An optimization-based approach to parameter learning for fractional type nonlocal models

Burkovska, Olena; Glusa, Christian; D'Elia, Marta

doi:10.1016/j.camwa.2021.05.005

Title: An optimization-based approach to parameter learning for fractional type nonlocal models

Journal Article · Wed May 26 00:00:00 EDT 2021 · Computers and Mathematics with Applications (Oxford)

DOI:https://doi.org/10.1016/j.camwa.2021.05.005· OSTI ID:1787538

Burkovska, Olena ^[1];

^[2]; D'Elia, Marta ^[3]

Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sandia National Lab. (SNL-CA), Livermore, CA (United States)

Nonlocal operators of fractional type are a popular modeling choice for applications that do not adhere to classical diffusive behavior; however, one major challenge in nonlocal simulations is the selection of model parameters. In this work we propose an optimization-based approach to parameter identification for fractional models with an optional truncation radius. We formulate the inference problem as an optimal control problem where the objective is to minimize the discrepancy between observed data and an approximate solution of the model, and the control variables are the fractional order and the truncation length. For the numerical solution of the minimization problem we propose a gradient-based approach, where we enhance the numerical performance by an approximation of the bilinear form of the state equation and its derivative with respect to the fractional order. Several numerical tests in one and two dimensions illustrate the theoretical results and show the robustness and applicability of our method.

View Accepted Manuscript (DOE)

View Accepted Manuscript (Publisher)

Cite

Export

Save

Research Organization:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: AC04-94AL85000; AC05-00OR22725

OSTI ID:: 1787538

Alternate ID(s):: OSTI ID: 1885396; OSTI ID: 1960986

Report Number(s):: SAND-2021-6450J; 696581

Journal Information:: Computers and Mathematics with Applications (Oxford), Vol. 116; ISSN 0898-1221

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (58)

Fractional vector calculus for fractional advection–dispersion Meerschaert, Mark M.; Mortensen, Jeff; Wheatcraft, Stephen W. Physica A: Statistical Mechanics and its Applications, Vol. 367 https://doi.org/10.1016/j.physa.2005.11.015	journal	July 2006
How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems Cortazar, Carmen; Elgueta, Manuel; Rossi, Julio D. Archive for Rational Mechanics and Analysis, Vol. 187, Issue 1 https://doi.org/10.1007/s00205-007-0062-8	journal	November 2007
Multiscaling fractional advection-dispersion equations and their solutions: MULTISCALING FRACTIONAL ADES Schumer, Rina; Benson, David A.; Meerschaert, Mark M. Water Resources Research, Vol. 39, Issue 1 https://doi.org/10.1029/2001WR001229	journal	January 2003
An energy-based coupling approach to nonlocal interface problems Capodaglio, Giacomo; D’Elia, Marta; Bochev, Pavel Computers & Fluids, Vol. 207 https://doi.org/10.1016/j.compfluid.2020.104593	journal	July 2020
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
Sparse Optimal Control for Fractional Diffusion Otárola, Enrique; Salgado, Abner J. Computational Methods in Applied Mathematics, Vol. 18, Issue 1 https://doi.org/10.1515/cmam-2017-0030	journal	September 2017
Numerical methods for nonlocal and fractional models D’Elia, Marta; Du, Qiang; Glusa, Christian Acta Numerica, Vol. 29 https://doi.org/10.1017/S096249292000001X	journal	May 2020
Mathematical and Numerical Analysis of Linear Peridynamic Models with Nonlocal Boundary Conditions Zhou, Kun; Du, Qiang SIAM Journal on Numerical Analysis, Vol. 48, Issue 5 https://doi.org/10.1137/090781267	journal	January 2010
Fractional Diffusion on Bounded Domains Defterli, Ozlem; D’Elia, Marta; Du, Qiang Fractional Calculus and Applied Analysis, Vol. 18, Issue 2 https://doi.org/10.1515/fca-2015-0023	journal	January 2015
Uniquely identifying the variable order of time-fractional partial differential equations on general multi-dimensional domains Zheng, Xiangcheng; Wang, Hong Inverse Problems in Science and Engineering https://doi.org/10.1080/17415977.2020.1849182	journal	November 2020
Identification of the Diffusion Parameter in Nonlocal Steady Diffusion Problems D’Elia, M.; Gunzburger, M. Applied Mathematics & Optimization, Vol. 73, Issue 2 https://doi.org/10.1007/s00245-015-9300-x	journal	May 2015
Numerical solution of the heat equation with nonlocal boundary conditions Liu, Yunkang Journal of Computational and Applied Mathematics, Vol. 110, Issue 1 https://doi.org/10.1016/S0377-0427(99)00200-9	journal	October 1999
Modeling non-Fickian transport in geological formations as a continuous time random walk Berkowitz, Brian; Cortis, Andrea; Dentz, Marco Reviews of Geophysics, Vol. 44, Issue 2 https://doi.org/10.1029/2005RG000178	journal	January 2006
Peridynamics and Material Interfaces Alali, Bacim; Gunzburger, Max Journal of Elasticity, Vol. 120, Issue 2 https://doi.org/10.1007/s10659-014-9512-3	journal	January 2015
Affine Approximation of Parametrized Kernels and Model Order Reduction for Nonlocal and Fractional Laplace Models Burkovska, Olena; Gunzburger, Max SIAM Journal on Numerical Analysis, Vol. 58, Issue 3 https://doi.org/10.1137/19M124321X	journal	January 2020
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
Aspects of an adaptive finite element method for the fractional Laplacian: A priori and a posteriori error estimates, efficient implementation and multigrid solver Ainsworth, Mark; Glusa, Christian Computer Methods in Applied Mechanics and Engineering, Vol. 327 https://doi.org/10.1016/j.cma.2017.08.019	journal	December 2017
fPINNs: Fractional Physics-Informed Neural Networks Pang, Guofei; Lu, Lu; Karniadakis, George Em SIAM Journal on Scientific Computing, Vol. 41, Issue 4 https://doi.org/10.1137/18M1229845	journal	January 2019
What is the fractional Laplacian? A comparative review with new results Lischke, Anna; Pang, Guofei; Gulian, Mamikon Journal of Computational Physics, Vol. 404 https://doi.org/10.1016/j.jcp.2019.109009	journal	March 2020
Error Estimates for the Optimal Control of a Parabolic Fractional PDE Glusa, Christian; Otárola, Enrique SIAM Journal on Numerical Analysis, Vol. 59, Issue 2 https://doi.org/10.1137/19M1267581	journal	January 2021
Bilevel Parameter Learning for Nonlocal Image Denoising Models D’Elia, M.; De Los Reyes, J. C.; Miniguano-Trujillo, A. Journal of Mathematical Imaging and Vision, Vol. 63, Issue 6 https://doi.org/10.1007/s10851-021-01026-2	journal	April 2021
On Fractional Laplacians Musina, Roberta; Nazarov, Alexander I. Communications in Partial Differential Equations, Vol. 39, Issue 9 https://doi.org/10.1080/03605302.2013.864304	journal	August 2014
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
Is Gauss Quadrature Better than Clenshaw–Curtis? Trefethen, Lloyd N. SIAM Review, Vol. 50, Issue 1 https://doi.org/10.1137/060659831	journal	January 2008
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
Machine Learning of Space-Fractional Differential Equations Gulian, Mamikon; Raissi, Maziar; Perdikaris, Paris SIAM Journal on Scientific Computing, Vol. 41, Issue 4 https://doi.org/10.1137/18M1204991	journal	January 2019
A New Type of Identification Problems: Optimizing the Fractional Order in a Nonlocal Evolution Equation Sprekels, Jürgen; Valdinoci, Enrico SIAM Journal on Control and Optimization, Vol. 55, Issue 1 https://doi.org/10.1137/16M105575X	journal	January 2017
A Physically Consistent, Flexible, and Efficient Strategy to Convert Local Boundary Conditions into Nonlocal Volume Constraints D'Elia, Marta; Tian, Xiaochuan; Yu, Yue SIAM Journal on Scientific Computing, Vol. 42, Issue 4 https://doi.org/10.1137/19M1266617	journal	January 2020
Discovering variable fractional orders of advection–dispersion equations from field data using multi-fidelity Bayesian optimization Pang, Guofei; Perdikaris, Paris; Cai, Wei Journal of Computational Physics, Vol. 348 https://doi.org/10.1016/j.jcp.2017.07.052	journal	November 2017
Stabilized linear semi-implicit schemes for the nonlocal Cahn–Hilliard equation Du, Qiang; Ju, Lili; Li, Xiao Journal of Computational Physics, Vol. 363 https://doi.org/10.1016/j.jcp.2018.02.023	journal	June 2018
Perspective on theories of non-Fickian transport in heterogeneous media Neuman, Shlomo P.; Tartakovsky, Daniel M. Advances in Water Resources, Vol. 32, Issue 5 https://doi.org/10.1016/j.advwatres.2008.08.005	journal	May 2009
A spectral Galerkin approximation of optimal control problem governed by fractional advection–diffusion–reaction equations Wang, Fangyuan; Zhang, Zhongqiang; Zhou, Zhaojie Journal of Computational and Applied Mathematics, Vol. 386 https://doi.org/10.1016/j.cam.2020.113233	journal	April 2021
First passage times for symmetric stable processes in space Getoor, R. K. Transactions of the American Mathematical Society, Vol. 101, Issue 1 https://doi.org/10.1090/S0002-9947-1961-0137148-5	journal	January 1961
The Dirichlet problem for nonlocal operators Felsinger, Matthieu; Kassmann, Moritz; Voigt, Paul Mathematische Zeitschrift, Vol. 279, Issue 3-4 https://doi.org/10.1007/s00209-014-1394-3	journal	November 2014
Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains Seleson, Pablo; Gunzburger, Max; Parks, Michael L. Computer Methods in Applied Mechanics and Engineering, Vol. 266 https://doi.org/10.1016/j.cma.2013.05.018	journal	November 2013
The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator D’Elia, Marta; Gunzburger, Max Computers & Mathematics with Applications, Vol. 66, Issue 7 https://doi.org/10.1016/j.camwa.2013.07.022	journal	October 2013
A review of applications of fractional advection–dispersion equations for anomalous solute transport in surface and subsurface water Sun, Liwei; Qiu, Han; Wu, Chuanhao WIREs Water, Vol. 7, Issue 4 https://doi.org/10.1002/wat2.1448	journal	May 2020
Regularity analyses and approximation of nonlocal variational equality and inequality problems Burkovska, O.; Gunzburger, M. Journal of Mathematical Analysis and Applications, Vol. 478, Issue 2 https://doi.org/10.1016/j.jmaa.2019.05.064	journal	October 2019
A FEM for an optimal control problem subject to the fractional Laplace equation Dohr, Stefan; Kahle, Christian; Rogovs, Sergejs Calcolo, Vol. 56, Issue 4 https://doi.org/10.1007/s10092-019-0334-3	journal	September 2019
A meshfree method based on the peridynamic model of solid mechanics Silling, S. A.; Askari, E. Computers & Structures, Vol. 83, Issue 17-18 https://doi.org/10.1016/j.compstruc.2004.11.026	journal	June 2005
A direct O(Nlog2N) finite difference method for fractional diffusion equations Wang, Hong; Wang, Kaixin; Sircar, Treena Journal of Computational Physics, Vol. 229, Issue 21 https://doi.org/10.1016/j.jcp.2010.07.011	journal	October 2010
Optimizing the Fractional Power in a Model with Stochastic PDE Constraints Geldhauser, Carina; Valdinoci, Enrico Advanced Nonlinear Studies, Vol. 18, Issue 4 https://doi.org/10.1515/ans-2018-2031	journal	March 2018
Persistence of anomalous dispersion in uniform porous media demonstrated by pore-scale simulations: ANOMALOUS DISPERSION IN UNIFORM POROUS MEDIA Zhang, Xiaoxian; Lv, Mouchao Water Resources Research, Vol. 43, Issue 7 https://doi.org/10.1029/2006WR005557	journal	July 2007
A fast solver for spectral elements applied to fractional differential equations using hierarchical matrix approximation Li, Xianjuan; Mao, Zhiping; Wang, Nan Computer Methods in Applied Mechanics and Engineering, Vol. 366 https://doi.org/10.1016/j.cma.2020.113053	journal	July 2020
Eulerian derivation of the fractional advection–dispersion equation Schumer, Rina; Benson, David A.; Meerschaert, Mark M. Journal of Contaminant Hydrology, Vol. 48, Issue 1-2 https://doi.org/10.1016/S0169-7722(00)00170-4	journal	March 2001
An adaptive finite element method for the sparse optimal control of fractional diffusion Otárola, Enrique Numerical Methods for Partial Differential Equations, Vol. 36, Issue 2 https://doi.org/10.1002/num.22429	journal	October 2019
Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints Du, Qiang; Gunzburger, Max; Lehoucq, R. B. SIAM Review, Vol. 54, Issue 4 https://doi.org/10.1137/110833294	journal	January 2012
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
Measurement and analysis of non-Fickian dispersion in heterogeneous porous media Levy, Melissa; Berkowitz, Brian Journal of Contaminant Hydrology, Vol. 64, Issue 3-4 https://doi.org/10.1016/S0169-7722(02)00204-8	journal	July 2003
Asymptotically Compatible Fourier Spectral Approximations of Nonlocal Allen--Cahn Equations Du, Qiang; Yang, Jiang SIAM Journal on Numerical Analysis, Vol. 54, Issue 3 https://doi.org/10.1137/15M1039857	journal	January 2016
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
Optimization with Respect to Order in a Fractional Diffusion Model: Analysis, Approximation and Algorithmic Aspects Antil, Harbir; Otárola, Enrique; Salgado, Abner J. Journal of Scientific Computing, Vol. 77, Issue 1 https://doi.org/10.1007/s10915-018-0703-0	journal	March 2018
The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator D'Elia, Marta; Gunzburger, Max arXiv https://doi.org/10.48550/arxiv.1303.6934	preprint	January 2013
A new type of identification problems: optimizing the fractional order in a nonlocal evolution equation Sprekels, Jürgen; Valdinoci, Enrico arXiv https://doi.org/10.48550/arxiv.1601.00568	preprint	January 2016
Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver Ainsworth, Mark; Glusa, Christian arXiv https://doi.org/10.48550/arxiv.1708.03912	text	January 2017
A priori error estimates for the optimal control of the integral fractional Laplacian D'Elia, Marta; Glusa, Christian; Otarola, Enrique arXiv https://doi.org/10.48550/arxiv.1810.04262	text	January 2018
Numerical methods for nonlocal and fractional models D'Elia, Marta; Du, Qiang; Glusa, Christian arXiv https://doi.org/10.48550/arxiv.2002.01401	text	January 2020
On a nonlocal Cahn-Hilliard model permitting sharp interfaces Burkovska, Olena; Gunzburger, Max arXiv https://doi.org/10.48550/arxiv.2004.14379	text	January 2020

Cited By (2)

Data-driven learning of nonlocal models: from high-fidelity simulations to constitutive laws. D'Elia, Marta; Silling, Stewart; You, Huaiqian Proposed for presentation at the Workshop on Mathematical Machine Learning and Application held December 14-16, 2020 in Virtual. https://doi.org/10.2172/1835229	conference	December 2020
Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anomalous Transport D'Elia, Marta; Gulian, Mamikon arXiv https://doi.org/10.48550/arxiv.2101.04289	preprint	January 2021

Similar Records

An optimization-based approach to parameter learning for fractional type nonlocal models

Technical Report · Thu Oct 15 00:00:00 EDT 2020 · OSTI ID:1787538

Burkovska, Olena; Glusa, Christian Alexander; D'Elia, Marta

Bilevel parameter optimization for learning nonlocal image denoising models

Technical Report · Wed May 06 00:00:00 EDT 2020 · OSTI ID:1787538

D'Elia, Marta; De los Reyes, Juan Carlos; Trujillo, Andres

Affine Approximation of Parametrized Kernels and Model Order Reduction for Nonlocal and Fractional Laplace Models

Journal Article · Thu May 14 00:00:00 EDT 2020 · SIAM Journal on Numerical Analysis · OSTI ID:1787538

Burkovska, Olena; Gunzburger, Max

Related Subjects

97 MATHEMATICS AND COMPUTING
Identification problem
Optimal control
Nonlocal diffusion
Fractional Laplacian
Affine approximation

Title: An optimization-based approach to parameter learning for fractional type nonlocal models

Citation Formats

References (58)

Cited By (2)

Similar Records

Related Subjects