September 25, 2019
Jose Antonio Carrillo
▦ Primal dual methods for Wasserstein gradient flows ▦
   Combining the classical theory of optimal transport with modern operator splitting techniques, I will present a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media,materials science, and biological swarming. Using the JKO scheme, along with the Benamou-Brenier dynamical characterization of the Wasserstein distance, we reduce computing the solution of these evolutionary PDEs to solving a sequence of fully discrete minimization problems, with strictly convex objective function and linear constraint. We compute the minimizer of these fully discrete problems by applying a recent, provably convergent primal dual splitting scheme for three operators. By leveraging the PDE’s underlying variational structure, ourmethod overcomes traditional stability issues arising from the strong nonlinearity and degeneracy, and it is also naturally positivity preserving and entropy decreasing. Furthermore, by transforming the traditional linear equality constraint, as has appeared in previous work, into a linear inequality constraint, our method converges in fewer iterations without sacrificing any accuracy. Remarkably, our method is also massively parallelizable and thus very efficient in resolving high dimensional problems. We prove that minimizers of the fully discrete problem converge to minimizers of the continuum JKO problem as the discretization is refined, and in the process, we recover convergence results for existing numerical methods for computing Wasserstein geodesics. Finally, we conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our numerical method.  

October 16, 2019
Rupert Frank
▦ A `Liquid-Solid' Phase Transition in a Simple Model for Swarming ▦
   We consider a non-local shape optimization problem, which is motivated by a simple model for swarming and other self-assembly/aggregation models, and prove the existence of different phases. In particular, we show that in the large mass regime the ground state density profile is the characteristic function of a round ball. An essential ingredient in our proof is a strict rearrangement inequality with a quantitative error estimate.

The talk is based on joint work with E. Lieb.  

October 23, 2019
Steven Low
▦ Mitigation of Cascading Failures in Power Systems ▦
   Line failure in power grid propgates in non-local, intricate and counterintuitive ways because of the interplay between power flow physics and network topology, making the mitigation of cascading failure difficult. The conventional approach to grid reliability is through building redundant lines. In this talk, we present an opposite approach to grid reliability through failure localization, by judiciously removing lines and adopting a new class of frequency control algorithms at real time. The topology design partitions the network into regions that are connected in a tree structure. The frequency control automatically adjusts controllable generators and loads to minimize disruption and localize failure propagation. This approach is derived from a spectral view of power flow equations that relates failure propagation to the graphical structure of the grid through its Laplacian matrix. We summarize the underlying theory and present simulation results that demonstrate that our approach not only localizes failure propagation, as promised by the theory, but also improves overall grid reliability even though it reduces line redundancy.

(Joint work with Daniel Guo, Chen Liang, Alessandro Zocca, and Adam Wierman)

October 30, 2019
Gianluca Favre
▦ Kinetic model with thermalization for a gas with total energy conservation ▦
   We consider the thermalization of a gas towards a Maxwellian velocity distribution which depends locally on the temperature of the background. The exchange of kinetic and thermal energy between the gas and the background drives the system towards a global equilibrium with constant temperature. The heat flow is governed by the Fourier's law. Mathematically we consider a coupled system of nonlinear kinetic and heat equations where in both cases we add a term that describes the energy exchange. For this problem we are able to prove existence of the solution in 1D, exponential convergence to the equilibrium through a hypocoercivity technique, macroscopic limit toward a cross-diffusion system. In the last two cases a perturbative approach is taken into account. It's worth noticing that also without heat conductivity we can show the temperature diffusion thanks to the transport of energy. It is also interesting to show that the thermalization is highly influenced by the background temperature. All these aspects have been investigated also from a numerical viewpoint in order to provide simulations in 2D.

January 22, 2020
Richard Kueng
▦ TBA ▦
   

January 29, 2020
Speaker TBD
▦ TBA ▦
   

February 26, 2020
Speaker TBD
▦ TBA ▦
   

April 15, 2020
Speaker TBD
▦ TBA ▦
   

April 22, 2020
Speaker TBD
▦ TBA ▦
   

April 29, 2020
Niles Pierce
▦ TBA ▦
   

May 20, 2020
Speaker TBD
▦ TBA ▦
   

August 22, 2019
• Special CMX Seminar •

Annenberg 213
12:00pm

Giacomo Garegnani
▦ Bayesian Inference of Multiscale Differential Equations ▦

      Inverse problems involving differential equations defined on multiple scales naturally arise in several engineering applications. The computational cost due to discretization of multiscale equations can be reduced employing homogenization methods, which allow for cheaper computations. Nonetheless, homogenization techniques introduce a modelling error, which has to be taken into account when solving inverse problems. In this presentation, we consider the treatment of the homogenization error in the framework of inverse problems involving either an elliptic PDE or a Langevin diffusion process. In both cases, theoretical results involving the limit of oscillations of vanishing amplitude are provided, and computational techniques for dealing with the modelling error are presented.

November 19, 2019
• Special CMX Seminar •

Annenberg 213
4:30pm

Matthew Thorpe
▦ How Many Labels Do You Need For Semi-Supervised Learning? ▦

      Given a data set of which a small subset are labelled, the goal of semi-supervised learning is to find the unknown labels. A popular method is to minimise a discrete p-Dirichlet energy defined on a graph constructed from the data. As the size of the data set increases one hopes that solutions of the discrete problem converge to a continuum variational problem with the continuum p-Dirichlet energy. It follows from Sobolev regularity that one cannot impose constraints if p is less than the dimension of the data hence, in this regime, one must also increase the number of labels in order to avoid labels "dissappearing" in the limit. In this talk I will address the question of what is the minimal number of labels. To compare labelling functions on different domains we use a metric based on optimal transport which then allows for the application of methods from the calculus of variation, in particular Gamma-convergence, and methods from PDE's, such as constructing barrier functions in order to apply the maximum principle. We can further show rates of convergence. This is joint work with Jeff Calder (Minnesota) and Dejan Slepcev (CMU).

December 6, 2019
• CMX Special Seminar •

Annenberg 213
4:00pm

Jose Antonio Carrillo
▦ Consensus Based Models and Applications to Global Optimization ▦

      We introduce a novel first-order stochastic swarm intelligence (SI) model in the spirit of consensus formation models, namely a consensus-based optimization (CBO) algorithm, which may be used for the global optimization of a function in multiple dimensions. The CBO algorithm allows for passage to the mean-field limit, which results in a nonstandard, nonlocal, degenerate parabolic partial differential equation (PDE). Exploiting tools from PDE analysis we provide convergence results that help to understand the asymptotic behavior of the SI model. We further present numerical investigations underlining the feasibility of our approach with applications to machine learning problems.

December 12, 2019
• CMX Special Seminar •

Annenberg 213
4:30pm

Zhenzhen Li
▦ A New Non-convex Optimization Framework For Low-rank Matrix Recovery with Mathematical Guarantee  ▦

      Recent years have witnessed growing importance of non-convex methods in many industrial and practical problems. Many low-rank related problems can be solved by reformulating them as nonconvex optimization problems. Surprisingly, these optimization problems usually do not have spurious local minima and all saddle points are strict under the Euclidean parameterized setting. Although a dimension-free polynomial convergence can be guaranteed for many problems, numerical experiments have demonstrated much better performance than what the current theory predicts. Different from previous non-convex methods (with weaker theoretical convergence guarantee) or convex relaxation (with heavier computational cost), in this talk, we will discuss a new global non-convex optimization framework for solving a general inverse problem of low-rank matrices under the Riemannian manifold setting. Given some random measurements of a low-rank matrix, how does a least square loss work via the Riemannian gradient descent (light computational cost version) with random initialization? Our analysis gives rigorous mathematical analysis with respect to both asymptotic convergence behavior and fast convergence rate under isometry or weaker isometry condtions. More specifically, under isometry case a low-rank matrix manifold with rank r (r << n) consists of 2^r branches. We will show that a random initialization with probability 1 falls into an intrinsic branch. Further, it needs O(log n+log(1/epsilon)) iterations to generate an epsilon-accuracy solution. Similar results also hold for low rank matrix recovery by given some random information under mild conditions (weaker isometry case). Potential applications include but not limited to low-rank matrix related problems, such as matrix sensing, matrix completion, low-rank Hankel matrix recovery, phase retrieval, robust PCA, and non-convex flow, etc.

January 10, 2020
• CMX Special Seminar •

Annenberg 213
12:00pm

Mitch Luskin
▦ Modeling and Computation for Trilayer Graphene  ▦

     

January 13, 2019
• CMX Special Seminar •

Annenberg 213
3:00pm

Dongho Chae
▦ On Type I blow-up for the incompressible Euler equations ▦

      In this talk we discuss the Type I blow-up and the related problems of the Euler equations. Small Type I conditions are easily excluded. In the case of the Euler equations Type I condition without smallness implies exclusion of atomic concentration of energy. As an important corollary we remove discretely self-similar blow-up in the energy conserving scale. We also discuss the Type I condition for the 2D Boussinesq system. The localized version of this result leads to removing some of Type II blow-up in the axisymmetric Euler equations off the axis.

(These results are joint works with J. Wolf.)

Southern California Applied Mathematics Symposium (SOCAMS 2019)

• Meeting Poster