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Time & Location*:
Tuesdays from 11am – 12pm in Hill 525
(*Some talks may be scheduled for different times or locations. Such details will be provided additionally.)

 

Organizers:
Narek Hovsepyan
nh507@math.rutgers.edu
Gokul Nair
gokul.nair@rutgers.edu
(For inquiries, e.g. to be added to the mailing list, please contact either one of the organizers.)

 

 

30 September
David Hien, Rutgers University
“Cycling Signatures: Identification and Analysis of Cycling Motions in Time Series“
Abstract
Recurrence is a fundamental characteristic of dynamical systems with complicated behavior.
Understanding the inner structure of recurrence is challenging, especially if the system has many
degrees of freedom and is subject to noise. The cycling signature is an algebraic topological notion
for identifying and classifying elementary recurrent motions — called cycling — and the transitions
between them. Statistics on these cycling motions can be computed from sampled trajectories
(time series data).
They provide a coarse global description of the structure of the recurrent behavior.

 

*13 October (Monday) from 11am – 12 pm in Hill 705
Dominic Blanco, McGill University
“A general approach for proving the symmetry of localized patterns in a class of PDEs posed on unbounded domains.“
Abstract
In this talk, we will show a general method for constructively proving the existence of localized patterns in a class of semilinear autonomous partial differential equations (PDEs) posed on unbounded domains along with any symmetry they may possess.  We will summarize our main approach for proving solutions on unbounded domains. We use a Newton-Kantorovich argument involving quantities to estimate partially by hand and partially on the computer. This makes our approach computer assisted. Following this, we will discuss previous methods that use computer assisted proofs (CAPs) for proving certain symmetries of solutions using Fourier series. Combining these methods can lead to proofs of some symmetries of localized patterns, but not all possible symmetries. The goal of this talk is to bridge this gap and demonstrate our general approach for proving any symmetry. This will include the construction of the approximate solution, approximate inverse, and the necessary Newton-Kantorovich argument. We will use dihedral symmetries in the 2D Swift Hohenberg PDE as our example. To conclude, we will discuss future improvements we would like to investigate with regards to the method.

 

21 October
Kathrin Smetana, Stevens Institute of Technology
“Randomized Multiscale Methods for Heterogeneous Nonlinear Partial Differential Equations.”
Abstract
To construct localizable multiscale methods for nonlinear partial differential equations we consider a transfer operator that maps arbitrary admissible boundary data on the boundary of an oversampling domain to the respective (local) solution on the target subdomain; here the boundary of the latter must have a distance greater than zero from the boundary of the oversampling domain. Then, we try to approximate the set of all local solutions on the target subdomain. Interpreting the boundary data as some input parameter, we can view this set of local solutions as a set of solutions depending on a parameter. This motivates using methods from model order reduction such as the proper orthogonal decomposition (POD) or the Greedy algorithm to approximate this set. However, both the POD and the Greedy algorithm rely on a training set of finite cardinality that is chosen such that every point in the admissible parameter set is close to a point in the training set. Therefore, both algorithms suffer from the curse of dimensionality. We thus employ randomization and consider the parameter (here: boundary data) as a random variable with values in a Hilbert space. By choosing a suitable distribution we can then exploit the concentration of measure phenomenon, which is also sometimes called the “blessing of dimensionality” to break the curse.
In detail, we will present a randomized greedy algorithm that provides with high probability a certification for the whole parameter set rather than only for the parameters in the training set. Moreover, we will present a randomized POD and a corresponding error analysis that shows that for exponentially decaying eigenvalues of the randomized POD which uses the exact correlation operator (integral in the expectation) the approximation error between any solution corresponding to a parameter in the admissible parameter set and the approximation with the POD that uses a Monte-Carlo approximation converges exponentially as well.

 

4 November
Miroslav Kramar, University of Oklahoma
“Using Persistent Homology to Detect Shadowing in Turbulent Flows.”
Abstract
The idea that turbulence can be described as a deterministic walk through a repertoire of patterns goes back to Eberhard Hopf. Over the years it was established that these patterns closely correspond to exact coherent structures (ECS) which are often formed by unstable (relative) periodic orbits. Over recent years, a large body of numerical simulations and experiments indicated that the turbulent trajectory moves through the phase space from one ECS to another. The turbulent trajectory approaches the ECS along its stable manifold and leaves along its unstable manifold. It is therefore natural to ask whether these results are coincidental or whether some collection of ECSs can in fact provide a dynamical and statistical description of fluid turbulence. In order to properly answer this question one needs to be able to identify when the turbulent trajectory follows (shadows) a given ECS. However, in systems with continuous symmetries, detecting when the turbulent trajectory shadows a given ECS remains challenging and computationally very expensive. In this talk, we present a novel and computationally efficient approach for detecting the shadowing based on persistent homology. To demonstrate the potential of our method we apply it to the Kuramoto-Sivashinsky equation, which serves as a simple model that mimics some of the properties of fluid turbulence, such as spatiotemporal chaos, in a more accessible setting.

 

11 November
Narek Hovsepyan, Rutgers University
“Scattering of waves.“
Abstract
We study the scattering of waves from a planar inclusion with constant refractive index, governed by the Helmholtz equation. It is well understood that singular inclusions—those whose boundary has a singular point—generically scatter every incident wave. Far less is known about the scattering behavior of regular inclusions. We consider a large class of regular inclusions and show that they generically scatter any (complex-analytic) incident wave. Focusing on incident plane waves, we prove that they always scatter from any regular inclusion whose refractive index is less than one. For inclusions with refractive index greater than one, under an additional convexity assumption, we obtain explicit wavenumber intervals in which scattering occurs. These intervals drift to infinity and expand, and are given by explicit formulas in terms of the geometry (directional widths) of the inclusion. If the inclusion is elongated (its maximal width is at least twice its minimal width), the union of these intervals covers the entire spectrum, and hence such inclusions scatter every incident plane wave. Our approach makes use of a connection between this scattering problem and the Schiffer/Pompeiu problem.
This is based on a joint work with Michael Vogelius.

 

18 November
Benjamin Zhang, University of North Carolina at Chapel Hill
“A mean-field games laboratory for generative artificial intelligence: from foundations to applications in scientific computing.”
Abstract
We demonstrate the versatility of mean-field games (MFGs) as a mathematical framework for explaining, enhancing, and designing generative models. We establish connections between MFGs and major classes of flow- and diffusion-based generative models by deriving continuous-time normalizing flows and score-based models through different choices of particle dynamics and cost functions. We study the mathematical structure and properties of each generative model by examining their associated MFG optimality conditions, which consist of coupled forward-backward nonlinear partial differential equations (PDEs). We present this framework as an MFG laboratory, a platform for experimentation, invention, and analysis of generative models. Through this laboratory, we show how MFG structure informs new normalizing flows that robustly learn data distributions supported on low-dimensional manifolds. In particular, we show that Wasserstein proximal regularizations inform the well-posedness and robustness of generative flows for singular measures, enabling stable training with less data and without specialized architectures.
We then apply these principled generative models to operator learning, where the goal is to learn solution operators of differential equations. We present a probabilistic framework that reveals certain classes of operator learning approaches, such as in-context operator networks (ICON), as implicitly performing Bayesian inference. ICON computes the mean of the posterior predictive distribution of solution operators conditioned on example condition-solution pairs. By extending ICON to a generative setting, we enable sampling from the posterior predictive distribution. This provides principled uncertainty quantification for predicted solutions, demonstrating how mathematical foundations translate to trustworthy applications in scientific computing.

 

2 December
Mallory Gaspard, Princeton University
“When is Camouflage Useful? A Case Study in Hover Fly Pursuit-Evasion Interactions.”
Abstract
Camouflaging is a widely used concealment tactic across the animal kingdom, but when is it actually beneficial for an organism to use? In this talk, we focus on analyzing when it is worthwhile for a pursuer to utilize motion camouflage (MC) amidst uncertainty in when an evader will feel threatened and attempt to escape. Using MC movement techniques to trick an evader’s visual system into believing that a pursuer is less threatening than they actually are has been observed in hover flies during mating rituals and in dragonflies during territorial disputes. To ground our discussion in a concrete example, I will focus on mathematically modeling biologically observed MC behaviors exhibited in hover fly pursuit-evasion interactions. In this model, the evader’s escape attempt time occurs as the result of a nonhomogeneous Poisson point process governed by a rate function that is dependent on the pursuer’s state and the evader’s position. I will then present a general mathematical framework to determine when MC tactics may be worthwhile for an energy-optimizing pursuer, and I will highlight the resulting sequence of Hamilton-Jacobi-Bellman (HJB) partial differential equations which encode the pursuer’s optimal trajectories. After presenting the model and mathematical approach, I will show a selection of numerical simulations and statistics that reveal the existence of a specific parameter regime for the rate function in which MC tactics are useful. Finally, I will wrap up our discussion with some suggestions for future expansions of the framework. 

 

9 December
Aakash Parikh, Rutgers University
“Bifurcations from finite vector field data via TDA.”
Abstract
The traditional approach to understanding the changes a parameterized physical system undergoes as the parameters are varied consists of (1) writing a model family of differential equations (oftentimes in an ad hoc fashion) and (2) numerically or analytically identifying bifurcations in this family. While this is a time honored and fruitful pipeline, we instead take the approach of identifying bifurcations via a Conley index approach to topological data analysis, effectively circumventing the oft dubious step (1). In this talk, we will discuss the details of this method in the case of the saddle node bifurcation. This is joint work with Konstantin Mischaikow.