State Estimation -- The Role of Reduced Models

TL;DR

This paper reviews recent advances in state estimation for complex systems using reduced models, emphasizing deterministic approaches over Bayesian methods, and highlights the importance of model adaptation to sensor configurations.

Contribution

It introduces the concept of adapting reduced models to sensor systems for efficient state estimation in small-data scenarios, moving beyond traditional Bayesian inversion.

Findings

Reduced models enhance computational efficiency in state estimation.

Model adaptation to sensor configurations improves accuracy.

Deterministic methods offer viable alternatives to Bayesian inversion.

Abstract

The exploration of complex physical or technological processes usually requires exploiting available information from different sources: (i) physical laws often represented as a family of parameter dependent partial differential equations and (ii) data provided by measurement devices or sensors. The amount of sensors is typically limited and data acquisition may be expensive and in some cases even harmful. This article reviews some recent developments for this "small-data" scenario where inversion is strongly aggravated by the typically large parametric dimensionality. The proposed concepts may be viewed as exploring alternatives to Bayesian inversion in favor of more deterministic accuracy quantification related to the required computational complexity. We discuss optimality criteria which delineate intrinsic information limits, and highlight the role of reduced models for developing efficient computational strategies. In particular, the need to adapt the reduced models -- not to a specific (possibly noisy) data set but rather to the sensor system -- is a central theme. This, in turn, is facilitated by exploiting geometric perspectives based on proper stable variational formulations of the continuous model.