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Continuous Data Assimilation with a Moving Cluster of Data Points for a Reaction Diffusion Equation: A Computational Study

Continuous Data Assimilation with a Moving Cluster of Data Points for a Reaction Diffusion Equation: A Computational Study
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Year:    2021

Author:    Adam Larios, Collin Victor

Communications in Computational Physics, Vol. 29 (2021), Iss. 4 : pp. 1273–1298

Abstract

Data assimilation is a technique for increasing the accuracy of simulations of solutions to partial differential equations by incorporating observable data into the solution as time evolves. Recently, a promising new algorithm for data assimilation based on feedback-control at the PDE level has been proposed in the pioneering work of Azouani, Olson, and Titi (2014). The standard version of this algorithm is based on measurement from data points that are fixed in space. In this work, we consider the scenario in which the data collection points move in space over time. We demonstrate computationally that, at least in the setting of the 1D Allen-Cahn reaction diffusion equation, the algorithm converges with significantly fewer measurement points, up to an order or magnitude in some cases. We also provide an application of the algorithm to the estimation of a physical length scale in the case of a uniform static grid.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2018-0315

Communications in Computational Physics, Vol. 29 (2021), Iss. 4 : pp. 1273–1298

Published online:    2021-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    26

Keywords:    Continuous data assimilation Allen-Cahn reaction-diffusion moving mesh synchronization.

Author Details

Adam Larios

Collin Victor

  1. Data Assimilation Using Time-Delay Nudging in the Presence of Gaussian Noise

    Celik, Emine | Olson, Eric

    Journal of Nonlinear Science, Vol. 33 (2023), Iss. 6

    https://doi.org/10.1007/s00332-023-09967-1 [Citations: 1]

  2. Data Assimilation for the Navier--Stokes Equations Using Local Observables

    Biswas, Animikh | Bradshaw, Zachary | Jolly, Michael S.

    SIAM Journal on Applied Dynamical Systems, Vol. 20 (2021), Iss. 4 P.2174

    https://doi.org/10.1137/20M136058X [Citations: 12]

  3. Super-Exponential Convergence Rate of a Nonlinear Continuous Data Assimilation Algorithm: The 2D Navier–Stokes Equation Paradigm

    Carlson, Elizabeth | Larios, Adam | Titi, Edriss S.

    Journal of Nonlinear Science, Vol. 34 (2024), Iss. 2

    https://doi.org/10.1007/s00332-024-10014-w [Citations: 3]

  4. The bleeps, the sweeps, and the creeps: Convergence rates for dynamic observer patterns via data assimilation for the 2D Navier–Stokes equations

    Franz, Trenton | Larios, Adam | Victor, Collin

    Computer Methods in Applied Mechanics and Engineering, Vol. 392 (2022), Iss. P.114673

    https://doi.org/10.1016/j.cma.2022.114673 [Citations: 9]

  5. Continuous data assimilation for the 3D and higher-dimensional Navier–Stokes equations with higher-order fractional diffusion

    Larios, Adam | Victor, Collin

    Journal of Mathematical Analysis and Applications, Vol. 540 (2024), Iss. 1 P.128644

    https://doi.org/10.1016/j.jmaa.2024.128644 [Citations: 1]

  6. Identifying the body force from partial observations of a two-dimensional incompressible velocity field

    Farhat, Aseel | Larios, Adam | Martinez, Vincent R. | Whitehead, Jared P.

    Physical Review Fluids, Vol. 9 (2024), Iss. 5

    https://doi.org/10.1103/PhysRevFluids.9.054602 [Citations: 2]

  7. Recovering critical parameter for nonlinear Allen–Cahn equation by fully discrete continuous data assimilation algorithms *

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    Inverse Problems, Vol. 40 (2024), Iss. 1 P.015008

    https://doi.org/10.1088/1361-6420/ad0e25 [Citations: 0]

  8. Analysis of continuous data assimilation with large (or even infinite) nudging parameters

    Diegel, Amanda E. | Li, Xuejian | Rebholz, Leo G.

    Journal of Computational and Applied Mathematics, Vol. 456 (2025), Iss. P.116221

    https://doi.org/10.1016/j.cam.2024.116221 [Citations: 1]

  9. Dynamically learning the parameters of a chaotic system using partial observations

    Carlson, Elizabeth | Hudson, Joshua | Larios, Adam | Martinez, Vincent R. | Ng, Eunice | Whitehead, Jared P.

    Discrete and Continuous Dynamical Systems, Vol. 42 (2022), Iss. 8 P.3809

    https://doi.org/10.3934/dcds.2022033 [Citations: 14]

  10. Sensitivity Analysis for the 2D Navier–Stokes Equations with Applications to Continuous Data Assimilation

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    Journal of Nonlinear Science, Vol. 31 (2021), Iss. 5

    https://doi.org/10.1007/s00332-021-09739-9 [Citations: 13]

  11. Determining the viscosity of the Navier–Stokes equations from observations of finitely many modes

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    Inverse Problems, Vol. 39 (2023), Iss. 12 P.125012

    https://doi.org/10.1088/1361-6420/ad065f [Citations: 3]