Observability and Effective Region of Partial Differential Equations with Application to Data Assimilation

Abstract

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page C249-C271, June 2024.
Abstract. In this work, we introduce a new definition of observability for dynamical systems, formulated on the principles of dynamic optimization. This definition gives rise to the concept of an effective region, specifically designed for partial differential equations (PDEs). The usefulness of these concepts is demonstrated through examples of state estimation using observational information for PDEs in a limited area. The findings empower a more efficient analysis of PDE observability. By confining computations to an effective region significantly smaller than the overall region in which the PDE is defined, we demonstrate a substantial reduction in computational demand of evaluating observability. As an application of observability and effective region, we propose a learning-based surrogate data assimilation (DA) model for efficient state estimation in a limited area. Our model employs a feedforward neural network for online computation, eliminating the need for integrating high-dimensional limited-area models. This approach offers significant computational advantages over traditional DA algorithms. Furthermore, our method avoids the requirement of lateral boundary conditions for the limited-area model in both online and offline computations.