Computational and Applied Mathematics (CAM) Seminar

Fall, 2016 – Spring, 2017

CAM seminar talks are held on Wednesday from 2:00-3:00PM in Snow Hall 306, unless otherwise noted.

Previous CAM talks: Fall, 2006; Spring, 2007; Fall, 2007; Spring, 2008; Fall, 2008; Spring, 2009;

2009-2010; 2010-2011; 2011-2012 2012-2013 2013-2014 2014-2015 2015-2016

Date	Speaker/Institution	Title and Abstract
Aug 31	Organizational meeting
Sept 7	Xiang Wang (Jilin University, China)	Title: L2 error estimates for high order finite volume methods on triangular meshes Abstract: Due to the local conservation property, the finite volume method (FVM) has enjoyed great popularity among scientific and engineering computation. However, compared to its wide applications, the development of FVM theory lags far behind, especially for high order schemes on triangular meshes. We establish a unified framework for L2 error analysis for high order Lagrange finite volume methods on triangular meshes. Orthogonal conditions are proposed to construct dual partitions on triangular meshes so that the corresponding FVMs hold optimal L2 norm convergence order.
Sept 14	Ashish Pandey (UIUC)	Title: Modulational instability in nonlinear dispersive equations. Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM and regularized Boussinesq type equations.
Sept 21	Geng Chen	Title: Large solutions of compressible Euler equations Abstract: Compressible Euler equations (introduced by Euler in 1757) model the motion of compressible inviscid fluids such as gases. It is well-known that solutions of compressible Euler equations often develop discontinuities, i.e. shock waves. Successful theories have been established in the past 150+ years for small solutions in one space dimension. The theory on large solutions is widely open for a long time, even in one space dimension. In this talk, I will discuss some recent exciting progresses in this direction. In the first part of this talk, I will discuss our complete resolution of shock formation problem, which extends the celebrated work of Peter Lax in 1964. Our result relies on the discovery of a sharp time-dependent lower bound on density, when solutions approach vacuum in infinite time. In the second part, I will show our recent negative example concerning the failure of current available frameworks on approximate solutions in order to establish large BV (bounded total variation) theory. The talk is based on my joint works with A. Bressan, H.K. Jenssen, R. Pan, R. Young, Q. Zhang, and S. Zhu. This talk is accessible for graduate students. The connection between our results and Riemann’s original paper in 1859 will also be mentioned.
Sept 28	Weizhang Huang	Title: Conditioning of finite element equations with arbitrary nonuniform meshes Abstract: Mesh adaptation has become an indispensable tool for use in the numerical solution of partial differential equations to improve computational accuracy and efficiency. However, mesh adaptation often leads to nonuniform meshes and their nonuniformity can have significant impacts on the conditioning and the efficient solution of the underlying algebraic systems. In this talk we will present some new results in the studies of those impacts for the finite element approximation of boundary value and initial-boundary value problems of linear diffusion equations.
Oct 5	Hang Si (WIAS, Berlin)	Title: An Introduction to Delaunay-based Mesh Generation and Adaptation Abstract: Mesh generation and adaptation are key steps in many applications such as numerical methods like finite element and finite volume methods. It is itself a research topic with background in mathematics, computer science, and engineering. Delaunay triangulation has many nice properties and is popularly used in many mesh generation methods. In this talk, we will begin with triangle mesh generation in the plane. This problem has been very well studied. Efficient algorithms are developed. We will then move to tetrahedral mesh generation in 3d, which is still challenged by many theoretical and practical issues. We will introduce classical and recent algorithms that are both theoretically correct and efficient in practice. Various examples are illustrated using open source software Triangle and TetGen.
Oct 12	Andrew Roberts (Cerner Corporation)	Title: Mathematics in Healthcare and The Challenges of Real-World Data Abstract: You may be familiar with methods like regression, decision trees, survival analysis, and time-series analysis. In academia, I was accustomed to implementing methods with nice data. What do you do when you have missing or mislabeled data? How do you analyze a time-series where the time between observations could range from 30 minutes to multiple hours? This presentation will highlight some of the challenges that the Cerner Math team faces and the nuances of using familiar methods in real-world examples.
Oct 19	German Lozada-Cruz (Sao Paulo State University- UNESP, Brazil)	Title: Continuity of the set of equilibria Abstract: In this talk we will treat about the continuity of the set of equilibria of a parabolic PDE with homogeneous Dirichlet boundary conditions via the discretization of finite element method.
Oct 26	Mat Johnson	Title: Stability and Long Time Modulational Dynamics of Periodic Waves in Dissipative Systems Abstract: The capability of spatially periodic waves to cary modulation signals makes their dynamics under perturbation rich in multi-scale phenomena and essentially infinite dimensional. Here, I will discuss recent progress in the understanding of the stability and local dynamics of periodic waves capable of carrying multiple modulation signals in dissipative models, and in particular how (locally) the long time dynamics are approximately governed by an averaged system of equations obtained through a nonlinear WKB process.
Nov 2	Junbo Cheng (Institute of Applied Physics and Computational Mathematics, Beijing, China)	Title: Approximate Riemann solvers and the high-order cell-centered Lagrangian schemes for elastic-plastic flows Abstract: For elastic-plastic flows with the hypo-elastic constitutive model and von-Mises' yielding condition, the non-conservative character of the hypo-elastic constitutive model and the von-Mises' yielding condition make the construction of the solution to the Riemann problem a challenging task. In this talk, I will first present analysis for the wave structure of the Riemann problem and develop accordingly a two-rarefaction Riemann solver with elastic waves for 1D elastic-plastic flows (TRRSE) and a four-rarefaction wave approximate Riemann solver with elastic waves (FRRSE) for 2D elastic-plastic flows. Because of the complexities of the equation of state and the discontinuities around the elastic limit, we have to use Gaussian quadrature method to evaluate the integral term of Riemann invariant variables. Moreover, it is impossible to get the exact solution of Riemann solvers, it is necessary to use Newton Iteration method to obtain the convergent Riemann solvers. Besides, for the 2D elastic-plastic flows, in the construction of FRRSE one needs to use a nested iterative method. A direct iteration procedure for four variables is complex and computationally expensive. In order to simplify the solution procedure we develop an iteration based on two nested iterations upon two variables, and our iteration method is simple in implement and efficient. Because the iteration is used during constructing TRRSE, it is expensive in CPU time. So, we build a HLLC approximate Riemann solver with elastic waves (HLLCE) for one-dimensional elastic-plastic flows. Based on TRRSE and HLLCE, we build the third-order cell-centered Lagrangian numerical schemes; Based on FRRSE as a building block, we therefore propose a 2nd-order cell-centered Lagrangian numerical scheme. A numerical result with smooth solutions shows our scheme achieves the desired convergent order. Moreover, a number of numerical experiments with shock and rarefaction waves demonstrate the scheme is essentially non-oscillatory and appears to be convergent. Moreover, for shock waves the present scheme has comparable accuracy to that of the scheme developed by Maire et al, while it is more accurate in resolving rarefaction waves.
Nov 9	Fola Agosto (KU Ecology and Evolutionary Biology)	Title: Mathematical Model for Zika Virus Dynamics with Sexual Transmission Pathway
Nov 16	Lam Mountaga (Cheikh Anta Diop University, Senegal)	Title: Optimal intervention strategies of a SI-HIV models with r-differential infectivity and two time delays Abstract: Retarded optimal control theory is applied to a system of delays ordinary differential equations modeling a HIV model with differential infectivity. Seeking to reduce the infective individuals with high viral load, we use control representing the fraction of infective individuals that is identified and will be put under treatment. The optimal controls are characterized in terms of the optimality system, which is solved numerically.
Nov 23	Thanksgiving
Nov 30	Weishi Liu	Title: Nonlocal nature of excess potentials and boundary value problems of Poisson-Nernst-Planck systems
Dec 7	Hongguo Xu	Title: Sign characteristics of Hermitian matrix polynomials Abstract: Sign characteristic is a concept that is essential for the stability analysis in Hamiltonian systems and the perturbation behavior of eigenvalues under structured perturbations. In this talk, we define sign characteristic for infinite eigenvalues. We show the behavior of sign characteristics under changes of variables and also a signature constraint theorem. This is a joint work with Volker Merman, Vanni Noferini, and Francois Tisseur.
Jan 18	Organizational meeting
Jan 25	Tao Huang (NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai)	Title: Singularities and nonuniqueness of nematic liquid crystal flows Abstract: In this talk, we will discuss two examples of finite time blowups for the simplified nematic liquid crystal flows in dimension three. The first example is constructed within the class of axisymmetric solutions, while the second one is constructed for initial data with small energy but large topology. We will also construct infinitely many weak solutions to the system with suitable initial and boundary data. These weak solutions have bounded energy and ‘backward bubbling’ at a large time.
Feb 1
Feb 8
Feb 15	Paul Cazeaux (University of Minnesota)	Title: C* Algebras, Modeling and Simulations for Incommensurate Van Der Waals 2D Heterostructures Abstract: The recently discovered family of two-dimensional materials has generated enormous interest as highlighted by the 2010 Nobel Prize for the discovery of graphene. These atomically-thin crystals include insulators (boron nitride), semiconductors (transition metal dichalcogenides), and conductors (graphene). Vertical stacks of a few such layers, interacting through van der Waals forces, create a venue to explore and tune desired electronic properties: new emergent orders and physical phenomena are expected, leading to novel functionalities. However, the lack of periodicity (incommensurability) of these systems represents a significant hurdle for theoretical understanding and in particular for numerical simulations. In this talk, we discuss a mathematical framework and multi-scale calculations aimed at predicting macroscopic properties of Van der Waals 2D heterostructures. We present an original methodology lying at the intersection between computational mathematics, powerful algebraic concepts such as C*-algebras and non-commutative geometry, and modern physics applications. With a one-dimensional example, we illustrate the various rich phenomena arising in such structures and their possible connections to non-trivial topological behavior in materials.
Feb 22	Agnieszka Miedlar	Title: On matrix nearness problems: distance to delocalization. Abstract: Numerous problems in mechanics, mathematical physics, and engineering can be formulated as eigenvalue problems where the focus is to determine whether the eigenvalues are inside a specific desirable domain, and later on to detect an admissible size of a perturbation which will not move the eigenvalues away from that domain. The most frequent of such domains in use are connected to the stability of dynamical systems: the open left half-plane of the complex plane (continuous dynamical systems) and the open unit disk (discrete dynamical systems). In this talk we introduce two new matrix nearness problems, i.e., distance to delocalization and the distance to localization, to analyze the robustness of eigenvalues with respect to arbitrary localization sets (domains) in the complex plane. Following the theoretical framework of Hermitian functions and the Lyapunov-type localization approach, we present a new Newton-type algorithm for the distance to delocalization (D2D). Since our investigations are motivated by several practical applications, we will illustrate our approach on some of them.
Mar 1	Chenchen Mou (UCLA)	Title: Uniqueness and existence of viscosity solutions for a class of integro-differential equations. Abstract: We present comparison theorems and existence of viscosity solutions for a class of nonlocal equations. This class of equations includes Bellman-Isaacs equations containing operators of Levy type with measures depending on the state variable and control parameters.
Mar 8	Geng Chen	Title: Global well-posedness for scalar integrable systems with cusp singularity Abstract: The cusp singularity is a common type of singularity for nonlinear waves. In this talk, we discuss the global well-posedenss for Camassa-Holm and Hunter-Saxton equations. We will focus on the existence and uniqueness. We will also discuss the regularity of solutions for this type of equations and other type of equations such as hyperbolic conservation laws and short-pulse equation.
Mar 15	Cuong Ngo	Title: A Moving Mesh Method for the Porous Medium Equation with Compactly Supported Solutions. Abstract: A moving mesh finite element method is considered for solving the porous medium equation (PME), where the original equation is reformulated in terms of its pressure. Such reformulation is a common strategy in mathematical analysis of PME, since the pressure solution has a higher regularity than the original solution. The method is based on the moving mesh partial differential equation (MMPDE) method, and is applicable to solutions with compact supports and/or having free boundaries. The method discretizes only on the support of such a solution and employs Darcy’s law for moving the free boundary. Numerical results in 2D are presented.
Mar 22	Spring break
Mar 29	Xuemin Tu	Title: BDDC methods for Darcy flows Abstract: In this talk, BDDC algorithms will be developed for the saddle point problems arising from mixed formulations of Darcy flow in porous media. In addition to the standard no-net-flux constraints on each edge/face, adaptive primal constraints obtained from the solutions of local generalized eigenvalue problems are included to control the condition number. Special deluxe scaling and local generalized eigenvalue problems are designed in order to make sure that these additional primal variables lie in a benign subspace in which the preconditioned operator is positive definite. Condition number estimates will be discussed and some numerical experiments will provide to confirm the theoretical estimates.
Apr 5	JiCong (Jack) Shi (KU Department of Physics & Astronomy)	Title: Strain-Energy Model for the Configuration of Self-Assembled Nanostructures in Nanocomposite Films Abstract: Self-assembled nanostructures in epitaxial films provide a unique approach to design and tailor physical properties of nanocomposite films by controlling the configuration of the nanostructures. High-temperature superconducting films with secondary phase nanostructures is an excellent example and the formation of vertically-aligned secondary phase nanorods has been extensively studied experimentally for the enhancement of magnetic pinning properties of the films. To achieve an optimal pinning efficiency for superconducting film applications, it is important to control nanostructure configuration with a desired nanostructure density through selecting compatible dopant materials or film fabrication conditions. Such a control requires an understanding of the underlying physics of the formation of the nanostructures. In the formation of secondary phase nanostructures in epitaxial films, the lattice strain due to the lattice mismatch between the film matrix and dopant has been recognized as a primary driving force determining the configuration of the nanostructures. In this talk, I will discuss how to model the elastic energy of the coherently strained lattice and the non-coherent interfacial energy on the nanostructure surface for studying the configuration of nanostructures in epitaxial nanocomposite films. Especially, the mathematical difficulties of this problem will be discussed.
Apr 12	Avary Kolasinski	Title: A new functional for variational mesh generation and adaptation based on equidistribution and alignment for bulk meshes Abstract: We will introduce a new meshing functional for variational mesh generation and adaptation with minimal parameters based on the equidistribution and alignment conditions. We will discuss the theoretical properties of this functional including its coercivity and the nonsingularity and existence of limiting meshes. We will then present a comparative numerical study of this new functional with one well known functional, which is also based on the equidistribution and alignment conditions. Finally, we will introduce the theory of variational mesh generation and adaptation on surface meshes.
Apr 19	Hongguo Xu	Title: Transforming an LTI passive system to a port-Hamiltonian system Abstract: Port-Hamiltonian system arises naturally from various modeling problems. A port-Hamiltonian system is a passive system but with a special structure. In this talk, we try to answer the following questions. Can a passive system be transformed to a port-Hamiltonian system? If the answer is yes, how to construct a port-Hamiltonian system? This is a joint work with Christopher Beattie and Volker Merman.
Apr 26	Bob Eisenberg (Rush University Medical Center)	Title: Electricity is different Abstract: here
May 3	Weishi Liu	Title: Flux ratios and channel structures Abstract: We investigate Ussing’s unidirectional fluxes and flux ratios of charged tracers motivated particularly by the insightful proposal of Hodgkin and Keynes on a relation between flux ratios and channel structure. The study is based on analysis of quasi-one-dimensional Poisson-Nernst-Planck type models for ionic flows through membrane channels. This class of models includes the Poisson equation that determines the electrical potential from the charges present and is in that sense consistent. Ussing’s flux ratios generally depend on all physical parameters involved in ionic flows, particularly, on bulk conditions and channel structures. Certain setups of ion channel experiments result in flux ratios that are universal in the sense that their values depend on bulk conditions but not on channel structures; other setups lead to flux ratios that are specific in the sense that their values depend on channel structures too. Universal flux ratios could serve some purposes better than specific flux ratios in some circumstances and worse in other circumstances. We focus on two treatments of tracer flux measurements that serve as estimators of important properties of ion channels. The first estimator determines the flux of the main ion species from measurements of the flux of its tracer. Our analysis suggests a better experimental design so that the flux ratio of the tracer flux and the main ion flux is universal. The second treatment of tracer fluxes concerns ratios of fluxes and experimental setups that try to determine some properties of channel structure. We analyze the two widely used experimental designs of estimating flux ratios and show that the most widely used method depends on the spatial distribution of permanent charge so this flux ratio is specific and thus allows estimation of (some of) the properties of that permanent charge, even with ideal ionic solutions. The talk is based on a joint work with Shuguan Ji from Jilin University and Bob Eisenberg from Rush Medical School at Chicago.
May 10	Brendan Gavin (University of Massachusetts Amherst)	Title: The FEAST Algorithm: Using Complex Contour Integration for Solving Large Eigenvalue Problems Abstract: One of the most challenging tasks in contemporary numerical linear algebra is the efficient solution of large eigenvalue problems. FEAST is an algorithm that uses a Cauchy integral representation of the indicator function in order to selectively obtain the eigenvalues in user-specified regions of the complex plane, making it possible to rapidly solve for the eigenvalues of interest in a naturally parallel fashion. We describe the theory and application of the FEAST eigenvalue solver, with an emphasis on recent developments that make it possible to implement FEAST by using only matrix-vector multiplication.