Applied Mathematics Colloquium
Tuesday,
April 9, 2019
2:45 PM - 3:45 PM
Facebook
TwitterHongkai Zhao
Chancellor’s Professor and Chair
Department of Mathematics, University of California, Irvine
"Intrinsic complexity and its scaling law: from approximation of random vectors and random fields to high frequency waves"
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 using Kolmogorov n-width, the distance from the set to the best n-dimensional linear space. We study the approximation of a set of random vectors (via principal component analysis a.k.a. singular value decomposition) and random fields (via Karhunen–Loève expansion) as well as high frequency waves (via the Helmholtz equation). We provide lower bounds and upper bounds for the intrinsic complexity and their asymptotic scaling laws in terms of an appropriate length scale, e.g., correlation length and wavelength.
Host: Kui Ren
Chancellor’s Professor and Chair
Department of Mathematics, University of California, Irvine
"Intrinsic complexity and its scaling law: from approximation of random vectors and random fields to high frequency waves"
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 using Kolmogorov n-width, the distance from the set to the best n-dimensional linear space. We study the approximation of a set of random vectors (via principal component analysis a.k.a. singular value decomposition) and random fields (via Karhunen–Loève expansion) as well as high frequency waves (via the Helmholtz equation). We provide lower bounds and upper bounds for the intrinsic complexity and their asymptotic scaling laws in terms of an appropriate length scale, e.g., correlation length and wavelength.
Host: Kui Ren
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