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

Journal Article · · Computers and Mathematics with Applications (Oxford)
 [1];  [2];  [3]
  1. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
  2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  3. Sandia National Lab. (SNL-CA), Livermore, CA (United States)
+ Show Author Affiliations
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 study 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.
Research Organization:
Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States); Sandia National Laboratories (SNL-NM), Albuquerque, NM (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: 1960986
OSTI ID: 1885396
Report Number(s):
SAND--2021-6450J; 696581
Journal Information:
Computers and Mathematics with Applications (Oxford), Journal Name: Computers and Mathematics with Applications (Oxford) Vol. 116; ISSN 0898-1221
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English

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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 preprint January 2021

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