Applied/ACMS

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Applied and Computational Mathematics Seminar


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) A Spatiotemporal Model of GPCR-G protein Interactions Craciun
Oct 13 Da Yang (University of Chicago) The Incredible Lightness of Water Vapor Smith
Oct 19 (*Thursday* *1:25pm*) Jiaxin Jin (The Ohio State University) On the Dimension of the R-Disguised Toric Locus of a Reaction Network Craciun
Oct 20 Yuehaw Khoo (University of Chicago) Randomized tensor-network algorithms for random data in high-dimensions 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) Wave propagation in inhomogeneous media with quasi-Trefftz methods Rycroft
Nov 10 Timothy Atherton (Tufts) Shape optimization and shapeshifting in soft matter Chandler, Spagnolie
Nov 17 Daphne Klotsa A touch of non-linearity: mesoscale swimmers and active matter in fluids

Zoom link: https://uwmadison.zoom.us/j/94647635488?pwd=eWNJSm5Xc2VyWklJSnY4Nkx2YVhMQT09 (Passcode: 247963)

Rycroft
Nov 24 Thanksgiving break
Dec 1 Adam Stinchcombe (University of Toronto) Cochran
Pending Invite sent to Talea Mayo Fabien
Pending Invite sent to Tony Kearsley 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.


Peter Jan van Leeuwen (Colorado State University)

Nonlinear Causal Discovery, with applications to atmospheric science

Understanding cause and effect relations in complex systems is one of the main goals of scientific research. Ideally, one sets up controlled experiments in which different potential drivers are varied to infer their influence on a target variable. However, this procedure is impossible in many systems, for example the atmosphere, where nature is doing one experiment for us. An alternative is to build a detailed computer model of the system, and perform controlled experiments in model world. An issue there is that one can only control external drivers, because controlling an internal variable would kill all feedbacks to that variable, resulting in a study of ‘a different planet’. Because many natural systems cannot be controlled, or only partially, we focus on causal discovery in systems that are non-intervenable. I will describe a non-linear causal discovery framework that is based on (conditional) mutual information. It will be shown that conventional analysis of causal relations via so-called Directed Acyclic Graphs (DAGs, se e.g. Pearl and others) is not suitable for nonlinear systems, and an extension is provided that allows for interacting drivers. I prove that the interacting contributions and interaction informations, and provide a solid interpretation of those, in terms of buffering, hampering, and positive feedbacks. Also ways to infer completeness of the causal networks will be discussed, as well as causal relations that are invisible to our framework. The framework will be applied to simple idealized cloud models, and to real very detailed ground-based remote-sensing observations of cloud properties, where we contrast the causal structure of precipitating and non-precipitation strato-cumulus clouds.


Polly Yu (Harvard)

A Spatiotemporal Model of GPCR-G protein Interactions

G-protein coupled receptors (GPCRs) is a class of transmembrane receptors important to many signalling pathways and a common drug target. As its name suggests, the receptor, once activated, binds to a G-protein. Recent experiments suggests that GPCRs form dense tiny clusters. What are the effects of these "hotspots" on signalling kinetics? I will introduce a semi-empirical spatiotemporal model for GPCR-G protein interactions, and present some numerical evidence for how these clusters might locally increase signalling speed.


Da Yang (University of Chicago)

The Incredible Lightness of Water Vapor

Conventional wisdom suggests that warm air rises while cold air sinks. However, recent satellite observations show that, on average, rising air is colder than sinking air in the tropical free troposphere. This is due to the buoyancy effect of water vapor: the molar mass of water vapor is less than that of dry air, making humid air lighter than dry air at the same temperature and pressure. Unfortunately, this vapor buoyancy effect has been considered negligibly small and thereby overlooked in large-scale climate dynamics. Here we use theory, reanalysis data, and a hierarchy of climate models to show that vapor buoyancy has a similar magnitude to thermal buoyancy in the tropical free troposphere. As a result, cold air rises in the tropical free troposphere. We further show that vapor buoyancy enhances thermal radiation, increases subtropical stratiform low clouds, favors convective aggregation, and stabilizes Earth’s climate. However, some state-of-the-art climate models fail to represent vapor buoyancy properly. This flaw leads to inaccurate simulations of cloud distributions—the largest uncertainty in predicting climate change. Implications of our results on paleoclimate and planetary habitability will also be discussed.


Jiaxin Jin (The Ohio State University)

On the Dimension of the R-Disguised Toric Locus of a Reaction Network

The properties of general polynomial dynamical systems can be very difficult to analyze, due to nonlinearity, bifurcations, and the possibility for chaotic dynamics. On the other hand, toric dynamical systems are polynomial dynamical systems that appear naturally as models of reaction networks and have very robust and stable properties. A disguised toric dynamical system is a polynomial dynamical system generated by a reaction network and some choice of positive parameters, such that it has a toric realization with respect to some other network. Disguised toric dynamical systems enjoy all the robust stability properties of toric dynamical systems. In this project, we study a larger set of dynamical systems where the rate constants are allowed to take both positive and negative values. More precisely, we analyze the R-disguised toric locus of a reaction network, i.e., the subset in the space rate constants (positive or negative) for which the corresponding polynomial dynamical system is disguised toric. In particular, we construct homeomorphisms to provide an exact bound on the dimension of the R-disguised toric locus.


Yuehaw Khoo (Chicago)

Randomized tensor-network algorithms for random data in high-dimensions

Tensor-network ansatz has long been employed to solve the high-dimensional Schrödinger equation, demonstrating linear complexity scaling with respect to dimensionality. Recently, this ansatz has found applications in various machine learning scenarios, including supervised learning and generative modeling, where the data originates from a random process. In this talk, we present a new perspective on randomized linear algebra, showcasing its usage in estimating a density as a tensor-network from i.i.d. samples of a distribution, without the curse of dimensionality, and without the use of optimization techniques. Moreover, we illustrate how this concept can combine the strengths of particle and tensor-network methods for solving high-dimensional PDEs, resulting in enhanced flexibility for both approaches.


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


Lise-Marie Imbert-Gérard (University of Arizona)

Wave propagation in inhomogeneous media with quasi-Trefftz methods

Trefftz methods rely, in broad terms, on the idea of approximating solutions to Partial Differential Equation (PDEs) via Galerkin methods using basis functions which are exact solutions of the PDE, making explicit use of information about the ambient medium. But wave propagation in inhomogeneous media is modeled by PDEs with variable coefficients, and in general no exact solutions are available.

Quasi-Trefftz methods have been introduced, in the case of the Helmholtz equation with variable coefficients, to address this problem: they rely not on exact solutions to the PDE but instead of high order approximate solutions constructed locally. We will discuss the origin, the construction, and the properties of these so-called quasi-Trefftz functions. We will also discuss the consistency error introduced by this construction process.


Timothy Atherton (Tufts University)

Shape optimization and shapeshifting in soft matter

Soft materials are ubiquitous in everyday life and are ideal candidates for advanced engineering applications including soft, biomimetic robots, self-building machines, shape-shifters, artificial muscles, and chemical delivery packages. In many of these, the material must make a dramatic change in shape with an accompanying re-ordering of the material; in others changes in the ordering can be used to drive or even interrupt shape change. To optimize the materials and structures, it is necessary to have a detailed understanding of how the microstructure and macroscopic shape co-evolve. In this talk, I will therefore discuss the interactions between order and shape evolution with examples primarily drawn from my group's work on emulsions, liquid crystals and other soft materials. All of these efforts draw upon mathematics, including finite elements, optimization theory and beyond.


Daphne Klotsa (UNC Chapel Hill)

A touch of non-linearity: mesoscale swimmers and active matter in fluids

Living matter, such as biological tissue, can be seen as a nonequilibrium hierarchical assembly of assemblies of smaller and smaller active components, where energy is consumed at many scales. The remarkable properties of such living or “active-matter” systems make them promising candidates to study and synthetically design. While many active-matter systems reside in fluids (solution, blood, ocean, air), so far, studies that include hydrodynamic interactions have focussed on microscopic scales in Stokes flows. At those microscopic scales viscosity dominates, and inertia can be neglected. What happens as swimmers slightly increase in size (say ~0.1mm-100cm) or as they form larger aggregates and swarms? The system then enters the intermediate Reynolds regime where both inertia and viscosity play a role, and where nonlinearities are introduced in the fluid. In this talk, I will present a simple model swimmer used to understand the transition from Stokes to intermediate Reynolds numbers, first for a single swimmer, then for pairwise interactions and finally for collective behavior. We show that, even for a simple model, inertia can induce hydrodynamic interactions that generate novel phase behavior, steady states and transitions.

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