Data assimilation via model reference adaptation for linear and nonlinear dynamical systems

TL;DR

This paper introduces a practical model reference adaptive system approach for data assimilation in linear and nonlinear dynamical systems, demonstrating its effectiveness through four complex benchmark problems.

Contribution

It provides the first practical implementation of model reference adaptive systems for online parameter identification in dynamical systems.

Findings

Successful application to Darcy flow, Fisher-KPP, nonlinear potential, and Allen-Cahn equations.

Verified assumptions and demonstrated numerical stability.

Proven versatility and reliability for data assimilation and real-time inversion.

Abstract

We address data assimilation for linear and nonlinear dynamical systems via the so-called \emph{model reference adaptive system}. Continuing our theoretical developments in \cite{Tram_Kaltenbacher_2021}, we deliver the first practical implementation of this approach for online parameter identification with time series data. Our semi-implicit scheme couples a modified state equation with a parameter evolution law that is driven by model-data residuals. We demonstrate four benchmark problems of increasing complexity: the Darcy flow, the Fisher-KPP equation, a nonlinear potential equation and finally, an Allen-Cahn type equation. Across all cases, explicit model reference adaptive system construction, verified assumptions and numerically stable reconstructions underline our proposed method as a reliable, versatile tool for data assimilation and real-time inversion.