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ACMS Abstracts: Fall 2022

James Hanna (UN-Reno)

Title: A snapping singularity

Abstract: I will discuss our preliminary work (with A. Dehadrai) on the focusing of kinetic energy and the amplification of various quantities during the snapping motion of the free end of a string or chain. This brief but violent event, with its remarkably large spikes in velocity, acceleration, and tension, is an essentially unavoidable feature of flexible structure dynamics, induced by generic initial and boundary conditions. We are guided by an analytical solution for a geometrically singular limit that features a finite-time singularity in other quantities. Regularization of this singularity does not arise from discretization of the continuous string equations or, equivalently, from the physical discreteness of a chain. It is instead associated with a length scale arising from the geometry of the problem, which evolves according to an anomalously slow curvature scaling.

Thomas Chandler (UW)

Title: Fluid–body interactions in liquid crystals: A complex variable approach

Abstract: Fluid anisotropy, or direction-dependent response to deformation, can be observed in biofluids like mucus or, at a larger scale, self-aligning swarms of swimming bacteria. A model fluid used to investigate such environments is a liquid crystal. Large colloidal bodies undergo shape-dependent interactions when placed in such an environment, whilst deformable bodies like red blood cells tend to be stretched, offering a passive means of measuring cell material properties. While numerous methods exist for studying the liquid crystalline configurations and fluid–body interaction for a single body, there are exceedingly few analytical results for the interaction of two or more bodies. In this talk, we will bring the power of complex variables to bear on this problem, presenting a simple methodology to analytically solve for the interactions inside a liquid crystalline environment. This approach allows for the solution of a wide range of problems, opening the door to studying the role of body shape and orientation, liquid crystal anchoring conditions, and body deformability.

Jennifer Franck (UW)

Title: Predictive modeling of oscillating foil wake dynamics

Abstract: Swimming and flying animals rely on the fluid around them to provide lift or thrust forces, leaving behind a distinct vortex wake in the fluid. The structure and size of the vortex wake is a blueprint of the animal’s kinematic trajectory, holding information about the forces and also the size, speed and direction of motion. This talk will introduce a bio-inspired oscillating turbine, which can be operated to generate energy from moving water through lift generation, in the same manner as flapping birds or bats. This style of turbines offers distinct benefits compared with traditional rotation-based turbines such as the ability to dynamically shift its kinematics for changing flow conditions, thus altering its wake pattern. Current efforts lie in predicting the vortex formation and dynamics of the highly structured wake such that it can be utilized towards cooperative motion within arrays of oscillating foils. Using numerical simulations, this talk will discuss efforts towards linking the fluid dynamic wake signature to the underlying foil kinematics, and investigating how that effects the energy harvesting performance of downstream foils. Two machine learning methodologies are introduced to classify, cluster and identify complex vorticity patterns and modes of energy harvesting, and inform more detailed modeling of arrays of oscillating foils.

Jinlong Wu (UW)

Title: Data-Driven Closure Modeling Using Derivative-free Kalman Methods

Closure problems are critical in predicting complex dynamical systems, e.g., turbulence or cloud dynamics, for which numerically resolving all degrees of freedom remains infeasible in the foreseeable future. Although researchers have been advancing traditional closure models of those systems for decades, the performance of existing models is still unsatisfactory in many applications, mainly due to the limited representation power of existing models and the associated empirical calibration process. Recently, the rapid advance of machine learning techniques shows great potential for improving closure models of dynamical systems. In this talk, I will share some progress in data-driven closure modeling for complex dynamical systems. More specifically, I will demonstrate the use of derivative-free Kalman methods to learn closure models from indirect and limited amount of data. In addition to deterministic closures, examples of sparse identification of dynamical systems and the learning of stochastic closures will also be presented.

Jeffrey Weiss (CU Boulder)

Title: Vortex-gas models for 3d atmosphere and ocean turbulence

Abstract: Atmospheres and oceans self-organize into coherent structures such as fronts, jets, and long-lived vortices. It is useful to model vortex dominated geophysical flows as a vortex gas, where solutions are assumed to take the form of a population of interacting vortices. There are many vortex gas models of increasing complexity for both 2d flow and for purely horizontal, so-called quasigeostrophic, 3d flow. Atmospheres and oceans, however, have small, but important vertical velocities. The smallness of the vertical velocity is due to rapid planetary rotation, quantified by a small Rossby number. The asymptotic expansion of the governing equations for planetary turbulence capture this small vertical velocity when carried to second order in the Rossby number. Here we find a find a vortex gas solution to these equations in the form of point vortices. The nonlinear dynamics of small numbers of such vortices shows complex and geophysically interesting vertical transport. This new point vortex model provides a platform to revisit in 3d the myriad problems studied with 2d point vortices, and provides a tool for modeling important processes in atmospheres and oceans.

Kui Ren (Columbia)

Title: Some results on inverse problems to elliptic PDEs with solution data and their implications in operator learning

Abstract: In recent years, there have been great interests in discovering structures of partial differential equations from given solution data. Very promising theory and computational algorithms have been proposed for such operator learning problems in different settings. We will try to review some recent understandings of such a PDE learning problem from the perspective of inverse problems. In particularly, we will highlight a few analytical and computational understandings on learning a second-order elliptic PDE from single and multiple solutions.

Daniel Lecoanet (Northwestern)

Title: Wave generation by convective turbulence

Abstract: In nature, turbulent convective fluids are often found adjacent to stably stratified fluids. These stably stratified regions host internal gravity waves, which can be excited by convection. This process occurs in the Earth's atmosphere and oceans, as well as in stars and in other planets. The dynamical effects of these waves depend on the efficiency of the excitation process. I will describe a series of numerical simulations which help explain how internal waves are generated by convection. The simulations are run using Dedalus, an open-source pseudo-spectral code that can solve nearly arbitrary PDEs in a range of geometries. These simulations show good agreement with heuristic theories of wave generation by convection.

Michael Gastner (Yale-NUS)

Title: Remapping data: visualizing geospatial statistics using cartograms

Abstract: Cartograms are thematic maps in which the areas of geographic regions (e.g., states or provinces) appear in proportion to a quantitative mapping variable (e.g., population or gross domestic product). Unlike conventional bar or pie charts, cartograms can correctly represent which regions share common borders, resulting in insightful visualizations that can be the basis for further spatial statistical analysis. The construction of cartograms poses intriguing mathematical and algorithmic challenges. In this talk, I will first review previous cartogram designs and algorithms. Then, I will explain how cartograms can be constructed using data-dependent map projections. Afterward, I will describe some of the remaining challenges and explain how computational geometry can help to solve them.

Casian Pantea (WVU)

Title: Motifs of multistationarity in mass-action reaction networks

Abstract: The existence of multiple positive steady states in models of reaction networks, referred to as multistationarity, underlies switching behavior in biochemistry, and has been an important area of study over the last two decades. A recent approach to multistationarity of large networks relies on “lifting”  positive steady states from smaller network components which are themselves multistationary.  This led to an effort of cataloging small multistationary network structures (multistationary motifs). In this talk we introduce two new classes of multistationary networks (networks with 1D stoichiometric subspace, and networks with cyclic structure). As a consequence we prove a partial converse to the DSR graph theorem, i.e. a graph-theoretical sufficient condition for multistationarity based solely on the wiring diagram of the network.

Ian Tobasco (UIC)

Title: Two examples in the optimal design of heat transfer

Abstract: Heat, or any other passive tracer, travels through a moving fluid according to the advection-diffusion equation. We ask how the underlying velocity field should be chosen to optimize the transfer. Initially this question stemmed from the search for good bounds on natural, buoyancy driven convection; since then, the problem has taken on a life of its own, with several contributors. After a brief review, we shall analyze two setups in detail --- optimizing heat transfer between a distributed source and a boundary sink or, separately, between an internal point-like source and an internal point-like sink. The character of the (nearly) optimal velocities we construct changes dramatically between the two, with the former lending itself to a "branching flow" design, and the latter being solved by a self-similar "pinching flow". The proof of these claims is via matching upper and lower bounds on a suitable objective functional measuring the overall efficiency of the transfer. The work on pinching flows is joint with Binglin Song (U Michigan) and Giovanni Fantuzzi (FAU Erlangen-Nürnberg).

Matthew Colbrook (Cambridge University)

Title: Residual Dynamic Mode Decomposition: Rigorous Data-Driven Computation of Spectral Properties of Koopman Operators for Dynamical Systems

Abstract: Koopman operators are infinite-dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra, can lack finite-dimensional invariant subspaces, and approximations can suffer from spectral pollution (spurious modes). These issues make computing the spectral properties of Koopman operators a considerable challenge. This talk will detail the first scheme (ResDMD) with convergence guarantees for computing the spectra and pseudospectra of general Koopman operators from snapshot data. Furthermore, we use the resolvent operator and ResDMD to compute smoothed approximations of spectral measures (including continuous spectra), with explicit high-order convergence. ResDMD is similar to a Galerkin method, except it rigorously concurrently computes a residual from the same snapshot data, allowing practitioners to gain confidence in the computed results. Kernelized variants of our algorithms allow for dynamical systems with a high-dimensional state-space, and the error control provided by ResDMD allows a posteriori verification of learnt dictionaries. We apply ResDMD to compute the spectral measure associated with the dynamics of a protein molecule (20,046-dimensional state-space) and demonstrate several problems in fluid dynamics (with state-space dimensions > 100,000). For example, we compare ResDMD and DMD for particle image velocimetry data from turbulent wall-jet flow, the acoustic signature of laser-induced plasma, and turbulent flow past a cascade of aerofoils.

[1] M.J. Colbrook, A. Townsend. "Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems." arXiv:2111.14889 (2021).

[2] M.J. Colbrook, L. Ayton, M. Szőke. "Residual Dynamic Mode Decomposition: Robust and verified Koopmanism." arXiv.2205.09779 (2022).

[3] M.J. Colbrook, "The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems." arXiv.2209.02244 (2022). 

Laurel Ohm (Princeton)

Title:A PDE perspective on the hydrodynamics of flexible filaments

Abstract: Many fundamental biophysical processes, from cell division to cellular motility, involve dynamics of thin structures immersed in a very viscous fluid. Various popular models have been developed to describe this interaction mathematically, but much of our understanding of these models is only at the level of numerics and formal asymptotics. Here we seek to develop the PDE theory of filament hydrodynamics.

First, we propose a PDE framework for analyzing the error introduced by slender body theory (SBT), a common approximation used to facilitate computational simulations of immersed filaments in 3D. Given data prescribed only along a 1D curve, we develop a novel type of boundary value problem and obtain an error estimate for SBT in terms of the fiber radius. This places slender body theory on firm theoretical footing.

Second, we consider a classical elastohydrodynamic model for the motion of an immersed inextensible filament. We highlight how the analysis can help to better understand undulatory swimming at low Reynolds number. This includes the development of a novel numerical method to simulate inextensible swimmers.