Mathematics Colloquium Seminar
Friday, October 4, 2019, 12:00pm, 130 Physics
Hongkai Zhao (University of California, Irvine)
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 a low rank approximation to the associated (discretized) linear system, which is essential for dimension reduction and developing fast algorithms.

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