Optimal uncertainty quantification via convex optimization and relaxation

Many engineering applications face the problem of bounding the expected value of a quantity of interest (performance, risk, cost, etc.) that depends on stochastic uncertainties whose probability distribution is not known exactly. Optimal uncertainty quantification (OUQ) is a framework that aims at obtaining the best bound in these situations by explicitly incorporating available information about the distribution. Unfortunately, this often leads to non-convex optimization problems that are numerically expensive to solve.

This thesis emphasizes on efficient numerical algorithms for OUQ problems. It begins by investigating several classes of OUQ problems that can be reformulated as convex optimization problems. Conditions on the objective function and information constraints under which a convex formulation exists are presented. Since the size of the optimization problem can become quite large, solutions for scaling up are also discussed. Finally, the capability of analyzing a practical system through such convex formulations is demonstrated by a numerical example of energy storage placement in power grids.

When an equivalent convex formulation is unavailable, it is possible to find a convex problem that provides a meaningful bound for the original problem, also known as a convex relaxation. As an example, the thesis investigates the setting used in Hoeffding's inequality. The naive formulation requires solving a collection of non-convex polynomial optimization problems whose number grows doubly exponentially. After structures such as symmetry are exploited, it is shown that both the number and the size of the polynomial optimization problems can be reduced significantly. Each polynomial optimization problem is then bounded by its convex relaxation using sums-of-squares. These bounds are found to be tight in all the numerical examples tested in the thesis and are significantly better than Hoeffding's bounds.

A probabilistic treatment of uncertainty in nonlinear dynamical systems

In this work, computationally efficient approximate methods are developed for analyzing uncertain dynamical systems. Uncertainties in both the excitation and the modeling are considered and examples are presented illustrating the accuracy of the proposed approximations.

For nonlinear systems under uncertain excitation, methods are developed to approximate the stationary probability density function and statistical quantities of interest. The methods are based on approximating solutions to the Fokker-Planck equation for the system and differ from traditional methods in which approximate solutions to stochastic differential equations are found. The new methods require little computational effort and examples are presented for which the accuracy of the proposed approximations compare favorably to results obtained by existing methods. The most significant improvements are made in approximating quantities related to the extreme values of the response, such as expected outcrossing rates, which are crucial for evaluating the reliability of the system.

Laplace's method of asymptotic approximation is applied to approximate the probability integrals which arise when analyzing systems with modeling uncertainty. The asymptotic approximation reduces the problem of evaluating a multidimensional integral to solving a minimization problem and the results become asymptotically exact as the uncertainty in the modeling goes to zero. The method is found to provide good approximations for the moments and outcrossing rates for systems with uncertain parameters under stochastic excitation, even when there is a large amount of uncertainty in the parameters. The method is also applied to classical reliability integrals, providing approximations in both the transformed (independently, normally distributed) variables and the original variables. In the transformed variables, the asymptotic approximation yields a very simple formula for approximating the value of SORM integrals. In many cases, it may be computationally expensive to transform the variables, and an approximation is also developed in the original variables. Examples are presented illustrating the accuracy of the approximations and results are compared with existing approximations.

Future of earthquake early warning : quantifying uncertainty and making fast automated decisions for applications

Earthquake early warning (EEW) systems have been rapidly developing over the past decade. Japan Meteorological Agency (JMA) has an EEW system that was operating during the 2011 M9 Tohoku earthquake in Japan, and this increased the awareness of EEW systems around the world. While longer-time earthquake prediction still faces many challenges to be practical, the availability of shorter-time EEW opens up a new door for earthquake loss mitigation. After an earthquake fault begins rupturing, an EEW system utilizes the first few seconds of recorded seismic waveform data to quickly predict the hypocenter location, magnitude, origin time and the expected shaking intensity level around the region. This early warning information is broadcast to different sites before the strong shaking arrives. The warning lead time of such a system is short, typically a few seconds to a minute or so, and the information is uncertain. These factors limit human intervention to activate mitigation actions and this must be addressed for engineering applications of EEW. This study applies a Bayesian probabilistic approach along with machine learning techniques and decision theories from economics to improve different aspects of EEW operation, including extending it to engineering applications.

Existing EEW systems are often based on a deterministic approach. Often, they assume that only a single event occurs within a short period of time, which led to many false alarms after the Tohoku earthquake in Japan. This study develops a probability-based EEW algorithm based on an existing deterministic model to extend the EEW system to the case of concurrent events, which are often observed during the aftershock sequence after a large earthquake.

To overcome the challenge of uncertain information and short lead time of EEW, this study also develops an earthquake probability-based automated decision-making (ePAD) framework to make robust decision for EEW mitigation applications. A cost-benefit model that can capture the uncertainties in EEW information and the decision process is used. This approach is called the Performance-Based Earthquake Early Warning, which is based on the PEER Performance-Based Earthquake Engineering method. Use of surrogate models is suggested to improve computational efficiency. Also, new models are proposed to add the influence of lead time into the cost-benefit analysis. For example, a value of information model is used to quantify the potential value of delaying the activation of a mitigation action for a possible reduction of the uncertainty of EEW information in the next update. Two practical examples, evacuation alert and elevator control, are studied to illustrate the ePAD framework. Potential advanced EEW applications, such as the case of multiple-action decisions and the synergy of EEW and structural health monitoring systems, are also discussed.

Essays in optimal resource allocation under uncertainty with capacity constraints

This thesis brings together four papers on optimal resource allocation under uncertainty with capacity constraints. The first is an extension of the Arrow-Debreu contingent claim model to a good subject to supply uncertainty for which delivery capacity has to be chosen before the uncertainty is resolved. The second compares an ex-ante contingent claims market to a dynamic market in which capacity is chosen ex-ante and output and consumption decisions are made ex-post. The third extends the analysis to a storable good subject to random supply. Finally, the fourth examines optimal allocation of water under an appropriative rights system.