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.