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Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

Year Degree Awarded

2017

Month Degree Awarded

May

First Advisor

Markos A. Katsoulakis

Second Advisor

Luc Rey-Bellet

Third Advisor

Patrick Flaherty

Fourth Advisor

Arya Mazumdar

Subject Categories

Numerical Analysis and Computation | Statistical Models

Abstract

The ever-increasing complexity of the models used in predictive modeling and data science and their use for prediction and inference has made the development of tools for uncertainty quantification and model selection especially important. In this work, we seek to understand the various trade-offs associated with the simulation of stochastic systems. Some trade-offs are computational, e.g., execution time of an algorithm versus accuracy of simulation. Others are analytical: whether or not we are able to find tractable substitutes for quantities of interest, e.g., distributions, ergodic averages, etc.

The first two chapters of this thesis deal with the study of the long-time behavior of parallel lattice Kinetic Monte Carlo (PL-KMC) algorithms for interacting particle systems. We introduce the relative entropy rate (RER) as a measure of long-time loss of information and illustrate that it is a computable a posteriori quantity. The RER can act as an information criterion (IC), discriminating between different parameter choices for the schemes and allowing comparisons at equilibrium. We make explicit how the RER scales with the time-step and the size of the system and that it captures details about the connectivity of the original process.

Another feature of long-time behavior is time-reversibility, which some physical systems naturally exhibit. Unfortunately, due to the domain and time-discretization, PL-KMC cannot conserve this property. To quantify the loss of reversibility, we introduce the entropy production rate (EPR) as an IC for comparisons between different schemes. We show that the EPR shares a lot of the properties of the RER and can be estimated efficiently from data.

The last chapter discusses uncertainty quantification for model bias. By connecting a recently derived goal-oriented divergence and concentration bounds, we define new divergences that provide computable bounds for model bias. The new bounds scale appropriately with data and become progressively more accurate depending on available information about the models and the quantities of interest. We discuss how the bounds allow us to bypass computationally expensive Monte Carlo sampling or specialized methods, e.g., Multilevel Monte Carlo.

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