Abstract:

Multiscale problems are challenging for computational processing due to the large number of unknowns required to just represent a multiscale function. This is mathematically related to the Nyquist – Shannon theorem for spectrally band limited functions. We will review different methodologies for handling this challenge from computational and data science. We will then focus on recent developments in computational analogues to the analytical techniques for averaging of dynamical systems and homogenization of elliptic operators. We will study a class of algorithms, which exploits scale separation in the solution and an alternative methodology, which compresses the spectrum in the solution. The latter is sometimes labeled as seamless methods.

Bio:

Bjorn Engquist received his Ph.D. in numerical analysis from Uppsala University in 1975. He has been Professor of Mathematics at UCLA, and the Michael Henry Stater University Professor of Mathematics and Applied and Computational Mathematics at Princeton University. He was director of the Mathematical Research Institute for Industrial Applications and of the HPC Center at the Royal Institute of Technology, Stockholm. At Princeton University, he was director of the Program in Applied and Computational Mathematics and the Princeton Institute for Computational Science. Engquist came to The University of Texas at Austin in 2004, where he is Professor of Mathematics holding the Computational and Applied Mathematics Chair I and is Director of the Oden Institute Center for Numerical Analysis. Engquist is a member of the Royal Swedish Academy of Sciences, the Royal Swedish Academy of Engineering Sciences, the Norwegian Academy of Science and Letters and the American Academy of Arts and Sciences. He is a SIAM Fellow and has received the AMS Birkhoff Prize, the Peter Henrici Prize, the SIAM Pioneer Prize, the Celsius medal in Gold and the First SIAM Prize in Scientific Computing.

Engquist’s research focuses on development, analysis, and application of numerical methods for differential equations. His work includes the development of absorbing or far field boundary conditions, homogenization theory and nonlinear high-resolution schemes for compressible fluid dynamics. Application areas have been aerodynamics and flow in porous media, acoustic, seismic, and electromagnetic wave propagation. More recently he has been working on computational fast and multi-scale methods. Another recent focus is inverse problems and global optimization.

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