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.

Professor Zhao received his B.S., M.S. and Ph.D in Mathematics from Peking University (China), University of Southern California and UCLA in 1990, 1992 and 1996 respectively. He went to Stanford University as Garbor Szego Assistant Professor in 1996 before he joined UCI in 1999. He is currently Chancellor’s Professor in the Department of Mathematics with a joint appointment in the Department of Computer Science at UCI. Professor Zhao received A. P. Sloan Research Fellowship (2002-2004), Feng Kang Prize for Scientific Computing in 2007 and Chang-Jiang Guest Professorship at Peking University in 2009. Professor Zhao' research interest is in computational and applied mathematics that includes modeling, analysis and developing numerical methods for problems arising from science and engineering such as moving interface problem, level set method, fast sweeping method, image processing/computer vision, imaging and inverse problems.