlocal elliptic equations (−∆) s u = f in Ω, subject to some homogeneous boundary conditions B(u) = 0 on ∂Ω, where s ∈ (0, 1), Ω ⊂ R n is a bounded domain, and (−∆) s is the spectral fractional Laplacian associated to B on ∂Ω. We use the solution representation (−∆) −s f together with its singular integral expression given by the method of semigroups. By combining finite element discretizations for the heat semigroup with monotone quadratures for the singular integral we obtain accurate numerical solutions. Roughly speaking, given a datum f in a suitable fractional Sobolev space of order r ≥ 0 and the discretization parameter h > 0, our numerical scheme converges as O(h r+2s ), providing super quadratic convergence rates up to O(h 4 ) for sufficiently regular data, or simply O(h 2s ) for merely f ∈ L 2 (Ω). We also extend the proposed framework to the case of nonhomogeneous boundary conditions and support our results with some illustrative numerical tests.

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Fractional operators with inhomogeneous boundary conditions: analysis, control, and discretization

Cited by 57 publications

References 30 publications

Fractional Operators Applied to Geophysical Electromagnetics

Fractional Operators Applied to Geophysical Electromagnetics

Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials

Numerical approximations for fractional elliptic equationsviathe method of semigroups

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