Mathematics Colloquium - Professor Hongkai Zhao

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Date: October 14, 2019
Time: 2:45 p.m. - 3:45 p.m.
Location: Faculty/Administration #1140 | Map
656 W. Kirby
Detroit, MI 48202
Category: Lecture

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.


Department of Mathematics



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