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.

翻译：在本文中,我们考虑了对部分观测到的无限线性扩散过程联合进行在线参数估计和最佳传感器定位的问题。我们以连续时间、两度尺度的随机梯度梯度下降算法的形式提出了解决这一问题的新办法,这种算法反复寻求在未知模型参数方面最大限度地实现日志相似性,并尽可能减少传感器位置的隐藏状态估计的预期平均正方形错误。我们还提供了大量的数字结果,说明在隐藏信号受二维蒸汽蒸发方程式制约的情况下,拟议方法的性能。