Rahul Rameshbabu
(Advisor: Prof. Dimitri Mavris)

will propose a doctoral thesis entitled,

Solving Large-scale Inverse Problems Using Bayesian Inference and Surrogate Modeling

On

Monday, January 30 at 1:00 p.m.
Weber Space Science and Technology Building (SST II), Room 304

Abstract
In many applications and fields, important decisions need to be made based on quantities of interest (QOIs) that are often difficult to observe and/or quantify directly. Engine fatigue, for example, is known as the leading cause of fatigue-related aircraft accidents. However, while real-time, accurate measurements of fatigue cracks are desirable to inform maintenance needs, such direct measurements are difficult to obtain due to the extreme conditions (e.g. temperature and pressure) experienced during engine operation. Similar situations occur in other domains as well (e.g. medical). 

In scenarios where QOIs are difficult to observe directly, one typical relies on two main pieces of information: 1) a set of observations of a physical process, and 2) a simulator, which is parameterized by the QOIs, and which models and simulates the same physical process. From there, an inverse problem can be solved to infer the QOIs, using both the observations and the simulator. High-fidelity simulators and large amounts of observation data are traditionally preferred to ensure that QOIs are inferred accurately.  

In addition, the physical process of interest, along with the simulator, are often subject to many sources of uncertainty, either aleatory or epistemic. Hence, methodologies that leverage high-fidelity, black-box simulators and large datasets to efficiently solve inverse problems and quantify the uncertainty in the QOIs are of particular interest.

Many methods exist for the purpose of solving inverse problems and quantifying uncertainty. Among those, approximate Bayesian inference methods are preferred for obtaining posterior distributions of QOIs given observations. However, such methods generally require many queries of the high-fidelity, black-box simulator, leading to the need to develop surrogate models to make approximate Bayesian inference a feasible option. 

To accurately infer QOIs in the presence of epistemic uncertainty, it is important to infer both the QOIs and the simulator discrepancy. However, when inferring the joint posterior of QOIs and simulator discrepancy, it becomes difficult to distinguish between the effects of each, which can result in inaccurate inferences of QOIs. Modular Bayesian analysis, where simulator discrepancy is inferred separately from the QOIs, is commonly used to alleviate this issue. Consequently, the overarching objective of this thesis is to develop a modular Bayesian analysis methodology, accelerated by surrogate modeling, to efficiently solve inverse problems involving high-fidelity, black-box simulators and large datasets. 

When investigating current implementations of modular Bayesian analysis, the approximate Bayesian inference algorithms used in these applications scale poorly with increasing number of QOIs and dataset size. An algorithm that is relatively unexplored in modular Bayesian analysis is Variational Inference (VI), which is known to scale better with parameter vector and dataset size than currently employed methods in modular Bayesian analysis. The first proposed contribution of this thesis is the development and benchmarking of a Modular Bayesian analysis method that uses VI. 

When utilizing surrogate models to approximate the high-fidelity, black-box simulator, Gaussian Processes (GPs) are a natural choice due to producing accurate estimates of prediction error and being sample efficient. However, when using GPs to expedite variational inference, the optimization could become inefficient if the prediction error of the GP is high. This leads to the second proposed contribution of this thesis, which is a multi-fidelity, adaptive variational inference algorithm that utilizes a GP surrogate model to speed up the algorithm, but keeps the high-fidelity model in the loop to ensure efficient optimization. 

Finally, the two proposed contributions are combined to meet the aforementioned research objective. The proposed outcome is a modular Bayesian analysis methodology that uses multi-fidelity, adaptive variational inference to solve inverse problems. The effectiveness and superiority of this methodology will be demonstrated on a fatigue crack monitoring problem. 

Committee

• Prof. Dimitri Mavris – School of Aerospace Engineering (advisor)
• Prof. Graeme Kennedy– School of Aerospace Engineering
• Prof. Elizabeth Cherry – School of Computational Science and Engineering
• Dr. Olivia Pinon Fischer – School of Aerospace Engineering