Modern data sets are increasingly vast, not only in the number of samples, but also in the number of measurements, or features, that they contain. This high-dimensionality poses a unique set of problems for data analysis due to a set of phenomena known as ``the curse of dimensionality.'' This thesis...
Existing nonlinear optimization methods have proven reliable over the past few decades for a wide range of applications but have critically relied on accurate function and gradient evaluations. Modern nonlinear optimization problems arising from machine learning and scientific computing applications are increasingly complex and large scale, which make accurate evaluations...
The ever growing desire for accurate estimation and efficient learning necessitates the efforts to quantitatively characterize uncertainties for models. In this thesis, four problems pertaining to uncertainty quantification are discussed: A sequential stopping framework of constructing fixed-precision confidence regions is proposed for a class of multivariate simulation problems where variance...