Joint Online Parameter Estimation and Optimal Sensor Placement for the Partially Observed Stochastic Advection-Diffusion Equation

TL;DR

This paper introduces a continuous-time stochastic gradient descent method for online parameter estimation and sensor placement in a partially observed stochastic advection-diffusion system, demonstrating effective numerical results.

Contribution

It proposes a novel two-timescale stochastic gradient algorithm for joint online parameter estimation and sensor placement in infinite-dimensional diffusion processes.

Findings

Algorithm effectively estimates parameters in real-time.

Sensor placement optimizes state estimation accuracy.

Numerical results validate approach on 2D stochastic advection-diffusion.

Abstract

In this paper, we consider the problem of jointly performing online parameter estimation and optimal sensor placement for a partially observed infinite dimensional linear diffusion process. We present a novel solution to this problem in the form of a continuous-time, two-timescale stochastic gradient descent algorithm, which recursively seeks to maximise the log-likelihood with respect to the unknown model parameters, and to minimise the expected mean squared error of the hidden state estimate with respect to the sensor locations. We also provide extensive numerical results illustrating the performance of the proposed approach in the case that the hidden signal is governed by the two-dimensional stochastic advection-diffusion equation.