# Math Calendar

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May 1st, 2015 (01:30pm - 02:30pm)
Seminar: CCMA PDEs and Numerical Methods Seminar Series
Title: Finite elements and mimetic finite differences for some problems in H(div) and H(curl), and their multigrid solution
Speaker: Carmen Rodrigo, University of Zaragoza
Location: MB315

We consider mimetic finite difference discretizations of divergence, gradient and curl operators. These discrete grid operators satisfy compatibility conditions which naturally connect grad, div and curl operators on the continuous level. Our aim is to show how such mimetic finite difference schemes can be derived using standard finite element spaces in H(curl) and H(div). Once this relationship is understood, we can use the finite element framework to prove the convergence of the mimetic finite difference schemes, and to construct efficient multigrid methods for the solution of elliptic partial differential equations on H(curl) and H(div). In this way, we propose and analyze, via the local Fourier analysis framework, robust geometric multigrid algorithms for such problems on simplicial/Voronoi grids in a matrix-free fashion, which result in fast solvers with low memory requirements. Finally, we demonstrate the robustness and efficiency of such methods by presenting several numerical tests.

May 1st, 2015 (03:35pm - 04:35pm)
Seminar: Probability and Financial Mathematics Seminar
Title: A Poisson limit theorem arising from continued fractions
Speaker: Xuan Zhang, PSU
Location: MB106

Doeblin in 1940 found a Poisson limit theorem when he studied continued fractions. Roughly speaking, his theorem says that the number of large digits in the continued fraction representation of an irrational number under Lebesgue measure converges weakly to a Poisson distribution. We recover Doeblin’s theorem using dynamical method, which extends to more general settings.

May 5th, 2015 (01:00pm - 02:00pm)
Seminar: Theoretical Biology Seminar
Title: Robust exponential memory in Hopfield networks, NOTE: will be held on Wednesday, May 6th, at 2pm in MB 106
Speaker: Chris Hillar, Berkeley, Redwood Center
Location: MB106

The Hopfield recurrent neural network is a classical auto-associative model of memory, in which collections of symmetrically-coupled McCulloch-Pitts neurons interact to perform emergent computation. Although previous researchers have explored the potential of this network to solve combinatorial optimization problems and store memories as attractors of its deterministic dynamics, a basic open problem is to design a family of Hopfield networks with a number of noise-tolerant memories that grows exponentially with neural population size. Here, we discover such networks by minimizing probability flow, a recently proposed objective for estimating parameters in discrete maximum entropy models. By descending the gradient of the convex probability flow, our networks adapt synaptic weights to achieve robust exponential storage, even when presented with vanishingly small numbers of training patterns. In addition to providing a new set of error-correcting codes that achieve Shannon's channel capacity bound, these networks also efficiently solve a variant of the hidden clique problem in computer science, opening new avenues for real-world applications of computational models originating from biology. (joint work with N. Tran)

May 5th, 2015 (02:30pm - 03:30pm)
Seminar: GAP Seminar
Title: Irregular Riemann-Hilbert correspondence and Alekseev-Meinrenken dynamical r-matrix
Speaker: Xiaomeng Xu, University of Geneva
Location: MB106

In 2004, Enriquez-Etingof-Marshall suggested a new approach to the Ginzburg-Weinstein linearization theorem. This approach is based on solving a system of PDEs for a gauge transformation between the standard classical r-matrix and the Alekseev-Meinrenken dynamical r-matrix. In the talk, we explain that this gauge transformation can be constructed as a monodromy (connection matrix) for a certain irregular Riemann-Hilbert problem. Geometrically, this leads to a symplectic neighborhood version of the Ginzburg-Weinstein linearization theorem. Our construction is based on earlier works by Boalch. As an application, we give a new description of the Lu-Weinstein symplectic double.

May 6th, 2015 (02:00pm - 03:00pm)
Seminar: Theoretical Biology Seminar
Title: Robust exponential memory in Hopfield networks
Speaker: Chris Hillar (Hosts: Carina Curto & Vladimir Itskov), Berkeley, Redwood Center
Location: MB106
Abstract: http://

The Hopfield recurrent neural network is a classical auto-associative model of memory, in which collections of symmetrically-coupled McCulloch-Pitts neurons interact to perform emergent computation. Although previous researchers have explored the potential of this network to solve combinatorial optimization problems and store memories as attractors of its deterministic dynamics, a basic open problem is to design a family of Hopfield networks with a number of noise-tolerant memories that grows exponentially with neural population size. Here, we discover such networks by minimizing probability flow, a recently proposed objective for estimating parameters in discrete maximum entropy models. By descending the gradient of the convex probability flow, our networks adapt synaptic weights to achieve robust exponential storage, even when presented with vanishingly small numbers of training patterns. In addition to providing a new set of error-correcting codes that achieve Shannon's channel capacity bound, these networks also efficiently solve a variant of the hidden clique problem in computer science, opening new avenues for real-world applications of computational models originating from biology. (joint work with N. Tran)

May 6th, 2015 (03:00pm - 05:00pm)
Seminar: Ph.D. Oral Comprehensive Examination
Title: "Invariant sigma algebras under lattices"
Speaker: Oleg Rudenko, Adviser: Federico Rodriguez-Hertz, Penn State
Location: MB113

Margulis proved some results about algebras of Borel sets that are invariant under the action of a lattice in a semisimple group. I will discuss some possible extensions of this result as well as a related problem.

May 7th, 2015 (02:30pm - 03:30pm)
Seminar: Noncommutative Geometry Seminar
Title: Singular foliations and their C* algebras: calculations. 4.
Speaker: Iakovos Androulidaki, University of Athens
Location: MB106

Singular foliations are examples of dynamical systems. They are abundant in many branches of mathematics, for instance control theory and Poisson geometry. In fact singular foliations appear much more often than regular ones. In this series of talks we discuss how to deal with the leaf space of such foliations, including calculations of various examples. Information about this space is encapsulated in the holonomy groupoid of the foliation and the associated C*-algebra. A tentative program for these lectures is: (1) singular foliations and bisubmersions, with examples (foliation by the flow of a single vector field, by orbits of the SO(3) action, by orbits of the action of SL(2,R)), (b) calculation of the holonomy groupoid for the above examples, (c) construction of the foliation C*-algebra, and (d) K-theory calculation for the above examples (the right-hand side of the Baum-Connes assembly map).

May 11th, 2015 (09:00am - 11:00am)
Seminar: Ph.D. Thesis Defense
Title: "Stable Discretization and Robust Preconditioning for Fluid Structure Interactions"
Speaker: Kai Yang, Adviser: Jinchao Xu, Penn State
Location: MB106

In the simulation of multiphysics systems, we are often encountered with large scale linear systems arising from the implicit time discretization of coupled PDEs. Although it is possible to utilize the existing solvers for each field, systematic study has to be conducted in order to design fast solvers for the coupled systems. For large scale sparse linear systems, preconditioned Krylov subspace methods are usually the most efficient solvers. Preconditioning techniques are the key for these iterative solvers to have robust performance for various applications. Based on mapping property, we study the well-posedness of linear systems and then optimal preconditioners are developed. Using this procedure, we study the well-posedness of coupled linear systems of fluid-structure interactions and poroelasticity and propose robust block preconditioners. Besides the preconditioning techniques, we also introduce a new arbitrary Lagrangian Eulerian method for fluid-structure interactions with the structure part undergoing large rotation and small deformation. This technique provides a new approach to simulate hydroelectric power generator, artificial heart pump, etc.

May 13th, 2015 (03:00pm - 05:00pm)
Seminar: Ph.D. Thesis Defense
Title: "General Purpose Compositional Model, Simulation and Fast Linear Solver for Multiphase Reactive Flow"
Speaker: Changhe Qiao, Adviser: Jinchao Xu, Penn State
Location: MB106

Reservoir simulation is an important tool for petroleum engineers to optimize the management of oil fields and predict oil production. The real fluids in oil and gas reservoirs are complex. In many cases, the fluids are in three or more phases and contain over 100 species that can react with each other. Many existing models including black oil model, compositional model and reactive transport model were developed to simulate the different aspects of the fluids. In this dissertation, the author proposed a general modeling framework that covers different fluid types. Based on the general framework we developed a software package PennSim which requires minimal efforts for new fluid extension. Large scale reservoir simulation (more than 1 million grid blocks) spend over 90\% CPU time on linear solver part. The performance of iterative linear solver depends on the choice of preconditioner, the design of which depends on the knowledge of the PDEs. We degisned advanced multistage preconditioners based on the analysis on PDEs. New decoupling strategies were proposed for constraint pressure residual method and advantageous performance were demonstrated. We applied the general modeling framework to design noval models to explore the mechanisms of enhanced oil recovery (EOR). With coupled surface complexation reaction and multiphase flow, we proposed a numerical model for low salinity waterflooding for carbonates. The wettablility were predicted from detailed description of reactions and flow unit functions were predicted. Our model was the first predictive model for low salinity EOR in carbonates. We also proposed a novel model to predict the coupled compositional and reactive effects for water-alternating CO$2$ injection based on our framework. The altered injectivity were predicted for water with different compositions. We recommended simultaneous WAG (SWAG) as the best injection schemes for optimal injectivity.

May 18th, 2015 (03:00pm - 05:00pm)
Seminar: Ph.D. Thesis Defense
Title: "Nonlocal Models with Convection Effects"
Speaker: Zhan Huang, Adviser: Qiang Du, Penn State
Location: MB106

Our work consists of two parts: the first part works on the volume-constraint problems associated with nonlocal linear convection-diffusion equations, and the second part studies initial-value problems of scalar nonlocal hyperbolic conservation laws. In both cases, we show the consistency of nonlocal models with the corresponding local problems from various analytical and numerical aspects, and numerical experiments based on Finite Difference methods are performed to investigate the behavior of nonlocal solutions.