Applied/ACMS

Applied and Computational Mathematics Seminar

• When: Fridays at 2:25pm (except as otherwise indicated)
• Where: 901 Van Vleck Hall
• Organizers: Maurice Fabien, Chris Rycroft, and Saverio Spagnolie,
• To join the ACMS mailing list: Send mail to acms+join@g-groups.wisc.edu.

Fall 2023

date	speaker	title	host(s)
Sep 8	Erik Bollt (Clarkson University)	A New View on Integrability: On Matching Dynamical Systems through Koopman Operator Eigenfunctions	Chen
Sep 15 4:00pm B239	John Schotland (Yale University)	Nonlocal PDEs and Quantum Optics	Li
Sep 22	Balazs Boros (U Vienna)	Oscillatory mass-action systems	Craciun
Sep 29	Peter Jan van Leeuwen (Colorado State University)	Nonlinear Causal Discovery, with applications to atmospheric science	Chen
Wed Oct 4	Edriss Titi (Cambridge/Texas A&M)	Distringuished Lecture Series	Smith, Stechmann
Oct 6	Polly Yu (Harvard/UIUC)	TBA	Craciun
Oct 13	Da Yang (University of Chicago)		Smith
Oct 20	Yuehaw Khoo (University of Chicago)		Li
Oct 27	Shukai Du (UW)	Element learning: a systematic approach of accelerating finite element-type methods via machine learning, with applications to radiative transfer	Stechmann
Nov 3	Lise-Marie Imbert-Gérard (University of Arizona)		Rycroft
Nov 10	Timothy Atherton (Tufts)		Chandler, Spagnolie
Nov 17	Daphne Klotsa		Rycroft
Nov 24	Thanksgiving break
Dec 1	Adam Stinchcombe (University of Toronto)		Cochran
Dec 8
Pending	Invite sent to Talea Mayo		Fabien

Abstracts

Erik Bollt (Clarkson University)

A New View on Integrability: On Matching Dynamical Systems through Koopman Operator Eigenfunctions

Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of non- linear dynamic behavior (e.g. through normal forms). In this presentation we will argue that the use of the Koopman operator and its spectrum are particularly well suited for this endeavor, both in theory, but also especially in view of recent data-driven machine learning algorithmic developments. Recall that the Koopman operator describes the dynamics of observation functions along a flow or map, and it is formally the adjoint of the Frobenius-Perrron operator that describes evolution of densities of ensembles of initial conditions. The Koopman operator has a long theoretical tradition but it has recently become extremely popular through numerical methods such as dynamic mode decomposition (DMD) and variants, for applied problems such as coherence and also in control theory. We demonstrate through illustrative examples that we can nontrivially extend the applicability of the Koopman spectral theoretical and computational machinery beyond modeling and prediction, towards a systematic discovery of rectifying integrability coordinate transformations.

John Schotland (Yale University)

Nonlocal PDEs and Quantum Optics

Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe a real-space formulation of quantum electrodynamics with applications to many body problems. The goal is to understand the transport of nonclassical states of light in random media. In this setting, there is a close relation to kinetic equations for nonlocal PDEs with random coefficients.

Balazs Boros (U Vienna)

Oscillatory mass-action systems

Mass-action differential equations are probably the most common mathematical models in biochemistry, cell biology, and population dynamics. Since oscillatory behavior is ubiquitous in nature, there are several papers (starting with Alfred Lotka) that deal with showing the existence of periodic solutions in mass-action systems. The standard way of proving the existence of a limit cycle in a high-dimensional system is via Andronov-Hopf bifurcation. In this talk, we recall some specific oscillatory models (like glycolysis or phosphorylation), as well as more recent results that aim to systematically classify small mass-action reaction networks that admit an Andronov-Hopf bifurcation.

Shukai Du (UW)

Element learning: a systematic approach of accelerating finite element-type methods via machine learning, with applications to radiative transfer

In the past decade, (artificial) neural networks and machine learning tools have surfaced as game changing technologies across numerous fields, resolving an array of challenging problems. Even for the numerical solution of partial differential equations (PDEs) or other scientific computing problems, results have shown that machine learning can speed up some computations. However, many machine learning approaches tend to lose some of the advantageous features of traditional numerical PDE methods, such as interpretability and applicability to general domains with complex geometry.

In this talk, we introduce a systematic approach (which we call element learning) with the goal of accelerating finite element-type methods via machine learning, while also retaining the desirable features of finite element methods. The derivation of this new approach is closely related to hybridizable discontinuous Galerkin (HDG) methods in the sense that the local solvers of HDG are replaced by machine learning approaches. Numerical tests are presented for an example PDE, the radiative transfer equation, in a variety of scenarios with idealized or realistic cloud fields, with smooth or sharp gradient in the cloud boundary transition. Comparisons are set up with either a fixed number of degrees of freedom or a fixed accuracy level of $10^{-3}$ in the relative $L^2$ error, and we observe a significant speed-up with element learning compared to a classical finite element-type method. Reference: arxiv: 2308.02467

Future semesters

• Spring 2024

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