Uncertainty quantification with experimental data and complex system models

Abstract

This dissertation discusses uncertainty quantication as posed in the Data Collaboration framework. Data Collaboration is a methodology for combining experimental data and system models to induce constraints on a set of uncertain system parameters. The framework is summarized, including outlines of notation and techniques. The main techniques include polynomial optimization and surrogate modeling to ascertain the consistency of all data and models as well as propagate uncertainty in the form of a model prediction. One of the main methods of Data Collaboration is using techniques of nonconvex quadratically constrained quadratic programming to provide both lower and upper bounds on the various objectives. The Lagrangian dual of the NQCQP provides both an outer bound to the optimal objective as well as Lagrange multipliers. These multipliers act as sensitivity measures relaying the effects of changes to the parameter constraint bounds on the optimal objective. These multipliers are rewritten to provide the sensitivity to uncertainty in the response prediction with respect to uncertainty in the parameters and experimental data. It is often of interest to find a vector of parameters that is both feasible and representative of the current community work and knowledge. This is posed as the problem of finding the minimal number of parameters that must deviate from their literature value to achieve concurrence with all experimental data constraints. This problem is heuristically solved using the l1-norm in place of the cardinality function. A lower bound on the objective is provided through an NQCQP formulation. In order to use the NQCQP techniques, the system models need to have quadratic forms. When they do not have quadratic forms, surrogate models are fitted. Surrogate modeling can be difficult for complex models with large numbers of parameters and long simulation times because of the amount of evaluation-time required to make a good fit. New techniques are developed for searching for an active subspace of the parameters, and subsequently creating an experiment design on the active subspace that adheres to the original parameter constraints. The active subspace can have a dimension signicantly lower than the original parameter dimension thereby reducing the computational complexity of generating the surrogate model. The technique is demonstrated on several examples from combustion chemistry and biology. Several other applications of the Data Collaboration framework are presented. They are used to demonstrate the complexity of describing a high dimensional feasible set of parameter values as constrained by experimental data. Approximating the feasible set can lead to a simple description, but the predictive capability of such a set is significantly reduced compared to the actual feasible set. This is demonstrated on an example from combustion chemistry.

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