Research Seminars: Applied and Computational Mathematics

Spring 2025

Time & Location: Typically talks will be in Gibson Hall at 3:00 pm on a Friday.
Organizers: Chen, Hongfei and Gkogkou, Aikaterini

Archives

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: _____________

Ke Chen, University of Delaware

Abstract: ________