Abstract

Shannon-type expected information gain can be used to evaluate the
relevance of a proposed experiment subjected to uncertainty. The
estimation of such gain, however, relies on a double-loop integration.
Moreover, its numerical integration in multidimensional cases, e.g.,
when using Monte Carlo sampling methods, is therefore computationally
too expensive for realistic physical models, especially for those
involving the solution of partial differential equations. In this work,
we present a new methodology, based on the Laplace approximation for the
integration of the posterior probability density function (pdf), to
accelerate the estimation of the expected information gains in the model
parameters and predictive quantities of interest. We obtain a
closed-form approximation of the inner integral and the corresponding
dominant error term in the cases where parameters are determined by the
experiment, such that only a single-loop integration is needed to carry
out the estimation of the expected information gain. To deal with the
issue of dimensionality in a complex problem, we use a sparse quadrature
for the integration over the prior pdf. We demonstrate the accuracy,
efficiency and robustness of the proposed method via several nonlinear
numerical examples, including the designs of the scalar parameter in an
one-dimensional cubic polynomial function, the design of the same scalar
in a modified function with two indistinguishable parameters, the
resolution width and measurement time for a blurred single peak
spectrum, and the boundary source locations for impedance tomography in a square domain.