The Woodruff School

SUMMARY

Many engineering systems are challenging to model in simulation because of the difficulty of measuring elements of the environment or inherent randomness. One framework for modeling system uncertainty is through augmenting the state space to include state and parameter uncertainty in the model. This work utilizes the Koopman operator to establish computational methods for uncertainty quantification and decision-making for models under parameter and/or state uncertainty. The first application demonstrates the computational advantages of a Koopman operator approach to generating optimal control decisions for systems under uncertainty compared with Monte Carlo simulation and direct uncertainty propagation using the stochastic Liouville equation. The next application develops a computationally efficient approach to solving a probabilistic inverse problem subject to expected value targets or probabilistic constraints on the model. The constrained quadratic programming approach allows a set of integral equations to be solved simultaneously and generate an input probability distribution that satisfies the model constraints. The next application compares a Koopman operator approach for computing the statistical moments of propagated model uncertainty to the Polynomial Chaos framework and Monte Carlo simulation. The advantages of the Koopman approach are demonstrated through simulation examples that demonstrate the computational advantages over the other two approaches. Finally, an investigation into generating marginal distributions and estimating quantiles of propagated model uncertainty is to be completed. Generating marginal densities through the Koopman approach may offer computational advantages over Monte Carlo and kernel based methods because of the efficient expected value computation. Together, these novel applications of the Koopman operator to uncertainty propagation of models with state and/or parameter uncertainty provide an computationally efficient alternative method to existing methods of uncertainty propagation.

SUBJECT:	Ph.D. Proposal Presentation

BY:	Joseph Meyers

TIME:	Friday, January 14, 2022, 9:30 a.m.

PLACE:	n/a, Zoom

TITLE:	A Koopman Operator Approach to Uncertainty Quantification and Decision-Making

COMMITTEE:	Dr. Jonathan Rogers, Co-Chair (ME/AE) Dr. Aldo Ferri, Co-Chair (ME) Dr. Adam Gerlach (AFRL) Dr. Panagiotis Tsiotras (AE) Dr. Ye Zhao (ME)

SUMMARY Many engineering systems are challenging to model in simulation because of the difficulty of measuring elements of the environment or inherent randomness. One framework for modeling system uncertainty is through augmenting the state space to include state and parameter uncertainty in the model. This work utilizes the Koopman operator to establish computational methods for uncertainty quantification and decision-making for models under parameter and/or state uncertainty. The first application demonstrates the computational advantages of a Koopman operator approach to generating optimal control decisions for systems under uncertainty compared with Monte Carlo simulation and direct uncertainty propagation using the stochastic Liouville equation. The next application develops a computationally efficient approach to solving a probabilistic inverse problem subject to expected value targets or probabilistic constraints on the model. The constrained quadratic programming approach allows a set of integral equations to be solved simultaneously and generate an input probability distribution that satisfies the model constraints. The next application compares a Koopman operator approach for computing the statistical moments of propagated model uncertainty to the Polynomial Chaos framework and Monte Carlo simulation. The advantages of the Koopman approach are demonstrated through simulation examples that demonstrate the computational advantages over the other two approaches. Finally, an investigation into generating marginal distributions and estimating quantiles of propagated model uncertainty is to be completed. Generating marginal densities through the Koopman approach may offer computational advantages over Monte Carlo and kernel based methods because of the efficient expected value computation. Together, these novel applications of the Koopman operator to uncertainty propagation of models with state and/or parameter uncertainty provide an computationally efficient alternative method to existing methods of uncertainty propagation.