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

Abstract

We present a method for computing A-optimal sensor placements for infinitedimensional Bayesian linear inverse problems governed by PDEs with irreducible model uncertainties. Here, irreducible uncertainties refers to uncertainties in the model that exist in addition to the parameters in the inverse problem, and that cannot be reduced through observations. Specifically, given a statistical distribution for the model uncertainties, we compute the optimal design that minimizes the expected value of the posterior covariance trace. The expected value is discretized using Monte Carlo leading to an objective function consisting of a sum of trace operators and a binary-inducing penalty. Minimization of this objective requires a large number of PDE solves in each step. To make this problem computationally tractable, we construct a composite low-rank basis using a randomized range finder algorithm to eliminate forward and adjoint PDE solves. We also present a novel formulation of the A-optimal design objective that requires the trace of an operator in the observation rather than the parameter space. The binary structure is enforced using a weighted regularized 0 -sparsification approach. We present numerical results for inference of the initial condition in a subsurface flow problem with inherent uncertainty in the flow fields and in the initial times.

Joint Estimation of Robin Coefficient and Domain Boundary for the Poisson Problem

“…In the context of Bayesian inference for problems governed by partial differential equations (PDE), the most common Gaussian approximation to the posterior is the so-called Laplace approximation, see e.g. [7,48,59]. Assuming the parameter-to-observable mapping is Fréchet differentiable with respect to α and β, the Laplace approximation is πpost (α, β|y) = N ((α MAP , β MAP ), Γpost ), where (α MAP , β MAP ) denotes the maximum a posteriori (MAP) estimate of (α, β),…”

Optimal Experimental Design for Infinite-dimensional Bayesian Inverse Problems Governed by PDEs: A Review

“…In this example, we consider flow of a contaminant in a geological formation. The present example is adapted from [47]. Focusing on a horizontal cross-section of the medium, we consider a two dimensional domain D = [0, L 1 ] × [0, L 2 ].…”

“…The homogeneous pure Neumann condition on the rest of the boundary allows for advective flux. See [47], for more details on this model problem.…”

“…see [47]. Since the prior covariance operator does not depend on w, one can find an Aoptimal design by minimizing the trace of the second term in the right hand side of the above equation.…”

Learning physics-based models from data: perspectives from inverse problems and model reduction

“…, Attia, Alexanderian and Saibaba 2018, Feng and Marzouk 2019, Koval, Alexanderian and Stadler 2020, Wu et al 2020, Alexanderian 2020, Herman, Alexanderian and Saibaba 2020, Jagalur-Mohan and Marzouk 2020, Wu, Chen and Ghattas 2021, Alexanderian, Petra, Stadler and Sunseri 2021.…”