Large Deviations for Non-Markovian Diffusions and a Path-Dependent Eikonal Equation

This paper provides a large deviation principle for Non-Markovian, Brownian motion driven stochastic differential equations with random coefficients. Similar to Gao and Liu \cite{GL}, this extends the corresponding results collected in Freidlin and Wentzell \cite{FreidlinWentzell}. However, we use a different line of argument, adapting the PDE method of Fleming \cite{Fleming} and Evans and Ishii \cite{EvansIshii} to the path-dependent case, by using backward stochastic differential techniques. Similar to the Markovian case, we obtain a characterization of the action function as the unique bounded solution of a path-dependent version of the Eikonal equation. Finally, we provide an application to the short maturity asymptotics of the implied volatility surface in financial mathematics

Inexact Fixed-Point Proximity Algorithms for Nonsmooth Convex Optimization

The aim of this dissertation is to develop efficient inexact fixed-point proximity algorithms with convergence guaranteed for nonsmooth convex optimization problems encountered in data science. Nonsmooth convex optimization is one of the core methodologies in data science to acquire knowledge from real-world data and has wide applications in various fields, including signal/image processing, machine learning and distributed computing. In particular, in the context of image reconstruction, compressed sensing and sparse machine learning, either the objective functions or the constraints of the modeling optimization problems are nondifferentiable. Hence, traditional methods such as the gradient descent method and the Newton method are not applicable since gradients of the objective functions or the constraints do not exist. Fixed-point proximity algorithms were developed via subdifferentials of the objective function to address the challenges. The theory of nonexpansive averaged operators was successfully employed in the existing analysis of exact/inexact fixed-point proximity algorithms for nonsmooth convex optimization. However, this framework has imposed restricted constraints on the algorithm formulation, which slows down the convergence and conceals relations between different algorithms. In this work, we characterize the solutions of convex optimization as fixed-points of certain operators, and then adopt the matrix splitting technique to obtain a framework of fully implicit fixed-point proximity algorithms. This results in a new class of quasiaveraged operators, which extends the class of nonexpansive averaged operators. Such framework covers and generalizes most of the existing popular algorithms for nonsmooth convex optimization. To deal with the implicitness of this framework, we follow the inspiration of the Schur’s lemma on the uniform boundedness of infinite matrices and propose a framework of inexact fixed-point iterations of quasiaveraged operators. This framework generalizes the inexact iterations of nonexpansive averaged operators. A combination of the frameworks of inexact fixed-point iterations and the implicit fixed-point proximity algorithms leads to the framework of inexact fixed-point proximity algorithms, which further extends existing methods for nonsmooth convex optimization. Numerical experiments on image deblurring problems demonstrate the advantages of inexact fixed-point proximity algorithms over existing explicit algorithms

Investigation of the energy performance of a novel modular solar building envelope

The major challenges for the integration of solar collecting devices into a building envelope are related to the poor aesthetic view of the appearance of buildings in addition to the low efficiency in collection, transportation, and utilization of the solar thermal and electrical energy. To tackle these challenges, a novel design for the integration of solar collecting elements into the building envelope was proposed and discussed. This involves the dedicated modular and multiple-layer combination of the building shielding, insulation, and solar collecting elements. On the basis of the proposed modular structure, the energy performance of the solar envelope was investigated by using the Energy-Plus software. It was found that the solar thermal efficiency of the modular envelope is in the range of 41.78–59.47%, while its electrical efficiency is around 3.51% higher than the envelopes having photovoltaic (PV) alone. The modular solar envelope can increase thermal efficiency by around 8.49% and the electrical efficiency by around 0.31%, compared to the traditional solar photovoltaic/thermal (PV/T) envelopes. Thus, we have created a new envelope solution with enhanced solar efficiency and an improved aesthetic view of the entire building