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# Seminars in Purdue

• ## Spring 2014

• Title THE KREISS & YSTROM EQUATIONS: A ROADMAP FOR THE UNSTABLE 1-D TWO-FLUID MODEL Dr. William Fullmer , Purdue University. April 15, 2014 Consider the partial differential equation like \phi_t - A\phi_x + D \phi_{xx} + F =0. By analyzing the eigenvalues of matrix A, we can get informations, such as stabilities, oscillation, etc, of the solution. Verification is solving the equation right while validation is solving the right equation. Parabolic problems which are ill-posed in the zero dissipation limit, Linear and nonlinear analysis of an unstable, but well-posed, one-dimensional two-fluid model for two-phase flow based on the inviscid Kelvin-Helmholtz instability, An Artificial Viscosity for the III-Posed One-Dimensional Incompressible Two-Fluid Model.

• Title Finite Difference Weighted Essentially Non-Oscillatory Schemes with Constrained Transport for Ideal MHD Prof Andrew J. Christlieb,Department of Mathematics, Michigan State University 1/14/2014 While the scientific computing community has made substantial progress over the past 60 years, there is still an enormous class of problems that are out side the scope of what we can compute on current super computers with state of the art algorithms. Aproaches to control \nabla \cdot B = 0 in ideal MHD: 1. projection method; 2. 8-wave formulation; 3. divergence clean approach; 4. constrained transport methods (staggered and non-staggered). In parrell setting, the uniform meshes offer speed and scalability advantages over non-uniform meshes.

• ## Fall 2013

• Title Free Boundary Problems: When Partial Differential Equations meet Calculus of Variations Prof Danielli Donatella , Purdue University 10/28/2013 Well-posedness: (i) existence; (ii) uniqueness (iii) the solution depends continuously on the data. Weak solution are in fact Holder continuous, i.e. \sup_{x\neq y} \frac{u(x) - u(y)}{|x-y|^{\alpha}} , for some 0<\alpha<1. Classical obstacle problems, e.g. Signorini problem. See the papers written by Hans Wilhelm Alt, Luis A. Caffarelli and Avner Friedman in 1980's and 1990's.

• Title The FC(Gram) Algorithm and Fast PDE Solution Prof Mark Lyon,University of New Hampshire 9/27/2013 High-order unconditionally stable FC-AD solvers for general smooth domains

• Title A High Order A-Stable Wave Propagation Method Prof Andrew J. Christlieb,Department of Mathematics, Michigan State University 9/20/2013 Method of Lines Transpose: A Fast Implicit Wave Propagator

• Title A Minimum Sobolev Norm Technique for the Numerical Solution of PDEs Prof Shivkumar Chandrasekaran ,University of California, Santa Barbara 9/13/2013 The idea to treat the problems on irregular domains is very interesting. The resulting linear system has two parts: the first part is from the equation while the second part comes from the boundary conditions. Minimum Sobolev norm interpolation with trigonometric polynomials on the torus

• Title Aggregation-Based Multilevel Methods for the Graph Laplacian Prof James Brannick , Penn State University 8/23/2013 Given a graph G(V,E) , the matrix A is called the Laplacian of G , which satisfies the following For the ith row of A , A_{ii} is the number of egdes related to the point i , and the off-diagonal elements A_{ij} (i\neq j) are -1 or 0 . In short, there is the so called 'zero row sum' property for matrix A . The advantage of multigrid method is: highly efficient, integrated with finite element; The disadvantage of multigrid method is: limited with respect to geometry and parameters in PDE. Algebraic multilevel preconditioners for the graph Laplacian based on matching in graphs

• ## Spring 2013

• Title Algorithms for Recovering Smooth Functions from Equispaced Data Prof Rodrigo Platte, Arizona State University April 26 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.

• Title Advanced Algorithmic Approach To Optimization Prof Narendra Karmarkar Tuesday, April 23 and Thursday, April 25 at 4:30 pm in the Krannert Auditorium 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.

• Title Understanding LS Methods through the Galerkin Methods Prof JaEun Ku ,Oklahoma State University April 19 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. 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 Prof David Gleich ,Purdue University April 19 The application of ' Circulant matrices' in muilti-dimensional problem. The Power and Arnoldi Methods in an Algebra of Circulants

• Title A PDE Approach to Fractional Diffusion in General Domains: A Priori Error Analysis Dr. Abner Salgado ,University of Maryland March 22 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 . A PDE approach to fractional diffusion in general domains: a priori error analysis

• Title Finite Element Methods with Penalty Terms Prof Haijun Wu ,Nanjing University March 22 (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}.

• Title The "How to Give a Talk" Talk Prof Andrew Toms ,Purdue University March 8 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 Prof Tonglei Li ,Industrial and Physical Pharmacy, Purdue University Feb 27 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. Crystal Structure Prediction Blind Tests

• Title The Laplacian and Friends: Old, New and Conjectured Spectral Bounds Prof Richard S. Laugesen,University of Illinois at Urbana-Champaign Feb 19 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. 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 Prof Edriss S. Titi, University of California - Irvine Feb 15 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. Young Measure Approach to Computing Slowly Advancing Fast Oscillations

• Title The Hybridizable Discontinuous Galerkin Methods Prof Bernardo Cockburn, University of Minnesota Feb 8 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. 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 Dr. Songting Luo, Iowa State University Feb 5 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.

• Title A Kernel-free Boundary Integral Method for Variable Coefficient Elliptic PDE Dr. Wenjun Ying ,Shanghai Jiao Tong University Feb 1 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. A kernel-free boundary integral method for elliptic boundary value problems

• Title Nonsmooth Optimization and Lagrange Multiplier Kazufumi Ito , Professor, North Carolina State University Jan 18 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 Ms. Wenhui Shi, Purdue University Jan 17 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. Prof A. Petrosyan's book

• Title Taming the butterfly effect --- towards computational engineering of chaotic systems Qiqi Wang, Assistant Professor of Aeronautics and Astronautics, MIT Jan 16 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. Forward and adjoint sensitivity computation of chaotic dynamical systems, Sensitivity computation of periodic and chaotic limit cycle oscillations.

'A man must love a thing very much if he not only practices it without any hope of fame and money, but even ... without any hope of doing it well.' --- Oliver Herford

'No human investigation can be called real science if it cannot be demonstrated mathematically.'--- 'A Treatise on Painting' by Leonardo da Vinci