91PORN

February 6— "Finding scattering resonances via generalized colleague matrices"

• Presenter: Vladimir Rokhlin, Arthur K Watson Professor of Computer Science & Mathematics, Yale University
• �������ٰ�������:��Locating scattering resonances is a standard task in certain areas of physics and engineering. This often can be reduced to finding zeros of complex analytic functions. In this talk, I will discuss a scheme for finding all roots of a complex analytic function in a square domain in C. The scheme can be viewed as a generalization of the classical approach to finding roots of a function on an interval by first approximating it by a polynomial in the Chebyshev basis, followed by diagonalizing the so-called “colleague matrices.” This extension to the complex domain is based on several observations that enable the construction of polynomial bases that satisfy three-term recurrences and are reasonably well-conditioned, giving rise to “generalized colleague matrices.” We also introduce a special-purpose QR algorithm for finding eigenvalues of the resulting structured matrices stably and efficiently. I will demonstrate the effectiveness of the approach via several numerical examples.

March 6 — "Tailored Forecasts from Short Time Series Using Meta-Learning and Reservoir Computing"

• Presenter: Michelle Girvan, Department of Physics, University of Maryland
• �������ٰ�������:��Machine learning (ML) models can be effective for forecasting the dynamics of unknown systems from time-series data, but they often require large datasets and struggle to generalize—that is, they fail when applied to systems with dynamics different from those seen during training. Combined, these challenges make forecasting from short time series particularly difficult. To address this, we introduce Meta-learning for Tailored Forecasting from Related Time Series (METAFORS), which supplements limited data from the system of interest with longer time series from systems that are suspected to be related. By leveraging a library of models trained on these potentially related systems, METAFORS builds tailored models to forecast system evolution with limited data. Using a reservoir computing implementation and testing on simulated chaotic systems, we demonstrate METAFORS’ ability to predict both short-term dynamics and long-term statistics, even when test and related systems exhibit significantly different behaviors, highlighting its strengths in data-limited scenarios.

April 3 — ��Layer potentials - quadrature error estimates and approximation with error control

• Presenter: Anna-Karin Tornberg, Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden

• Abstract: When numerically solving PDEs reformulated as integral equations, so called layer potentials must be evaluated. The quadrature error associated with a regular quadrature rule for evaluation of such integrals increases rapidly when the evaluation point approaches the surface and the integrand becomes sharply peaked. Error estimates are needed to determine when the accuracy becomes insufficient, and then, a sufficiently accurate special quadrature method needs to be employed.

  In this talk, we motivate the use of integral equations for some problems in microfluidics. We then discuss how to estimate quadrature errors, building up from simple integrals in one dimension to layer potentials over smooth surfaces in three dimensions. We also discuss a new special quadrature technique for axisymmetric surfaces with error control.�� The underlying technique is so-called interpolatory semi-analytical quadrature in conjunction with a singularity swap technique. Here, adaptive discretizations and parameters are set automatically given an error tolerance, utilizing further quadrature and interpolation error estimates derived for this purpose. Several different examples are shown, including examples with rigid particles in Stokes flow.

May 1— Geostrophic turbulence and the formation of large scale structure

• Presenter: Edgar Knobloch; Department of Physics; University of California, Berkeley

• Abstract: Rotating convection is a prototypical system at the core of geophysical fluid dynamics. However, the parameter values for geophysical flows take values that are far outside those that can be studied in the laboratory or via state of the art numerical simulations. In this talk I will describe a formal asymptotic procedure that leads to a reduced system of equations valid in the limit of very strong rotation. These equations describe four regimes as the Rayleigh number Ra increases: a disordered cellular regime near threshold, a regime of weakly interacting convective Taylor columns at larger Ra, followed for yet larger Ra by a breakdown of the convective Taylor columns into a disordered plume regime characterized by reduced heat transport efficiency, and finally by a type of turbulence called geostrophic turbulence. Properties of these states will be described and illustrated using direct numerical simulations of the reduced equations.

  These simulations reveal that geostrophic turbulence is unstable to the formation of large scale barotropic vortices or jets, via a process known as spectral condensation. The details of this process will be quantified and its implications explored. The predictions from the reduced equations have been corroborated via direct numerical simulations of the Navier-Stokes equations, albeit at much more modest rotation rates, confirming that the reduction procedure captures the essence of the problem. Moreover, rescaling the Navier-Stokes equations using the aforementioned scales serves as an excellent preconditioner that allows us to perform fully resolved direct numerical simulations of very rapidly rotating convection at unprecedented Ekman numbers, some six orders of magnitude smaller than the current state of the art, approaching geophysically realistic values for the very first time. The computations reveal a new transition at Ek\sim 10^{-9} towards symmetry between cyclonic and anticyclonic large scale vortex structures, and show that in the statistically stationary state the solutions converge to the predictions of the asymptotically reduced equations.