Spring 2025
Time & Location: Typically talks will be in Gibson Hall Gibson Hall 325 at 3:00 pm on a Friday.
Organizers: Chen, Hongfei and Gkogkou, Aikaterini
January 17
Title: Unbounded Hamiltonian Simulation: Quantum Algorithm and Superconvergence
Di Fang - Duke University
Abstract: Simulation of quantum dynamics, emerging as the original motivation for quantum computers, is widely viewed as one of the most important applications of a quantum computer. Quantum algorithms for Hamiltonian simulation with unbounded operators Recent years have witnessed tremendous progress in developing and analyzing quantum algorithms for Hamiltonian simulation of bounded operators. However, many scientific and engineering problems require the efficient treatment of unbounded operators, which may frequently arise due to the discretization of differential operators. Such applications include molecular dynamics, electronic structure, quantum differential equations solver and quantum optimization. We will introduce some recent progresses in quantum algorithms for efficient unbounded Hamiltonian simulation, including Trotter type splitting and Magnus expansion based algorithms in the interaction picture. (The talk does not assume a priori knowledge on quantum computing.)
Time: 3:00 pm
Location: Gibson Hall 126
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January 24
Title: Random solitons and soliton gasses for the Korteweg de Vries equation
Manuela Girotti - Emory University Host: (Aikaterini Gkogkou)
Abstract: N. Zabusky coined the word "soliton" in 1965 to describe a curious feature he and M. Kruskal observed in their numerical simulations of the initial-value problem for a simple nonlinear PDE. The first part of the talk will be a broad introduction to the theory of solitons/solitary waves and integrable PDEs (the KdV and modified KdV equation in particular), describing classical results in the field. The second (and main) part of the talk will focus on some new developments and growing interest into a special case of solutions defined as "soliton gas".
I will describe a collection of works done in collaborations with K. McLaughlin (Tulane U.), T. Grava (SISSA/Bristol), R. Jenkins (UCF) and A. Minakov (U. Karlova).
We analyze the case of a regular, dense KdV soliton gas and its large time behaviour with the presence of a single trial soliton travelling through it.
We are able to derive a series of physical quantities that precisely describe the dynamics, such as the local phase shift of the gas after the passage of the soliton, and the velocity of the soliton peak, which is highly oscillatory and it satisfies the kinetic velocity equation analogous to the one posited by V. Zakharov and G. El (at leading order).
I will finally present some ongoing work where we establish that the soliton gas is the universal limit for a large class of N-solutions with random initial data.
Time: 3:00 pm
Location: Gibson Hall 325
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January 31
Title: Towards efficient deep operator learning for forward and inverse PDEs: theory and algorithms
Ke Chen - University of Delaware Host: (Hongfei Chen)
Abstract: Deep neural networks (DNNs) have been a successful model across diverse machine learning tasks, increasingly capturing the interest for their potential in scientific computing. This talk delves into efficient training for PDE operator learning in both the forward and inverse PDE settings. Firstly, we address the curse of dimensionality in PDE operator learning, demonstrating that certain PDE structures require fewer training samples through an analysis of learning error estimates. Secondly, we introduce an innovative DNN, the pseudo-differential auto-encoder integral network (pd-IAE net), and compare its numerical performance with baseline models on several inverse problems, including optical tomography and inverse scattering.
Time: 3:00 pm
Location: Gibson Hall 325
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February 7
Title: Admissible behavior of Krylov subspace iterative methods for non-symmetric systems
Kirk Soodhalter - Trinity College Dublin (Lisa Fauci)
Abstract: In this talk, we introduce the audience Krylov subspace iterative methods, the work-horse matrix-free methods for solving linear systems and eigenvalue problems in the case that matrix is large and sparse or otherwise not available to be solved by accessing all entries of the matrix. These methods are built on the core assumption that we only have access to a procedure that multiplies the matrix times vectors at relatively low computational cost. After discussing some basic convergence theory related to polynomial interpolation, we demonstrate the complicated nature of estimating rate of convergence for such methods based on quantities such as eigenvalues when the matrix is non-normal. In particular, we discuss and contextualize a constructively proven theorem that in pathological cases, the eigenvalues need not be at all descriptive with regard to convergence behavior [Greenbaum, Pták, Strakoš 1996]. We build the language needed to describe the mechanics of how to construct such cases. We then discuss our currently running project on developing more robust theory of convergence for these methods applied to non-symmetric Toeplitz systems, the type of which arise in a variety of problems from the computational sciences. Toeplitz matrices are constant along each diagonal; thus the linear system is governed by a matrix with fewer degrees of freedom than in general. We demonstrate that tools developed in the aformentioned constructive proof can be repurposed to develop refine previous convergence theory for Toeplitz systems. This is ongoing work; thus we finish by discussing what remains to be proven and how we intend to extend this theory to generalizations of Toeplitz matrices encompassing a larger class of matrix structures often arising in the computational sciences.
Time: 3:00 pm
Location: Gibson Hall 325
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February 21
Title: The unreasonable utility of symmetric three-term recurrences
Tom Trogdon - University of Washington (Host: Aikaterini Gkogkou)
Abstract: Symmetric three-term recurrences (STRs) naturally arise in the study of orthogonal polynomials, iterative methods for symmetric matrices and numerical complex analysis. While deceptively simple, STRs allow for many extremely effective numerical methods. This talk will review some classical methods and uses and connect to more recent developments related to the computation of Cauchy integrals, computing matrix functions and spectral density estimation for random matrices.
Time: 3:00 pm
Location: Gibson Hall 325
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March 21
Title: Inverse Problems and In-Context Operator Learning for Mean-Field Games
Speaker: Siting Liu - UC Riverside Host: (Hongfei Chen)
Abstract: Mean-field game (MFG) systems provide a powerful framework for modeling the collective behavior of multi-agent systems with diverse applications. However, unknown model parameters pose challenges. In the first part of the talk, we address an inverse problem in MFGs, recovering running cost and interaction energy from noisy boundary observations. We formulate it as a constrained optimization problem and solve it efficiently using an operator-splitting algorithm. In the second part, we introduce In-Context Operator Networks (ICON), a neural framework that learns solution operators from prompts to solve MFG problems. ICON demonstrates strong few-shot learning across forward and inverse differential equation tasks, including mean-field control.
Time: 3:00 pm
Location: Gibson Hall 325
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March 28
Title: Infinitesimal Homeostasis in Mass-Action Systems
Jiaxin Jin - University of Louisiana-- Lafayette
Abstract: Homeostasis occurs in a biological system when a chosen output variable remains approximately constant despite changes in an input variable. In this work we specifically focus on biological systems which may be represented as chemical reaction networks and consider their infinitesimal homeostasis, where the derivative of the input-output function is zero. The specific challenge of chemical reaction networks is that they often obey various conservation laws complicating the standard input-output analysis. We derive several results that allow to verify the existence of infinitesimal homeostasis points both in the absence of conservation and under conservation laws where conserved quantities serve as input parameters. In particular, we introduce the notion of infinitesimal concentration robustness, where the output variable remains nearly constant despite fluctuations in the conserved quantities. We provide several examples of chemical networks which illustrate our results both in deterministic and stochastic settings.
Time: 3:00 pm
Location: Gibson Hall 325
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April 4
Title: Solitons + Periodic Traveling Waves = Breathers, Dispersive Shock Waves, Oh My!
Mark Hoefer - University of Colorado Boulder Host: (Aikaterini Gkogkou)
Abstract: Completely integrable evolutionary partial differential equations admit a plethora of fascinating, nonlinear-only wave solutions that are realized in a variety of physical systems. Solitons and nonlinear periodic traveling waves TWs are perhaps the most well-known. By nonlinear superposition and modulation, these two basic solution components give rise to new, intriguing solutions and physical phenomena. This talk will use the Darboux transformation on cnoidal TW solutions of the Korteweg-de Vries (KdV) equation to construct exact dark and bright breather solutions. These solutions are then used to describe the transmission and trapping of a soliton by a dispersive shock wave, an expanding modulated nonlinear wavetrain, using multiphase Whitham modulation theory. Finally, experiments in a miscible, viscous core-annular fluid demonstrate the creation, interaction, and properties of breathers including their nonlinear dispersion relation. The experiments are shown to be in qualitative agreement with KdV breather solutions and in quantitative agreement with numerical simulations of a strongly nonlinear, long-wavelength model equation.
Short bio: Mark Hoefer is Professor and Chair of the Department of Applied Mathematics at the University of Colorado, Boulder, drawn back to the Rockies in 2014 after obtaining his PhD from the same department in 2006. In between, he held National Research Council and National Science Foundation postdoc positions at NIST and Columbia University before becoming an Assistant Professor in the Mathematics Department of North Carolina State University. Mark's research encompasses the mathematics and physics, including in-house experiments, of multiscale nonlinear waves. He received the National Science Foundation's Career award in 2013 and the T. Brooke Benjamin Prize in Nonlinear Waves from SIAM in 2020. In 2022, he was the principal organizer for a six-month research program on Dispersive Hydrodynamics at the Isaac Newton Institute in Cambridge, UK. He is deputy editor of the recently established Journal of Nonlinear Waves, published by Cambridge University Press.
Time: 3:00 pm
Location: Gibson Hall 325
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April 11
Title: How fluid rheology shapes micororganism swimming gait
Robert Guy - UC Davis Host: (Lisa Fauci)
Abstract: Many microorganisms propel through complex media by deformations of their flagella. The beat is thought to emerge from interactions between forces of the surrounding fluid, passive elastic response from deformations of the flagellum, and active forces from internal molecular motors. The beat varies in response to changes in the fluid rheology, including elasticity, but there is limited data on how systematic changes in rheology alters the beat. In this talk we develop mathematical and computational models in order to understand observations from experimental data of the flagellar gait and swimming speed of the bi-flagellated alga Chlamydomonas reinhardtii in different fluids
In the first part of the talk we use shape mode analysis to quantify changes in the flagellar beat, and we use computational models to vary the stroke and fluid rheology independently. This analysis shows that changes in the mean shape are primarily responsible for the observed change in swimming speed and the variations of the beat around the mean change very little with fluid rheology. In the second part of the talk we examine how fluid elasticity affects the emergent frequency of the beat. We analyze the emergence of beating of an elastic planar filament driven by a follower force at the tip in a viscoelastic fluid. This analysis predicts that the frequency increases with the fluid relaxation time, and using numerical simulations, the model predictions are compared with experimental data. The model shows the same trends in response to changes in both fluid viscosity and fluid elasticity, and thus provides a possible mechanistic explanation of the experimental observations.
Time: 3:00 pm
Location: Gibson Hall 325
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April 23
Title: Central limit theorems in classical Hamiltonian dynamics
Alexander Moll - Reed College
Abstract: In work with Robert Chang (Rhodes College), we derive certain central limit theorems in the context of classical Hamiltonian dynamical systems with random initial data. In this talk, I'll state our general CLT in the framework of Gaussian measures, then illustrate it in two concrete examples: (i) in the simple linear harmonic oscillator and (ii) in the nonlinear inviscid Burgers equation on the circle. This second example of our CLT implies a new relationship between fractional Brownian motions on the circle and certain Gaussian processes on the line found by Kerov in 1993.
Time: 3:00 pm
Location: Dinwiddie (DW) Hall 102
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