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College of Liberal Arts and Sciences | Mathematics

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

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.

For more information about this event, please contact Department of Mathematics at 313-577-2479 or math@wayne.edu.