PDE & Applied math seminar
Wednesday 10:00 -10:50 am, on zoom
Organizers : Weitao Chen / Heyrim Cho / Yat Tin Chow / Qixuan Wang / Jia Guo / Mykhailo Potomkin
In fall 2021, the PDE & Applied math seminar will be held both through zoom and in person. Specific information about format of each talk will be provided in the email announcement and posted below. If you're interested in attending the seminar, please contact Dr. Yat Tin Chow (yattinc@ucr.edu) or Mykhailo Potomkin (mykhailp@ucr.edu).
Fall 2021 Schedule
Sep 29 10:00 AM (Wed) Organization meeting
Oct 6 10:00 AM (Wed) Dr. Alpár Richárd Mészáros, Durham University, United Kingdom
Oct 14 4:40 PM (Thu) Dr. Ernest Ryu, Seoul National University, Korea (!!! Note unusual day & time)
Oct 20 10:00 AM (Wed) Dr. Samy Wu Fung, Colorado School of Mine
Nov 3 10:00 AM (Wed) Dr. Glenn Young, Kennesaw State University
Nov 10 4:00 PM (Wed) Dr. Alexander Gavrikov, Pennsylvania State University (!!! Note unusual time)
Nov 17 10:00 AM (Wed) Dr. Franziska Weber, Carnegie Mellon University
Nov 24 10:00 AM (Wed) Dr. Jose Rodrigues, University of Memphis
Dec 1 4:00 PM (Wed) Dr. Wojciech Ozanski, University of Southern California (!!! Note unusual time)
September 29, 2021, 10:00-10:50 AM PT
Dr. Yat Tin Chow (UC Riverside) and Dr. Mykhailo Potomkin (UC Riverside)
Title: Organizational meeting
October 6, 2021, 10:00 - 10:50 AM PT
Dr. Alpár Richárd Mészáros, Durham University, United Kingdom
Title: Mean Field Games and Master Equations
Abstract: The theory of mean field games has been initiated around 15 years ago by Lasry-Lions on the one hand and by Huang-Malhamé-Caines on the other hand. The main goal of both groups (inspired by the mean field models from statistical physics) was to characterize limits of Nash equilibria of stochastic differential games, when the number of agents tends to infinity. Since then, this theory has witnessed a great success, both theoretically and from the point of view of applications.
In this talk we take a journey into this field, starting with the derivation of the main systems of PDEs, which characterize the mentioned limits of the equilibria. Then, we present the so-called master equation, which was first introduced by Lions. This is an infinite dimensional PDE set on the space of Borel probability measures, which encodes all the properties of the underlying game. Because of their infinite dimensional nature, many new challenges arise regarding the solvability of these equations. In the second half of the talk, we will discuss how different notions of convexity/monotonicity on the data could lead to the global in time well-posedness of these equations. Our main results in this direction have been obtained recently in collaboration with W. Gangbo (UCLA) on the one hand and with W. Gangbo, C. Mou (City U, Hong Kong) and J. Zhang (USC) on the other hand.
October 14 (!Thursday!), 2021, 4:40 - 5:30 PM PT (Note unusual evening time & day!!!)
Dr. Ernest Ryu, Seoul National University, Korea
Title: Accelerated Algorithms for Smooth Convex-Concave Minimax Problems with O(1/k^2) Rate on Squared Gradient
Abstract: In this work, we study the computational complexity of reducing the squared gradient magnitude for smooth minimax optimization problems. First, we present algorithms with accelerated O(1/k^2) last-iterate rates, faster than the existing O(1/k) or slower rates for extragradient, Popov, and gradient descent with anchoring. The acceleration mechanism combines extragradient steps with anchoring and is distinct from Nesterov's acceleration. We then establish optimality of the O(1/k^2) rate through a matching lower bound.
October 20, 2021, 10:00 - 10:50 AM PT
Dr. Samy Wu Fung, Colorado School of Mines
Title: Efficient Training of Infinite-Depth Neural Networks via Jacobian-Free Backpropagation
Abstract: A promising trend in deep learning replaces fixed depth models by approximations of the limit as network depth approaches infinity. This approach uses a portion of network weights to prescribe behavior by defining a limit condition. This makes network depth implicit, varying based on the provided data and an error tolerance. Moreover, existing implicit models can be implemented and trained with fixed memory costs in exchange for additional computational costs. In particular, backpropagation through implicit networks requires solving a Jacobian-based equation arising from the implicit function theorem. We propose a new Jacobian-free backpropagation (JFB) scheme that circumvents the need to solve Jacobian-based equations while maintaining fixed memory costs. This makes implicit depth models much cheaper to train and easy to implement. Numerical experiments on classification are provided.
November 3, 2021, 10:00 - 10:50 AM PT
Dr. Glenn Young, Kennesaw State University
Title: The interplay between costly reproduction and unpredictable environments shapes the dynamics and stability of cooperative breeding
Abstract: All sexually reproducing organisms are faced with a fundamental decision: to invest valuable resources and energy in reproduction or in their own survival. This trade-off between reproduction and survival represents the 'cost of reproduction' and occurs across a diverse range of organisms. It is widely assumed that cooperative breeding behavior in vertebrates — when individuals care for young who are not their own — results in part from costly parental care. When caring for young is too costly, parents need help from related or unrelated individuals to successfully raise their offspring. Cooperatively breeding birds and mammals are also more commonly found in unpredictable environments than non-cooperative species, suggesting that decisions about when to breed or help may represent complex yet critical choices that depend on the energy individuals have available to dedicate to reproduction given the harshness of the current environment. Here, we introduce a novel, socially-tiered model of a cooperatively breeding species that incorporates the influence environmental stochasticity. Through numerical and analytical methods, we use this model to show that costly reproduction and environmental variability are compounding factors in the evolution and maintenance of cooperation.
November 10, 2021, 4:00 - 4:50 PM PT (Note unusual evening time!!!)
Dr. Alexander Gavrikov, Pennsylvania State University
Title: Modeling the Bacterial Motion in Viscoelastic Liquid Crystals
Abstract: We study numerically the motion of a self-propelled elliptic particle in a viscoelastic liquid-crystalline media modeling bacterial motion in biofluids. The liquid crystal is described via the Beris–Edwards system coupled with PDEs for the conformation tensor taking into account viscoelastic behavior. The computational model is studied in a periodic setting that allows for the implementation of the fast Fourier transform for finding the director field and viscoelastic stresses. The Stokes equation describing the flow of the media is solved using the boundary integral method. We compare our numerical results with known experimentally observed as well as numerically predicted behavior. This is the joint work with Hai Chi, Igor Aronson, and Leonid Berlyand.
November 17, 2021, 10:00 - 10:50 AM Pacific Time
Dr. Franziska Weber, Carnegie Mellon University
Title: A Convergent Numerical Method for a Model of Liquid Crystal Director Coupled to An Electric Field
Abstract: Starting from the Oseen-Frank theory, we derive a simple model for the dynamics of a nematic liquid crystal director field under the influence of an electric field. The resulting nonlinear system of partial differential equations consists of the electrostatics equations for the electric field coupled with the damped wave map equation for the evolution of the liquid crystal director field, which is a normal vector pointing in the direction of the main orientation of the liquid crystal molecules. The liquid crystal director field enters the electrostatics equations in the constitutive relations while the electric field enters the wave map equation in the form of a nonlinear source term. Since it is a normal vector, the variable for the liquid crystal director field has to satisfy the constraint that it takes values in the unit sphere. We derive an energy-stable and constraint preserving numerical method for this system and prove convergence of a subsequence of approximations to a weak solution of the system of partial differential equations. In particular, this implies the existence of weak solutions for this model.
November 24, 2021, 10:00 - 10:50 AM Pacific Time
Dr. Jose Rodrigues, University of Memphis
Title: On the attractors for a nonlinear wave equation with critical exponents
Abstract: In this talk, we are going to present the main results concerning the well-posedness and long-time dynamics for the solutions of a nonlinear dissipative wave equation. The nonlinearities presented here are taken up to a critical level and, therefore, require more sensitive and elaborated approach. Our purpose here is to promote the main tools used in the study of attractors of such very interesting PDE models, such as asymptotic compactness/smoothness and ultimately dissipativity.
December 1, 2021, 4:00 - 4:50 PM Pacific Time (!!! Note unusual evening time)
Dr. Wojciech Ozanski, University of Southern California
Title: Well-posedness of logarithmic spiral vortex sheets
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydround Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spirals: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals’ arms with respect to a given point in the plane.