Solving Large Systems of Equations from Data- Driven Models

This lecture will familiarize the participants with the methods of translating a continuous model (such as those coming from partial differential equations, integral equations,  systems of ODEs, or systems of nonlinear equations) to a discrete system, i.e., a large system of linear equations. Then several methods for solving such systems will be introduced. This will include the concept of the Moore-Penrose Pseudoinverse highlighting its ubiquity and its limitations. The highlight of the talk will be on the use of regularization techniques that can address the instability of solutions obtained from standard methods.

Optimization for Data-Driven Research and Policymaking

This course will provide the participants an overview of the various optimization methods that are commonly used in the data enabled- sciences. Particular emphasis will be given to optimization principles over function spaces that are commonly used as the theoretical backbone of regularization techniques. 

Theoretical and computational aspects of optimal control will also be given emphasis as these are crucial in data-driven research and policy making.

Inverse Problems

This course will give the participants first-hand experience in tackling inverse problems using the linear algebra and optimization tools discussed in the first two courses. This research process will be exemplified through inverse problems in field synthesis and cloaking of acoustic and electromagnetic fields. These problems require the formation of large linear systems emanating from integral equations which are then solved using regularization methods. 

Fundamentals of Machine Learning

This course will feature the fundamentals of machine learning and their applications to various fields, such as imaging, machine vision and classification, and the biological and environmental sciences. This will include discussions about problem formulation, data preprocessing, classification and clustering algorithms, model validation, and result communication (data visualization and story- telling).