A fast and scalable method for A-optimal design of experiments for infinite-dimensional Bayesian nonlinear inverse problems

We address the problem of optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs). The inverse problem seeks to infer an infinite-dimensional parameter from experimental data observed at a set of sensor locations and from the governing PDEs. The goal of the OED problem is to find an optimal placement of sensors so as to minimize the uncertainty in the inferred parameter field. Specifically, we seek an optimal subset of sensors from among a fixed set of candidate sensor locations. We formulate the OED objective function by generalizing the classical A-optimal experimental design criterion using the expected value of the trace of the posterior covariance. This expected value is computed through sample averaging over the set of likely experimental data. To cope with the infinite-dimensional character of the parameter field, we construct a Gaussian approximation to the posterior at the maximum a posteriori probability (MAP) point, and use the resulting covariance operator to define the OED objective function. We use randomized trace estimation to compute the trace of this covariance operator, which is defined only implicitly. The resulting OED problem includes as constraints the system of PDEs characterizing the MAP point, and the PDEs describing the action of the covariance (of the Gaussian approximation to the posterior) to vectors. We control the sparsity of the sensor configurations using sparsifying penalty functions. Variational adjoint methods are used to efficiently compute the gradient of the PDE-constrained OED objective function. We elaborate our OED method for the problem of determining the optimal sensor configuration to best infer the coefficient of an elliptic PDE. Furthermore, we provide numerical results for inference of the log permeability field in a porous medium flow problem. Numerical results show that the number of PDE solves required for the evaluation of the OED objective function and its gradient is essentially independent of both the parameter dimension and the sensor dimension (i.e., the number of candidate sensor locations). The number of quasi-Newton iterations for computing an OED also exhibits the same dimension invariance properties.