Efficient D-Optimal Design of Experiments for Infinite-Dimensional Bayesian Linear Inverse Problems

Abstract

We develop a computational framework for D-optimal experimental design for PDEbased Bayesian linear inverse problems with infinite-dimensional parameters. We follow a formulation of the experimental design problem that remains valid in the infinite-dimensional limit. The optimal design is obtained by solving an optimization problem that involves repeated evaluation of the logdeterminant of high-dimensional operators along with their derivatives. Forming and manipulating these operators is computationally prohibitive for large-scale problems. Our methods exploit the lowrank structure in the inverse problem in three different ways, yielding efficient algorithms. Our main approach is to use randomized estimators for computing the D-optimal criterion, its derivative, as well as the Kullback-Leibler divergence from posterior to prior. Two other alternatives are proposed based on low-rank approximation of the prior-preconditioned data misfit Hessian, and a fixed low-rank approximation of the prior-preconditioned forward operator. Detailed error analysis is provided for each of the methods, and their effectiveness is demonstrated on a model sensor placement problem for initial state reconstruction in a time-dependent advection-diffusion equation in two space dimensions.we seek sensor placements that maximize the expected information gain in parameter inversion. Our approach, however, is general in that it is applicable to D-optimal experimental design for a broad class of linear inverse problems. ]. In particular, A-optimal experimental design for large-scale applications has been addressed in [3, 4, 17, 20-22, 24, 43]. Computing Aoptimal designs for large-scale applications faces similar challenges to the D-optimal designs. The use of Monte-Carlo trace estimators [7] has been instrumental in making A-optimal design computationally feasible for such applications.Fast algorithms for computing expected information gain, i.e., the D-optimal criterion, for nonlinear inverse problems, via Laplace approximations, were proposed in [31,32]. The focus of the aforementioned works was efficient computation of the D-optimal criterion. The optimal design was then computed by an exhaustive search over prespecified sets of experimental scenarios. We also mention [26-28] that use polynomial chaos (PC) expansions to build easy-to-evaluate surrogates of the forward model. The expected information gain is then evaluated using an appropriate Monte Carlo procedure. In [27,28], the authors use a gradient based approach to obtain the optimal design. In [44], the authors propose an alternate design criterion given by a lower bound of the expected information gain, and use PC representation of the forward model to accelerate objective function evaluations. However, the approaches based on PC representations remain limited in scope to problems with low to moderate parameter dimensions (e.g., parameter dimensions in order of tens).Efficient estimators for the evaluation of D-optimal criterion were developed in [38,39]; however, these works do not...

Optimal experimental design under irreducible uncertainty for linear inverse problems governed by PDEs

“…We build on previous work [3,13,15], which developed efficient methods for computing A-optimal designs for high-or infinite-dimensional linear inverse problems using either a Bayesian or a frequentist approach. Other approaches for computing optimal experimental designs for high/infinite-dimensional linear inverse problems are explored, for example, in [2,5,22]. In [22], the authors propose a measure-based OED formulation that does not choose sensor locations from a finite number of candidate locations but allows sensors to be placed anywhere on a closed subset of the domain.…”

“…In [22], the authors propose a measure-based OED formulation that does not choose sensor locations from a finite number of candidate locations but allows sensors to be placed anywhere on a closed subset of the domain. The articles [2,5] explore an alternate OED criterion-D-optimality-for infinite-dimensional Bayesian linear inverse problems.…”

Efficient Krylov subspace methods for uncertainty quantification in large Bayesian linear inverse problems

“…The D‐optimal criterion for optimal experimental design is related to the expected KL divergence, where a precise connection was derived in (Reference 28, theorem 1). In D‐optimal experimental design, the objective function is to maximize Similar to the KL divergence, we can estimate the D‐optimal criterion using the genGK bidiagonalization as From the proof of Theorem 3, it can be readily seen that a bound for the error in approximating is given by …”

Logical Inference Framework for Experimental Design of Mechanical Characterization Procedures

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