Diffusion Maps and its Applications to Time series Forecasting and Filtering and Second Order Elliptic PDEs
Open Access
Author:
Gilani, Faheem
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
May 05, 2021
Committee Members:
Alberto Bressan, Major Field Member John Harlim, Chair & Dissertation Advisor Bharath Kumar Sriperumbudur, Outside Unit & Field Member Xiantao Li, Major Field Member Carina Curto, Program Head/Chair
In this dissertation, we study generalizations of the diffusion maps algorithm, which is a diffusion based kernel method that allows for the construction of a weighted second order elliptic operator (a weighted Laplacian or a Kolmogorov operator) using data sampled from a Riemanninan manifold, thereby recovering its underlying geometry. In particular, we use an extension of diffusion maps via local kernels to solve second order elliptic PDEs on Riemannian manifolds and a variable bandwidth extension to predict and smooth time series observations generated from higher order dynamical systems. We also compare the efficacy of the proposed prediction methods with neural networks on real climate date. When possible, we present the mathematical framework necessary to ensure the proposed methods are on rigorous footing.
