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Volume 29, Issue 4
Continuous Data Assimilation with a Moving Cluster of Data Points for a Reaction Diffusion Equation: A Computational Study

Adam Larios & Collin Victor

DOI: 10.4208/cicp.OA-2018-0315

Commun. Comput. Phys., 29 (2021), pp. 1273-1298.

Published online: 2021-02

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  • 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.

  • Keywords

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

  • AMS Subject Headings

35K57, 35K40, 35K61, 37C50, 35Q93, 34D06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-29-1273, author = {Larios , Adam and Victor , Collin}, title = {Continuous Data Assimilation with a Moving Cluster of Data Points for a Reaction Diffusion Equation: A Computational Study}, journal = {Communications in Computational Physics}, year = {2021}, volume = {29}, number = {4}, pages = {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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0315}, url = {http://global-sci.org/intro/article_detail/cicp/18652.html} }
TY - JOUR T1 - Continuous Data Assimilation with a Moving Cluster of Data Points for a Reaction Diffusion Equation: A Computational Study AU - Larios , Adam AU - Victor , Collin JO - Communications in Computational Physics VL - 4 SP - 1273 EP - 1298 PY - 2021 DA - 2021/02 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2018-0315 UR - https://global-sci.org/intro/article_detail/cicp/18652.html KW - Continuous data assimilation, Allen-Cahn, reaction-diffusion, moving mesh, synchronization. AB -

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.

Adam Larios & Collin Victor. (2021). Continuous Data Assimilation with a Moving Cluster of Data Points for a Reaction Diffusion Equation: A Computational Study. Communications in Computational Physics. 29 (4). 1273-1298. doi:10.4208/cicp.OA-2018-0315
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