Multidimensional sum-up rounding for integer programming in optimal experimental design
Here we present a numerical method for approximating the solution of convex integer programs stemming from optimal experimental design. The statistical setup consists of a Bayesian framework for linear inverse problems for which the direct relationship is described by a discretized integral equation. Specifically, we aim to find the optimal sensor placement from a set of candidate locations where data are collected with measurement error. The convex objective function is a measure of the uncertainty, described here by the trace or log-determinant of the posterior covariance matrix, for the discretized linear inverse problem solution. The resulting convex integer program is relaxed, producing a lower bound. An upper bound is obtained by extending the sum-up rounding approach to multiple dimensions. For this extension, we analyze its accuracy as a function of the discretization mesh size for a rectangular domain. We show asymptotic optimality of the integer solution defining the upper bound for different experimental design criteria (A- and D-optimal), by proving the convergence to zero of the gap between the upper and lower bounds as the mesh size goes to zero. The technique is illustrated on a two-dimensional gravity surveying problem for both A-optimal and D-optimal sensor placement where our designs yield better results compared with a thresholding rounding approach.
- Research Organization:
- Argonne National Laboratory (ANL), Argonne, IL (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)
- Grant/Contract Number:
- AC02-06CH11347; FP061151-01-PR; CNS-1545046
- OSTI ID:
- 1808030
- Journal Information:
- Mathematical Programming, Vol. 185, Issue 1-2; ISSN 0025-5610
- Publisher:
- SpringerCopyright Statement
- Country of Publication:
- United States
- Language:
- English
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