Error Estimates for the Optimal Control of a Parabolic Fractional PDE

Glusa, Christian; Otárola, Enrique

doi:10.1137/19m1267581

Error Estimates for the Optimal Control of a Parabolic Fractional PDE

Journal Article · 28 April 2021 · SIAM Journal on Numerical Analysis

DOI:https://doi.org/10.1137/19m1267581· OSTI ID:1828020

^[1]; Otárola, Enrique ^[2]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Universidad Tecnica Federico Santa Maria, Valparaiso (Chile)

In this work, we consider the integral definition of the fractional Laplacian and analyze a linear-quadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. To discretize the state equation we propose a fully discrete scheme that relies on an implicit finite difference discretization in time combined with a piecewise linear finite element discretization in space. We derive stability results and a novel $$L^2(0,T;L^2(\Omega))$$ a priori error estimate. On the basis of the aforementioned solution technique, we propose a fully discrete scheme for our optimal control problem that discretizes the control variable with piecewise constant functions, and we derive a priori error estimates for it. We illustrate the theory with one- and two-dimensional numerical experiments.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program; Spanish National Fund for Scientific and Technological Development (FONDECYT)

Grant/Contract Number:: NA0003525

OSTI ID:: 1828020

Report Number(s):: SAND--2020-14257J; 697280

Journal Information:: SIAM Journal on Numerical Analysis, Journal Name: SIAM Journal on Numerical Analysis Journal Issue: 2 Vol. 59; ISSN 0036-1429

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

References (30)

Regularity of spectral fractional Dirichlet and Neumann problems: Regularity of spectral problems Grubb, Gerd Mathematische Nachrichten, Vol. 289, Issue 7 https://doi.org/10.1002/mana.201500041	journal	October 2015
Application of a fractional advection-dispersion equation Benson, David A.; Wheatcraft, Stephen W.; Meerschaert, Mark M. Water Resources Research, Vol. 36, Issue 6 https://doi.org/10.1029/2000WR900031	journal	February 2000
An Extension Problem Related to the Fractional Laplacian Caffarelli, Luis; Silvestre, Luis Communications in Partial Differential Equations, Vol. 32, Issue 8 https://doi.org/10.1080/03605300600987306	journal	August 2007
Limited regularity of solutions to fractional heat and Schrödinger equations Grubb, Gerd Discrete & Continuous Dynamical Systems - A, Vol. 39, Issue 6 https://doi.org/10.3934/dcds.2019148	journal	January 2019
Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains Bonforte, Matteo; Sire, Yannick; Vázquez, Juan Luis Discrete and Continuous Dynamical Systems, Vol. 35, Issue 12 https://doi.org/10.3934/dcds.2015.35.5725	journal	May 2015
Fractional Laplace Operator and Meijer G-function Dyda, Bartłomiej; Kuznetsov, Alexey; Kwaśnicki, Mateusz Constructive Approximation, Vol. 45, Issue 3 https://doi.org/10.1007/s00365-016-9336-4	journal	April 2016
Optimal Distributed Control of Nonlocal Steady Diffusion Problems D'Elia, Marta; Gunzburger, Max SIAM Journal on Control and Optimization, Vol. 52, Issue 1 https://doi.org/10.1137/120897857	journal	January 2014
General edge asymptotics of solutions of second-order elliptic boundary value problems I Costabel, Martin; Dauge, Monique Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 123, Issue 1 https://doi.org/10.1017/S0308210500021272	journal	January 1993
A Priori Error Estimates for the Optimal Control of the Integral Fractional Laplacian D'Elia, Marta; Glusa, Christian; Otárola, Enrique SIAM Journal on Control and Optimization, Vol. 57, Issue 4 https://doi.org/10.1137/18M1219989	journal	January 2019
Ten equivalent definitions of the fractional laplace operator Kwaśnicki, Mateusz Fractional Calculus and Applied Analysis, Vol. 20, Issue 1 https://doi.org/10.1515/fca-2017-0002	journal	January 2017
Regularity in L Sobolev spaces of solutions to fractional heat equations Grubb, Gerd Journal of Functional Analysis, Vol. 274, Issue 9 https://doi.org/10.1016/j.jfa.2017.12.011	journal	May 2018
A speculative study of 2∕3-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures Chen, Wen Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 16, Issue 2 https://doi.org/10.1063/1.2208452	journal	June 2006
Image Recovery via Nonlocal Operators Lou, Yifei; Zhang, Xiaoqun; Osher, Stanley Journal of Scientific Computing, Vol. 42, Issue 2 https://doi.org/10.1007/s10915-009-9320-2	journal	August 2009
Numerical analysis of a transient non-linear axisymmetric eddy current model Bermúdez, Alfredo; Gómez, Dolores; Rodríguez, Rodolfo Computers & Mathematics with Applications, Vol. 70, Issue 8 https://doi.org/10.1016/j.camwa.2015.08.017	journal	October 2015
A Space-Time Fractional Optimal Control Problem: Analysis and Discretization Antil, Harbir; Otárola, Enrique; Salgado, Abner J. SIAM Journal on Control and Optimization, Vol. 54, Issue 3 https://doi.org/10.1137/15M1014991	journal	January 2016
Regularity of the obstacle problem for a fractional power of the laplace operator Silvestre, Luis Communications on Pure and Applied Mathematics, Vol. 60, Issue 1 https://doi.org/10.1002/cpa.20153	journal	January 2006
Optimal control of fractional semilinear PDEs Antil, Harbir; Warma, Mahamadi ESAIM: Control, Optimisation and Calculus of Variations, Vol. 26 https://doi.org/10.1051/cocv/2019003	journal	January 2020
Hitchhikerʼs guide to the fractional Sobolev spaces Di Nezza, Eleonora; Palatucci, Giampiero; Valdinoci, Enrico Bulletin des Sciences Mathématiques, Vol. 136, Issue 5 https://doi.org/10.1016/j.bulsci.2011.12.004	journal	July 2012
Reformulation of elasticity theory for discontinuities and long-range forces Silling, S. A. Journal of the Mechanics and Physics of Solids, Vol. 48, Issue 1 https://doi.org/10.1016/S0022-5096(99)00029-0	journal	January 2000
Fractional Laplacians on domains, a development of Hörmander's theory of μ-transmission pseudodifferential operators Grubb, Gerd Advances in Mathematics, Vol. 268 https://doi.org/10.1016/j.aim.2014.09.018	journal	January 2015
An integral equation theory for inhomogeneous molecular fluids: The reference interaction site model approach Ishizuka, R.; Chong, S. -H.; Hirata, F. The Journal of Chemical Physics, Vol. 128, Issue 3 https://doi.org/10.1063/1.2819487	journal	January 2008
Fractional elliptic equations, Caccioppoli estimates and regularity Caffarelli, Luis A.; Stinga, Pablo Raúl Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Vol. 33, Issue 3 https://doi.org/10.1016/j.anihpc.2015.01.004	journal	May 2016
Scaling laws of marine predator search behaviour Sims, David W.; Southall, Emily J.; Humphries, Nicolas E. Nature, Vol. 451, Issue 7182 https://doi.org/10.1038/nature06518	journal	February 2008
A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations Acosta, Gabriel; Borthagaray, Juan Pablo SIAM Journal on Numerical Analysis, Vol. 55, Issue 2 https://doi.org/10.1137/15M1033952	journal	January 2017
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary Ros-Oton, Xavier; Serra, Joaquim Journal de Mathématiques Pures et Appliquées, Vol. 101, Issue 3 https://doi.org/10.1016/j.matpur.2013.06.003	journal	March 2014
Numerical methods for fractional diffusion Bonito, Andrea; Borthagaray, Juan Pablo; Nochetto, Ricardo H. Computing and Visualization in Science, Vol. 19, Issue 5-6 https://doi.org/10.1007/s00791-018-0289-y	journal	March 2018
Numerical Methods for the Fractional Laplacian: A Finite Difference-Quadrature Approach Huang, Yanghong; Oberman, Adam SIAM Journal on Numerical Analysis, Vol. 52, Issue 6 https://doi.org/10.1137/140954040	journal	January 2014
A FEM for an Optimal Control Problem of Fractional Powers of Elliptic Operators Antil, Harbir; Otárola, Enrique SIAM Journal on Control and Optimization, Vol. 53, Issue 6 https://doi.org/10.1137/140975061	journal	January 2015
Nonlocal Operators with Applications to Image Processing Gilboa, Guy; Osher, Stanley Multiscale Modeling & Simulation, Vol. 7, Issue 3 https://doi.org/10.1137/070698592	journal	January 2009
A PDE Approach to Fractional Diffusion in General Domains: A Priori Error Analysis Nochetto, Ricardo H.; Otárola, Enrique; Salgado, Abner J. Foundations of Computational Mathematics, Vol. 15, Issue 3 https://doi.org/10.1007/s10208-014-9208-x	journal	August 2014

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97 MATHEMATICS AND COMPUTING
linear-quadratic optimal control problem
fractional diffusion
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Error Estimates for the Optimal Control of a Parabolic Fractional PDE

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