Math Colloquium - Fall 2024

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September 6

Danial Szyld, Temple University

Host: Yassine Boubendir

Randomized Householder-Cholesky QR Factorization with Multisketching

We present and analyze a new randomized algorithm called rand_cholQR for computing tall-and-skinny QR factorizations. Using one or two random sketch matrices, it is  proved that with high probability, its orthogonality error is bounded by a constant of the order of unit roundoff for any numerically full-rank matrix.An evaluation of the performance of rand_cholQR on a NVIDIA A100 GPUdemonstrates that for tall-and-skinny matrices, rand_cholQR} with multiple sketch matrices is nearly as fast as, or in some cases faster than, the state-of-the-art CholeskyQR2. Hence, compared to CholeskyQR2, rand_cholQR is more stable with almost no extra computational or memory cost, and therefore a superior algorithm both in theory and practice.

Joint work with Andrew J. Higgins, Erik G. Boman, and Ichitaro Yamazaki.

September 13

Shahriar Afkami, NJIT

Numerical methods for complex flows

I will give an overview of recent developments in numerical modeling of some complex flow problems. Examples include interfacial flows, fluid-solid interactions, viscoelastic fluids, and flows with surfactants. I will attempt to focus on methods for discretizing continuum-level differential constitutive equations for viscoelastic flows, and for interfacial flows, on an Eulerian description based on the Volume-of-Fluid method. Outstanding issues and avenues for further progress will be discussed.

September 20

Igor Aronson, Penn State University

Host: Kela Lushi

Confined Bacterial Suspensions

Previous experiments have shown [1,2] that the complex spatiotemporal vortex structures emerging in motile bacterial suspensions are susceptible to weak geometrical constraints. By a combination of continuum theory and experiments, we have shown how artificial obstacles guide the flow profile and reorganize topological defects, which enables the design of bacterial vortex lattices with tunable properties. In more recent studies, we observed the emergence of spatiotemporal chaos in a bacterial suspension confined in a cylindrical well. As the well radius increases, we observed a bifurcation sequence from a steady-state vortex to periodically reversing vortices, four pulsating vortices, and, finally, to spatiotemporal chaos (active turbulence). The results of experiments are rationalized by the analysis of the continuum model for bacterial suspensions based on the complex Swift-Hohenberg equations. Furthermore, the bifurcation sequence is explained by reduction to amplitude equations for the three lowest azimuthal modes. Equations of motion are then reconstructed from experimental data.  The results indicate that the vortex reversal precedes the onset of spatiotemporal chaos in confined active systems. 
 
[1] D Nishiguchi, IS Aranson, A Snezhko, A Sokolov, Engineering bacterial vortex lattice via direct laser lithography, Nature communications 9 (1), 4486        115       (2018)
[2] H Reinken, D Nishiguchi, S Heidenreich, A Sokolov, M Bär, S. H. L. Klapp & I. S. Aranson,  Organizing bacterial vortex lattices by periodic obstacle arrays. Commun Phys 3, 76 (2020)