AIM Seminar: Robert Krasny, Mathematics, University of Michigan

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Two topics in computational fluid dynamics

1. The Lamb dipole is a steady propagating solution of the inviscid fluid equations with opposite-signed vorticity in a circular disk. We compare finite-difference solutions of the Navier-Stokes equation (NSE) and the linear diffusion equation (LDE) using the Lamb dipole as the initial condition. We find some expected and some unexpected results; among the latter is that the maximum core vorticity decreases at the same rate for the NSE and LDE, but at higher Reynolds numbers, convection enhances the viscous cancellation of opposite-signed vorticity.
(This is joint work with Ling Xu.)

2. We discuss a new implementation of the vortex method for the incompressible Euler equations. The vorticity is carried by Lagrangian particles and the velocity is recovered by a regularized Biot-Savart integral. The new work employs remeshing and adaptive refinement to resolve small-scale features in the vorticity as well as a treecode for efficiency. The method is demonstrated for vortex dynamics on a rotating sphere (with Peter Bosler) and axisymmetrization of an elliptical vortex (with Ling Xu).

AIM Seminar: Alex Gorodetsky, Aerospace Engineering, University of Michigan

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Low-rank tensor approaches for adaptive function approximation: algorithms and examples

In this talk, we present an adaptive method for approximating high-dimensional low-rank functions. Taking advantage of low-rank structure in approximation problems has been shown to prove advantageous for scaling numerical algorithms and computation to higher dimensions by mitigating the curse-of-dimensionality. The method we describe is an extension of the tensor-train cross approximation algorithm to the continuous case of multivariate functions that enables both global and local adaptivity. Our approach relies on a new adaptive algorithm for computing the CUR/skeleton decomposition of bivariate functions. We then extend this technique to the multidimensional case of the function-train decomposition. We demonstrate the benefits of our approach compared with the standard methodology that computes low-rank approximations by decomposing coefficients of tensor-product basis functions. We finish by demonstrating a wide range of applications that include machine learning, uncertainty quantification, stochastic optimal control, and Bayesian filtering.

MICDE Seminar: Michael Shelley, Courant Institute, New York University

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Bio: Michael J. Shelley is an American applied mathematician who works on the modeling and simulation of complex systems arising in physics and biology. He holds a BA in Mathematics from the University of Colorado (1981) and a PhD in Applied Mathematics from the University of Arizona (1985). He was a postdoctoral researcher at Princeton University, and then joined the faculty of mathematics at the University of Chicago. In 1992 he joined the Courant Institute of Mathematical Sciences at New York University where he is the George and Lilian Lyttle Professor of Applied Mathematics. He is also a Professor of Neuroscience (NYU) and Professor of Mechanical Engineering (NYU-Poly).

Professor Shelley’s work includes free-boundary problems in fluids and materials science, singularity formation in partial differential equations, modeling visual perception in the primary visual cortex, dynamics of complex and active fluids, cellular biophysics, and fluid-structure interaction problems such as the flapping of flags, stream-lining in nature, and flapping flight. He is also the co-founder and co-director of the Courant Institute’s Applied Mathematics Lab.


Modeling and Simulating Active Mechanics in the Cell

Many fundamental phenomena in eukaryotic cells — nuclear migration, spindle positioning, chromosome segregation — involve the interaction of (often transitory) cellular structures with boundaries and fluids. Understanding the consequences of these interactions require specialized numerical methods for their large-scale simulation, as well as mathematical modeling and analysis. In this context, I will discuss the recent interactions of mathematical modeling and large-scale, detailed simulations with experimental measurements of activity-driven Biomechanical processes within the cell.