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Seminars in Purdue

Spring 2013

• Seminars calendar

  • CS&E seminars
  • CCAM seminars
  • CCAM lunch seminars
  • Math seminars

• Notes

  • Title	Algorithms for Recovering Smooth Functions from Equispaced Data
    Speaker	Prof Rodrigo Platte, Arizona State University
    Date	April 26
    Comments	Two cases that have exponential convergence rate for interpolation: (i) Analytic function + equispaced points + periodical boundary conditions => Fourier; (ii) Analytic function + Chebyshev points + non-periodical boundary conditions => Polynomial interpolation; However, analytic function + equispaced points + non-periodical BC leads to Runge phenomenon (wild oscillations near the endpoints). Several ways to get exponential convergence rate for equispaced interpolation: (i) Mock-Chebyshev points, e.g. from 100 equispaced points to 17 Chebyshev points; (ii) Discrete least square, e.g. the number of equispaced points is n while the degree of interpolation polynomial is \sqrt{n} ; (iii) Barycentric rational interpolation plus fast SVD trucation; (iv) Mapping: Chebyshev points \leftrightarrow equispaced points.
    Reference

  
  
  

  • Title	Advanced Algorithmic Approach To Optimization
    Speaker	Prof Narendra Karmarkar
    Date	Tuesday, April 23 and Thursday, April 25 at 4:30 pm in the Krannert Auditorium
    Comments	Dr. Karmarkar is a pioneer in the field of mathematical programming and optimization, and is renowned for his work in interior-point methods. Dr. Karmarkar gave this presentations from a strategically advantageous position, since he has a broad knowledge and profound understanding on this topic. I just keep a record of some key words here. (i) 'Difficult to predict' does not equal to 'predictably difficult'. (ii)'Unpredictability' does not equal to 'intractability'. (iii) Alan Turing is also an expert in mathematical biology. Alan Turing's achievements in the theory of computation led to his recognition as the father of modern computer science. The most prestigious award in the field is named after him. He is also widely recognized in cryptography for his work on 'Cryptology bombs' - code-breaking machines that were used by the Allies during the Battle of the Atlantic. It is less well known that he spent the last few years of his life developing mathematical theories to describe biological processes. (iv) Kurt Godel : purse 'constructism' but do away with 'finitism'. (v) Linear programming: both convexity and concavity. (vi) Continuum based method. (vii) The algebraic closure of meromorphic function. (viii) Good algorithm: finite representation and efficient operations. (ix) For the function of interest, we need to find suitably curved space so that the function behaves like linear or quadratic. This idea is based on the fact that some non-convex functions in Euclidean space could be convex in some kind of curved space. (x) Geometry: Euclidean --> Riemannian --> Lorentzian --> multi-cellular. It is the interdiscipline of Mathematics, Physics and Computer Science.
    Reference

  
  
  

  • Title	Understanding LS Methods through the Galerkin Methods
    Speaker	Prof JaEun Ku ,Oklahoma State University
    Date	April 19
    Comments	For the problem -\Delta u =f , in \Omega with homogeneous Dirichlet boundary condition u=0, on \partial \Omega , the 'asymptotically exact error estimators' are \frac{1}{1+m(h)}\|\sigma_h + \nabla u_h\|_0 \leq \|\nabla u - \nabla u_h \|_0 \leq \frac{1}{1-m(h)}\|\sigma_h + \nabla u_h\|_0, \frac{1}{1+m(h)}\|\sigma_h + \nabla u_h\|_0 \leq \|\sigma- \sigma_h \|_0 \leq \frac{1}{1-m(h)}\|\sigma_h + \nabla u_h\|_0, where m(h) = Ch^{\epsilon}\rightarrow 0, as h \rightarrow 0. Here 0<\epsilon \ll 1 if the solution u is of low regularity.
    Reference	Least-squares solutions as solutions of a perturbation form of the Galerkin methods: Interior pointwise error estimates and pollution effect, Local A Posteriori Estimates on a Nonconvex Polygonal Domain.

  
  
  

  • Title	The Power and Arnoldi Methods in an Algebra of Circulants
    Speaker	Prof David Gleich ,Purdue University
    Date	April 19
    Comments	The application of ' Circulant matrices' in muilti-dimensional problem.
    Reference	The Power and Arnoldi Methods in an Algebra of Circulants

  
  
  

  • Title	A PDE Approach to Fractional Diffusion in General Domains: A Priori Error Analysis
    Speaker	Dr. Abner Salgado ,University of Maryland
    Date	March 22
    Comments	Consider the fractional powers of the Dirichlet Laplace operator (-\Delta)^s , where s \in (0,1), which is called the fractional Laplacian for convenience. There are two ways to do the numerical discretization: (i) -\Delta \rightarrow -\Delta_h \rightarrow (-\Delta_h)^{s} ; (ii) -\Delta \rightarrow (-\Delta)^s \rightarrow ((-\Delta)^{s})_h .
    Reference	A PDE approach to fractional diffusion in general domains: a priori error analysis

  
  
  

  • Title	Finite Element Methods with Penalty Terms
    Speaker	Prof Haijun Wu ,Nanjing University
    Date	March 22
    Comments	(i) Error estimate for convection-dominante problem -\varepsilon \Delta u + \beta \nabla u + \sigma u =f: -\|u-u_h\|_{H^1} \leq C \varepsilon^{-1}h^2 |u|_{H^2}, \quad h \geq \varepsilon, p=1. (ii) hp-error estimate for Helmholtz equation -\Delta u + ku = f: |u-u_h|_{H^1}/|u|_{H^1} \leq C_1 \left(\frac{kh}{p}\right)^p + C_2 k \left(\frac{kh}{2p}\right)^{2p}.
    Reference

  
  
  

  • Title	The "How to Give a Talk" Talk
    Speaker	Prof Andrew Toms ,Purdue University
    Date	March 8
    Comments	You may have a nice result, but the way in which you present it could raise you from zero to hero or vice versa. (i) The seminar/conference talk: explain the results. State the main result near the begining. In proof, give a clear outline first. (ii) The colloquium talk: get people interested in this topic. (iii) The job talk: get people interested in what you are doing AND convince them that you are awesome. Attention should be paid on: 1. CLEARITY; 2. connect your work with some NAMES; 3. statement about your work in progress and your OWN research program; 4. make the audience like you and talk about their research.

  
  
  

  • Title	Crystal Structure Prediction: The Challenge and Our Approach
    Speaker	Prof Tonglei Li ,Industrial and Physical Pharmacy, Purdue University
    Date	Feb 27
    Comments	In computational chemistry, the Fukui function or frontier function is a function that describes the electron density in a frontier orbital, as a result of an small change in the total number of electrons. The Fukui function, denoted by f(r) , is defined as the differential change in electron density due to an infinitesimal change in the number of electrons. That is, f(r) = \left(\frac{\partial \rho(r)}{\partial N} \right)_{v_{r}}, where \rho(r) is the electron density and N = \int \rho(r) dr is the total number of electrons in the system.
    Reference	Crystal Structure Prediction Blind Tests

  
  
  

  • Title	The Laplacian and Friends: Old, New and Conjectured Spectral Bounds
    Speaker	Prof Richard S. Laugesen,University of Illinois at Urbana-Champaign
    Date	Feb 19
    Comments	Consider the Lapacian eigenvalue problem -\Delta u= \lambda u, x \in \Omega with Dirichlet boundary condition u = 0 on \partial\Omega. Suppose the area \Omega be A and \lambda_1 be the smallest eigenvalue. (i) \lambda_1 A is minimal for disk among all shapes; \lambda_1 A is minimal for the square among all rectangles; But \lambda_1 A is minimal for regluar N-gon among all N-gons ??? (ii) \lambda_2 A is minimal for double disk; But how about \lambda_j A, as j\rightarrow \infty (iii) Hermann Weyl's formular: \lambda_j \sim \frac{4\pi j}{A}, which means given the eigenvalues \lambda_j we can know the area of the domain A. Generally, this is the problem called 'hearing the shape of a drum'. (iv) For other kinds of boundary conditions, we have Neumann \leq Robin \leq Dirichlet.
    Reference	Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians, Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian

  
  
  

  • Title	A Numerical Algorithm for Advancing Slow Features in Fast-Slow Systems without Scale Separation --- A Young Measure Approach
    Speaker	Prof Edriss S. Titi, University of California - Irvine
    Date	Feb 15
    Comments	Suppose (x_{\varepsilon}(t), y_{\varepsilon}(t)) be a solution to the singularly pertubated problem \frac{dx}{dt} = f(x,y), \quad{\varepsilon}\frac{dy}{dt} = g(x,y), and (x_{0}(t), y_0(t)) be the solution of the reduced problem \frac{dx}{dt} = f(x,y), \quad 0 = g(x,y), then y_{\varepsilon}(t) does not converge to y_{0}(t) as {\varepsilon}\rightarrow 0. Instead,y_{\varepsilon}(t) converges to a measure, \mu_{x}, for every fixed t. Here, \mu_{x} is a probability distribution supported on some curve \Gamma_{x}, and is called in literature as Young Measure.
    Reference	Young Measure Approach to Computing Slowly Advancing Fast Oscillations

  
  
  

  • Title	The Hybridizable Discontinuous Galerkin Methods
    Speaker	Prof Bernardo Cockburn, University of Minnesota
    Date	Feb 8
    Comments	Advantages of DG (Discontinuous Galerkin): (i) locally conservative methods; (ii) suitable for hp-adaptivity; (iii) built-in stabilization mechanism, not degrade high-order accuracy. HDG (Hybridizable Discontinuous Galerkin): from the characterization of the exact solution, to local problems and transmission conditions.
    Reference	Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems

  
  
  

  • Title	High-Order Factorization of Source Singularities and High-Order Methods for Point-Source Eikonal Equations
    Speaker	Dr. Songting Luo, Iowa State University
    Date	Feb 5
    Comments	In order to achieve high order accuracy for singular problems, we need to do two things: (i) Factor out the singular part (just near the singular point): multiplicative factor u = wv or additive factor u = w+v. Here w is chosen such that v is smooth. (ii) Hybrid scheme: inside the neiborhood of singular points, solve the factored equation; while outside the singular region, just solve the original equation. The order of convergence rate is determined by \min(N,M), where N is the order of the numerical scheme (the order of polynomials) and M is the terms of asymtopic expasion of the singular part.
    Reference

  
  
  

  • Title	A Kernel-free Boundary Integral Method for Variable Coefficient Elliptic PDE
    Speaker	Dr. Wenjun Ying ,Shanghai Jiao Tong University
    Date	Feb 1
    Comments	Advantages of BIM (boundary integral method): (i) reduce the dimension from d to d-1; (ii) no volume of grid, suitable for moving interfaces. Disadvantages of BIM: (i) inaccurate for the points near the boundary;(ii) variable coefficients lead to dense matrices. Advantages of KFBIM (kernel-free boundary integral method): just need the matrix-vector multiplication, suitable for variable coefficients.
    Reference	A kernel-free boundary integral method for elliptic boundary value problems

  
  
  

  • Title	Nonsmooth Optimization and Lagrange Multiplier
    Speaker	Kazufumi Ito , Professor, North Carolina State University
    Date	Jan 18
    Comments	If h is convex, then the bi-conjugate function h^{**} is itself, h = h^{**}; If h is not convex, then the bi-conjugate function h^{**} is the convexification of h, and h^{**} <= h.

  
  
  

  • Title	Regularity of Free Boundaries for the Elliptic Thin Obstacle Problem
    Speaker	Ms. Wenhui Shi, Purdue University
    Date	Jan 17
    Comments	For the free boundary problem with thin obstacle, where the free boundary is located on the boundary of the domain, the solution usually behaves like r^{3/2} around the free boundary points.
    Reference	Prof A. Petrosyan's book

  
  
  

  • Title	Taming the butterfly effect --- towards computational engineering of chaotic systems
    Speaker	Qiqi Wang, Assistant Professor of Aeronautics and Astronautics, MIT
    Date	Jan 16
    Comments	This talk discusses the cause of the divergence of computed sensitivity in chaotic dynamical systems. The original non-linear Navier-Stokes equation is stable while the linearized NS equation is not for the problem with high Reynold numbers. The least squares solution of a linear differential equation automatically forces the divergent and convergent components of the solution propagate in different time directions.
    References	Forward and adjoint sensitivity computation of chaotic dynamical systems, Sensitivity computation of periodic and chaotic limit cycle oscillations.

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