Title: A Fast and Scalable Method for A-Optimal Design of
Experiments for Infinite-dimensional Bayesian Nonlinear Inverse
Problems
Abstract:
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
a parameter field (e.g., the log permeability field in a porous
medium flow model problem) from synthetic observations 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. 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. Due to the infinite-dimensional
character of the parameter field, we seek an optimization method
that solves the OED problem at a cost (measured in the number of
forward PDE solves) that is independent of both the parameter and
the sensor dimension.
To facilitate this goal, 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. 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, and solve the resulting
penalized bilevel optimization problem via an interior-point
quasi-Newton method, where gradient information is computed via
adjoints. We elaborate our OED method for the problem of
determining the optimal sensor configuration to best infer 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.
(This is joint work with Alen Alexanderian (NC State), Georg
Stadler (NYU) and Omar Ghattas (UT Austin).
Reference: http://arxiv.org/pdf/1410.5899.pdf).
Bio:
Noemi Petra is an assistant professor in the Applied Mathematics
department in the School of Natural Sciences at the University of
California, Merced. She earned her Ph.D. degree in Applied Mathe-
matics from the University of Maryland, Baltimore County in
2010. Prior to joining the University of California, Merced,
Noemi was the recipient of the ICES Postdoctoral Fellowship
during 2010- 2011 in the Institute for Computational Engineering
and Sciences (ICES) at The University of Texas at Austin. Her
research interests include PDE- and DAE-constrained optimization,
inverse problems, uncertainty quantification and optimal
experimental design.