Math Colloquium - Fall 2025 | Department of Mathematical Sciences

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September 5

Alberto Padovan, NJIT (Mechanical and Industrial Engineering)

Host: Michael Siegel

Leveraging oblique projections for model reduction: recent advances in Petrov-Galerkin and non-intrusive formulations

Developing accurate reduced-order models (ROMs) for high-dimensional physical systems (e.g., fluids, solids, systems in thermochemical kinetics, etc.) is essential to accelerate simulations and enable engineering tasks like design and optimization. In this talk, we emphasize the importance of oblique projections for the development of ROMs with long-time predictive accuracy, and we present recent advances on their computation using both intrusive
and non-intrusive methods.

Several systems in engineering, including transport-dominated fluid flows, exhibit high sensitivity to low-energy coordinates. These coordinates are typically truncated in ROMs obtained with orthogonal projections designed to capture the high-energy features of the system. As a consequence of this truncation, the resulting ROMs can fail to faithfully reproduce the dynamics of the original system. By contrast, oblique projections can be used to compute ROMs that capture the energetic features of the flow while also accounting for the low-energy coordinates that are necessary to accurately describe the dynamics of the full-order model. In this presentation, we present two novel formulations to compute oblique-projection-based (or Petrov-Galerkin) models for nonlinear systems. The first one is similar in spirit to the well-known balanced truncation formulation for linear systems, and it identifies a quasi-optimal oblique projector by balancing the state and gradient covariance matrices associated with the flow of the underlying full-order model. The second method is a refinement of the latter, where oblique projections for Petrov-Galerkin model reduction are further optimized against high-fidelity data to reduce forecasting error. While these two methods can work very well and outperform existing model reduction formulation, they are highly intrusive and require direct access to the underlying governing equations. They may therefore be inapplicable to systems that are simulated using black-box solvers (e.g., commercial software or legacy codes) whose source code might be proprietary or very difficult to modify. To address this issue, we introduce a third (non-intrusive) formulation, which we call “NiTROM: Non-intrusive Trajectory-based optimization of ROMs.” Specifically, given high-fidelity data from a black-box solver, NiTROM computes a ROM by simultaneously seeking the optimal projection and latent-space dynamics that minimize forecasting error. This formulation can achieve predictive accuracy comparable to that of the first two (intrusive) Petrov-Galekin methods, and it has been shown that it can significanly outperform state-of-the-art non-intrusive frameworks like Operator Inference. All the methods discussed in this talk are demonstrated on several systems, including incompressible fluid flows with states.

September 12

Anand Oza, NJIT

Waves and Interaction Modes of Capillary Surfers

We present the results of a combined experimental and theoretical investigation into “capillary surfers,” which are millimetric objects that self-propel while floating at the interface of a vibrating fluid bath. Recent experiments showed that surfer pairs may lock into one of seven bound states, and that larger collectives of surfers self-organize into coherent flocking states. Our theoretical model for the surfers’ horizontal positional and orientational dynamics approximates a surfer as a pair of vertically oscillating point sources of weakly viscous gravity-capillary waves. We derive an analytical solution for the associated interfacial deformation and thus the hydrodynamic force exerted by one surfer on another. Our model recovers the bound states found in experiments and exhibits good quantitative agreement with experimental data. We then study the vertical dynamics of a floating disc, recasting the elliptic boundary value problem with mixed boundary conditions as an integral equation that is solved numerically. We find that the disc exhibits resonant oscillations at a critical value of the forcing frequency, in agreement with experiment. Generally, our work shows that self-propelling objects coupled by interfacial flows constitute a promising platform for studying active matter systems in which both inertial and viscous effects are relevant.

September 19

Bruce Pitman, University of Buffalo

Host: Yuan-Nan Young

Inverse Problems via Gaussian Processes

One class of inverse problems consists of determining the properties of an inhomogeneous inclusion in an otherwise homogeneous domain from measurements at the domain boundary. This is the setting of imaging problems in geophysics and biomedicine. This inverse problem can be complicated to solve, and specialized methods have been developed to that end. Here we report on a different approach to solving the inverse problem, using Gaussian Process emulation. This emulation provides a fast solver with very modest computational cost. We illustrate the idea with three simple applications.

SPECIAL SEMINAR - September 25

Eliot Fried, Mechanics and Materials Unit Okinawa Institute of Science and Technology

Host: Jointly by Mechanical Engineering and Mathematical Sciences. For details, please contact Anthony Rosato, anthony.rosato@njit.edu

Location: Tiernan Hall, Lecture Hall 1 (10:00AM - 11:00AM)

Inside out without effort: Exceptional linkages and the geometry of isometric band eversions

We explore a unified framework linking the kinematics of underconstrained ring linkages to the isometric and isoenergetic eversion of continuous bands. The starting point is the M¨obius kaleidocycle—a closed chain of n ≥ 7 identical rigid links joined by revolute hinges at a critical twist angle. Despite being spatially underconstrained, such linkages exhibit only a single internal degree of freedom, manifested in a smooth everting motion. As n → ∞, this family of linkages converges to a ruled M¨obius band with three half-twists, uniform torsion midline, and trefoil-knotted boundary. Motivated by this limit surface, we consider the inverse problem: determining all stable isometric deformations from circular helicoids to M¨obius bands, and characterizing their shape-preserving periodic eversions. For isotropic elastic bands with bending energy depending only on mean curvature, these motions are isoenergetic, requiring no work input. Finally, we return to discrete systems by showing how the midline of such a band can be discretized via a nonuniform finite-difference scheme, yielding a class of linkages that preserve mobility for arbitrary link arrangements. This synthesis reveals deep geometric and mechanical connections between exceptional-mobility linkages and continuous surfaces capable of effortless eversion, with implications for deployable structures, robotics, and the design of topologically nontrivial elastic systems.

Biographical Sketch: The speaker received his Ph.D. in Applied Mechanics from the California Institute of Technology in 1991 and held a National Science Foundation postdoctoral fellowship. He has since held tenured faculty positions at the University of Illinois at Urbana–Champaign, Washington University in St. Louis, McGill University—where he was Tier I Canada Research Chair in Interfacial and Defect Mechanics—and the University of Washington. He is now Professor and Head of the Mechanics and Materials Unit at the Okinawa Institute of Science and Technology. His research spans mechanics, thermodynamics, and geometry, with current interests in surface instabilities, reaction–diffusion processes coupled to bulk strain, and underconstrained linkages. He is also co-inventor on two U.S. patents

September 26

Jingyi Jessica Li, Fred Hutchinson Cancer Center

Host: Chenlu Shi

Nullstrap: A Simple, High-Power, and Fast Framework for FDR Control in Variable Selection for Diverse High-Dimensional Model

Balancing false discovery rate (FDR) control with high statistical power remains a central challenge in high-dimensional variable selection. While several FDR-controlling methods have been proposed, many degrade the original data—by adding knockoff variables or splitting the data—which often leads to substantial power loss and hampers detection of true signals. We introduce Nullstrap, a novel framework that controls FDR without altering the original data. Nullstrap generates synthetic null data by fitting a null model under the global null hypothesis that no variables are important. It then applies the same estimation procedure in parallel to both the original and synthetic data. This parallel approach mirrors that of the classical likelihood ratio test, making Nullstrap its numerical analog. By adjusting the synthetic null coefficient estimates through a data-driven correction procedure, Nullstrap identifies important variables while controlling the FDR. We provide theoretical guarantees for asymptotic FDR control at any desired level and show that power converges to one in probability. Nullstrap is simple to implement and broadly applicable to high-dimensional linear models, generalized linear models, Cox models, and Gaussian graphical models. Simulations and real-data applications show that Nullstrap achieves robust FDR control and consistently outperforms leading methods in both power and efficiency.

October 3

Matt Holzer, George Mason University

Host: James MacLaurin

Persistence of steady-states for dynamical systems on large networks

The goal of this work is to identify steady-state solutions to dynamical systems defined on large, random families of networks. We do so by passing to a continuum limit where the adjacency matrix is replaced by a non-local operator with kernel called a graphon. Our work establishes a rigorous connection between steady-states of the continuum and network systems. Precisely, we show that if the graphon equation has a steady-state solution whose linearization is invertible, there exists related steady-state solutions to the finite-dimensional networked dynamical system over all sufficiently large graphs converging to the graphon. The proof involves setting up a Newton--Kantorovich type iteration scheme which is shown to be a contraction on a suitable metric space. We also discuss linear stability properties that carry over from the graphon system to the graph dynamical system. Our results are applied to twisted states in a Kuramoto model of coupled oscillators, steady-states in a model of neuronal network activity, and a Lotka--Volterra model of ecological interaction. We also discuss subsequent extensions of our work to nonhyperbolic points with saddle-node bifurcations leading to the onset of synchrony as our primary motivating example.

October 10

Liliana Borcea, Columbia University

Host: Amir Sagiv

Data driven reduced order modeling for first order hyperbolic systems with application to waveform inversion

Waveform inversion seeks to estimate an inaccessible heterogeneous medium by using sensors to probe the medium with signals and measure the generated waves. It is an inverse problem for a hyperbolic system of equations, with the sensor excitation modeled as a forcing term and the heterogeneous medium described by unknown, variable coefficients. The traditional formulation of the inverse problem, called full waveform inversion (FWI), estimates the unknown coefficients via nonlinear least squares data fitting. For typical band limited and high frequency data, the data fitting objective function has spurious local minima near and far from the true coefficients. This is why FWI implemented with gradient based optimization can fail, even for good initial guesses. We propose a different approach to waveform inversion: First, use the data to "learn" a good algebraic model, called a reduced order model (ROM), of how the waves propagate in the unknown medium. Second, use the ROM to obtain a good approximation of the wave field inside the medium. Third, use this approximation to solve the inverse problem. I will give a derivation of such a ROM for a general first order hyperbolic system satisfied by all linear waves in lossless media (sound, electromagnetic or elastic). I will describe the properties of the ROM and will use it to solve the inverse problem for sound waves.