Date of Award

4-30-2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

First Advisor

Xiaojing Ye

Second Advisor

Alexandra Smirnova

Abstract

We develop efficient numerical methods for solving inverse problems in a decentralized setting. First we consider decentralized optimization of a known objective with known gradient, analyzing the convergence of a decentralized consensus algorithm using delayed stochastic gradient information across the network. Each node privately holds a part of the objective function, and nodes collaboratively solve for the consensus optimal solution of the total objective while they can only communicate with their immediate neighbors. In real-world applications, it is often difficult and sometimes impossible to synchronize the nodes, and therefore they have to use stale gradient information during computations. We show that the iterates generated converge to a consensual optimal solution as long as the random delays are bounded in expectation and a proper diminishing step size policy is employed. Convergence rates of both objective and consensus are derived. Numerical results on a number of synthetic problems and real-world seismic tomography datasets in decentralized sensor networks are presented. We then consider inverse problems in epidemiology where the disease transmission rate (objective) is unknown, looking to develop an efficient decentralized method for its estimation. As the objective is unknown, we first consider the centralized setting, and then extend our work to the decentralized case. We use an SEIR compartmental model and we assume the transmission rate is time-dependent. Using observed incidence case data, we develop a method for estimating disease transmission rate, which may be used to forecast future incidence cases for cyclic disease epidemics. We test the method on synthetic and real-world datasets. We then investigate whether this method may be modified for extension to the decentralized case. We are motivated by the problem in which regions experience an outbreak of a common, cyclic disease epidemic, and we consider whether collaboration can allow for the recovery of a common transmission rate. We investigate whether the common estimate returned by the method produces accurate forecasts of each local region’s future incidence cases. The method is tested on a synthetic dataset using both full and partial data for transmission rate recovery.

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