Mathematics Colloquium - Professor Hongkai Zhao
This event is in the past.
Detroit, MI 48202
Speaker: Professor Hongkai Zhao, University of California-Irvine
Title: Intrinsic complexity: from approximation of random vectors and random fields to solutions of PDEs
Abstract: We characterize the intrinsic complexity of a set in a metric space by the least dimension of a linear space that can approximate the set to a given tolerance. This is dual to the characterization of the set using Kolmogorov n-width, the distance from the set to the best n-dimensional linear space. In this talk I will start with the intrinsic complexity of a set of random vectors (via principal component analysis) and random fields (via Karhunen–Loève expansion) and then characterize solutions to partial differential equations of various type. Our study provides a mathematical understanding of the complexity/richness and its mechanism of the underlying problem independent of representation basis. In practice, our study is directly related to the question of whether there is low dimensional structure or a low rank approximation to one can exploit for the underlying problem, which is essential for dimension reduction and developing fast algorithms.