(587b) A Scalable Design of Experiments Framework for Infinite-Dimensional Systems

The sensor placement problem seeks to determine the optimal number, location, and types of sensors that would maximize information about a dynamical system. Information is often expressed in terms of the posterior covariance matrix of the states of parameters of the system. The problem can thus be cast as an optimal design of the experiments problem. Such a problem is computationally challenging, particularly in the infinite-dimensional case, because one must solve a mixed-integer and bilevel optimization problem constrained by differential algebraic equations (DAEs) or by partial differential equations (PDEs).

The sensor placement problem has been addressed by using mixed-integer programming techniques in the context of contaminant detection in water networks. In these studies, an optimal set of sensor locations is selected from a set of candidate locations to minimize a certain engineering metric such as contaminant detection time, population exposea, or likelihood of detection. Likelihoods are assigned based on contamination scenarios and not on information content of the sensor data recorded, as in a traditional experimental design setting. Consequently, these approaches fail to provide statistically meaningful sensor network designs. Moreover, because the formulations capture flow dynamics by using surrogate representations such as transportation delays, they fail to capture nonlinear effects.

Sensor placement problems also have been addressed in a more general control setting where one weeks a measure of observability such as the covariance matrix, Kalman estimator gain, or so-called observability Grammian matrix. This problem again is a bilevel optimization problem. The covariance matrix approach in [1] bypasses this by assuming that the dynamic model is linear, thus allowing the inner minimization problem to be formulated as a linear matrix inequality. The approach in [2] models the dynamics of the covariance matrix directly as a Ricatti differential equation, which implicitly assumes linearity and thus enables the use of semidefinite programming algorithms. However, the work in [2] is focused on control policy design to extract maximum information, and not on sensor placement design. Consequently, the authors do not consider mixed-integer formulations. A rigorous treatment of nonlinear dynamics is presented in [3] by casting the problem as a mixed-integer nonlinear programming. The authors, however, use a genetic algorithm to deal with the inner minimization problem that computes the observability metric. A similar approach is used in [4] to address the inner minimization problem. Mixed-integer techniques have also been used in the context of information maximization for Gaussian processes and for designing Latin hypercube samples [5]. These approaches do not capture physical models.

Recently, the sensor placement problem for systems described by PDEs has been cast as an A-experimental design problem in which the number of sensors (i.e., the design cost) is controlled by using an l0 regularization norm that is in turn approximated using a smoothing function [6]. This approach was shown to be scalable and applicable to infinite-dimensional systems, but it requires tuning and can be numerically unstable. One can also formulate and solve the problem as a mixed-integer programming problem directly, but this approach can become computationally intractable because the PDEs are in general nonconvex and because the problem is bilevel in nature.

In this work we present a scalable design of an experiments framework for sensor placement that seeks to compute optimal sensor locations where observational data are collected, by minimizing the uncertainty in parameters estimated from Bayesian inverse problems, which are governed by PDEs. The resulting problem is a mixed- integer infinite-dimensional optimal control problem. We approach this problem using two efficient heuristics that have a potential to be scalable for such problems: a sparsity-inducing approach [6] and a sum-up rounding approach [7]. We investigate two objectives: the total flow variance and the A-optimal design criterion. We conclude that the sum-up rounding approach produces shrinking gaps with increased meshes. We also observe that convergence for the white noise measurement error is slower than for the colored noise case. For the A-optimal design, the solution is close to the uniform distribution, but for the total flow variance the pattern is noticeably different.

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