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Meeting-ID: 280 519 2121
Passwort: 584190

Abstract:

Spatiotemporal data occur in many fields such as air pollution or groundwater flow monitoring, and meteorology. Their collection is inevitably related to discrete spatial and temporal sampling of an inherently continuous system. This raises the question of how to locate a limited number of measurement sites so as the amount of information about the observed system be as high as possible. This is of special importance in parameter estimation of systems modelled by partial differential equations. A distinguishing feature of environmental data collection is the presence of correlations in the measurements from different sites and/or time instants. This is because deviations in the observed responses at different sites may be brought about by the same sources, e.g., weather fluctuations on the scale of the whole spatial region. But then the Fisher information matrix is no longer the sum of elemental information matrices stemming from single sites. As a result, powerful convex optimum experimental design theory cannot be directly applied and various heuristic approaches dominate construction of approximations to optimum designs. The aim of this talk is to demonstrate that, in spite of these difficulties, convex relaxed formulations can be invented, for which extremely efficient computational procedures can be set up fully exploiting the power of modern convex optimization algorithms.  

In the first part of the talk, the trace of the covariance matrix of the weighted least-squares estimator is employed as the measure of the estimation accuracy. The pivotal role in the new convex relaxation proposed here is played by the decomposition of the noise into uncorrelated and correlated components. Necessary and sufficient optimality conditions are then formulated and optimal solutions are found via simplicial decomposition which alternates between updating the design weights using the well-known multiplicative algorithm and computing a closed-form solution to a linear programming problem.  

In the second part of the talk, the setting is considered when the exact correlation structure may not be known exactly, so that the ordinary least squares method is supposed to be used for estimation and the determinant of the covariance matrix of the resulting estimator is the measure of estimation accuracy. This time, the relaxed formulation turns out to be non-convex, but this is overcome by application of the majorization-minimization algorithm. At each of its iterations, a convex tangent surrogate function that majorizes the original nonconvex design criterion is minimized using simplicial decomposition.  

As the resulting relaxed solution in both the cases are measures on the set of candidate sites and not specific subsets of selected sensors, various sequential conversions to a nearly optimal subset of selected sensors are discussed. Simulation experiments are also reported to demonstrate that the proposed approaches are highly competitive with the traditional approaches.