Applied Math Seminar

Synopsis:

Liquid crystal polymeric networks (LCNs) are materials where a nematic liquid crystal is coupled with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Thin bodies of LCNs are natural candidates for soft robotics applications. Starting from the classical 3D trace formula energy of Bladon, Warner and Terentjev (1994), we derive a 2D membrane energy as the formal asymptotic limit of the 3D energy and characterize the zero energy deformations. The membrane energy lacks certain convexity properties, which presents challenges for the design of a numerical method. We discretize the problem with a finite element method and add a higher order bending energy regularization to address the lack of convexity. We prove that minimizers of the discrete energy converge to zero energy states of the membrane energy in the spirit of Gamma convergence. For minimizing the discrete problem, we employ an energy stable gradient flow scheme. We present computations showing the geometric effects that arise from liquid crystal defects and as well as computations of nonisometric origami.

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We propose and analyze a gradient descent algorithm with added stochastic terms for finding the global optimizers of nonconvex
optimization problems, The main feature of the algorithm is that the size of the randomness is tuned adaptively based on the value of the
objective function. We show, in the setup of the discrete algorithm not its continuous limit, that it is possible for the algorithm to achieve
global convergence with an algebraic rate. We will discuss possible generalizations of the algorithm for applications. This talk is based on joint works with Bjorn Engquist and Yunan Yang.

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Interface and mixing and their non-equilibrium kinetics and dynamics couple micro to macro scales, and are ubiquitous to occur in fluids, plasmas and materials, in high energy density regimes. Stellar evolution, plasma fusion, reactive fluids, purification of water, and nano-fabrication are a few examples of many processes to which dynamics of interfaces is directly relevant. This talk presents the rigorous theory of the stability of the interface – a phase boundary broadly defined. We directly link the structure of macroscopic flow fields to microscopic interfacial transport, quantify the contributions of macro and micro stabilization mechanisms to interface stability, and discover the fluid instabilities never previously discussed. In ideal and realistic fluids, the interface stability is set primarily by the interplay of the macroscopic inertial mechanism balancing the destabilizing acceleration, whereas microscopic thermodynamics create vortical fields in the bulk. By linking micro to macro scales, the interface is the place where balances are achieved.

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Clearly, the answer is no! Nevertheless, the Gaussian assumption remains prevalent among theoreticians, particularly in high-dimensional statistics and physics, less so in traditional statistical learning circles. To what extent are Gaussian features merely a convenient choice for certain theoreticians, or genuinely an effective model for learning? In this talk, I will review recent progress on these questions, achieved using rigorous probabilistic approaches in high-dimension and techniques from mathematical statistical physics. I will demonstrate that, despite its apparent limitations, the Gaussian approach is sometimes much closer to reality than one might expect. In particular, I will discuss key findings from a series of recent papers that showcase the Gaussian equivalence of generative models, the universality of Gaussian mixtures, and the conditions under which a single Gaussian can characterize the error in high-dimensional estimation. These results illuminate the strengths and weaknesses of the Gaussian assumption, shedding light on its applicability and limitations in the realm of theoretical machine learning.

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"Epsilon-near-zero" devices are a class of photonic devices (novel waveguides, scatterers, resonators) that operate at a frequency at which one of their components has dielectric permittivity nearly zero. I'll discuss recent results, obtained in joint work with Bob Kohn, about their robustness, and their shape design.

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We consider 2 problems where resonances (or lack of an expected resonance) yields insight into problems of waves in fluids. The first problem concerns whether mode-2 solitary waves exist in stratified flows. Except in degenerate cases, they are not expected to exist because of a resonance with shorter wavelength mode-1 waves which would lead to energy radiation. We show that, in interfacial models, such waves do exist at particular (discrete) amplitudes. The second problem concerns surface gravity waves in a cylindrical container. While this is a classic problem, it appears that the existence of general triad resonances was unknown (although Miles found certain 1:2 resonances in circular cylinders) perhaps because they do not exist in unbounded problems and in rectangular domains. We give a complete characterisation of such resonances, given the spectrum of the Laplacian on the cross-section of the cylinder. This demonstrates how boundaries can alter fundamental wave resonance properties.

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Increasingly sophisticated optimization and machine learning methods are being applied to solve some of the outstanding problems in the field of nuclear fusion. I will discuss some advances in system identification, sparse regression, and physics-informed neural networks and illustrate how they can be used to design novel models and new devices.

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In-situ measurements by nearly every spacecraft to date have found that the solar wind plasma consists of turbulent velocity and magnetic fluctuations undergoing a turbulence cascade over a broad range of length scales. For scales much larger than the proton gyroradius, this turbulence is believed to result from an incompressible Magnetohydrodynamics (MHD) cascade. In this sense, the solar wind provides us with a natural wind tunnel that we can probe to investigate several fundamental questions in magnetized plasma turbulence theory. The Parker Solar Probe (PSP) mission, launched in 2018, is presently probing the near-Sun solar wind with an orbit that will reach its point of closest approach at a heliospheric radius of approximately 9.8 solar radii by 2025, exploring the outermost portion of the solar corona where the solar wind originates. In this talk I will present a brief overview of how solar wind observations combined with numerical simulations have helped us advance our understanding of magnetized plasma turbulence before and after the PSP mission, as well as the new opportunities that PSP will usher in the coming decades to help us answer important outstanding questions in plasma turbulence.

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Turbulence of gravity waves (waves with domination of gravity force, capillary effects are
neglected) on the surface of sea or ocean is described by Hasselmann's waves kinetic equation. It is
used for wave forecasting: prediction of evolution of waves distribution function dynamics over
time. Some applications of it to practical problems will be discussed in introduction of the talk.
Verification of Hasselmann's waves kinetic equation together with understanding of its range of
applicability are important problems which motivate numerous numerical and laboratory experiments.
In many of such experiments formation of strong long wave background is observed. Also, observed
waves spectra in some cases are different from the ones predicted by the Waves Turbulence Theory
based on Hasselmann's waves kinetic equation and introduced by V.E. Zakharov in late 60's. Results
of massive numerical experiment (arXiv:2211.16567) are considered and explained analytically. We
used scale separation technique together with diffusion approximation in the space of wave vectors
to derive a new inverse cascade spectrum slope which is in good agreement with numerical results.

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Recent progress in experimental quantum optics has facilitated the physical construction of systems of increasing complexity. Of particular importance are experiments involving the scattering of one or two photons from a collection of atoms. In this context a central question is to understand the time evolution of the entanglement between atoms, mediated by the field. In this talk we will discuss analytical and experimental results on the properties of these systems, and how those properties depend on disorder or distribution of the locations of the atoms.

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In this talk, we will discuss about one particular type of iinverse problem regarding reconstructing the evolutional PDE model from
the internal solution data. We will characterize the nature of the problem depending the whether the PDE model is parabolic or hyperbolic. Then we will focus on the PDE learning problem for the radiative transport model which can be viewed as the transition between the two types of models. The related fast algorithms solving the inverse problem and related applications will be discussed as well.

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Recent years have witnessed tremendous progress in developing and analyzing quantum computing algorithms for quantum dynamics simulation of bounded operators (Hamiltonian simulation). However, many scientific and engineering problems require the efficient treatment of unbounded operators, which frequently arise due to the discretization of differential operators. Such applications include molecular dynamics, electronic structure theory, quantum control and quantum machine learning. We will introduce some recent advances in quantum algorithms for efficient unbounded Hamiltonian simulation, including Trotter type splitting and the quantum highly oscillatory protocol (qHOP) in the interaction picture. The latter yields a surprising superconvergence result for regular potentials. In the end, I will discuss briefly how Hamiltonian simulation techniques can be applied to a quantum learning task achieving optimal scaling. (The talk does not assume a priori knowledge on quantum computing.)

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