Goal-Oriented Optimal Design of Experiments for Large-Scale Bayesian Linear Inverse Problems
For practitioners solving PDE-constrained inverse problems where the parameter is an intermediate step, GOODE offers a more efficient and targeted experimental design approach.
The paper introduces a goal-oriented optimal design of experiments (GOODE) framework for large-scale Bayesian linear inverse problems, which minimizes posterior uncertainty in a quantity of interest rather than the parameter itself. Numerical experiments on contaminant source identification show that GOODE reduces computational cost and avoids wasted resources compared to classical ODE.
We develop a framework for goal-oriented optimal design of experiments (GOODE) for large-scale Bayesian linear inverse problems governed by PDEs. This framework differs from classical Bayesian optimal design of experiments (ODE) in the following sense: we seek experimental designs that minimize the posterior uncertainty in the experiment end-goal, e.g., a quantity of interest (QoI), rather than the estimated parameter itself. This is suitable for scenarios in which the solution of an inverse problem is an intermediate step and the estimated parameter is then used to compute a QoI. In such problems, a GOODE approach has two benefits: the designs can avoid wastage of experimental resources by a targeted collection of data, and the resulting design criteria are computationally easier to evaluate due to the often low-dimensionality of the QoIs. We present two modified design criteria, A-GOODE and D-GOODE, which are natural analogues of classical Bayesian A- and D-optimal criteria. We analyze the connections to other ODE criteria, and provide interpretations for the GOODE criteria by using tools from information theory. Then, we develop an efficient gradient-based optimization framework for solving the GOODE optimization problems. Additionally, we present comprehensive numerical experiments testing the various aspects of the presented approach. The driving application is the optimal placement of sensors to identify the source of contaminants in a diffusion and transport problem. We enforce sparsity of the sensor placements using an $\ell_1$-norm penalty approach, and propose a practical strategy for specifying the associated penalty parameter.