Optimal Experimental Design of a Moving Sensor for Linear Bayesian Inverse Problems

We optimize the path of a mobile sensor to minimize the posterior uncertainty of a Bayesian inverse problem. Along its path, the sensor continuously takes measurements of the state, which is a physical quantity modeled as the solution of a partial differential equation (PDE) with uncertain parameters. Considering linear PDEs specifically, we derive the closed-form expression of the posterior covariance matrix of the model parameters as a function of the path, and formulate the optimal experimental design problem for minimizing the posterior's uncertainty. We discretize the problem such that the cost function remains consistent under temporal refinement. Additional constraints ensure that the path avoids obstacles and remains physically interpretable through a control parameterization. The constrained optimization problem is solved using an interior-point method. We present computational results for a convection-diffusion equation with unknown initial condition.