[Back]
Title  THE KREISS & YSTROM EQUATIONS: A ROADMAP FOR THE UNSTABLE 1D TWOFLUID MODEL 

Speaker  Dr. William Fullmer , Purdue University. 
Date  April 15, 2014 
Comments 
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. 
Reference  Parabolic problems which are illposed in the zero dissipation limit, Linear and nonlinear analysis of an unstable, but wellposed, onedimensional twofluid model for twophase flow based on the inviscid KelvinHelmholtz instability, An Artificial Viscosity for the IIIPosed OneDimensional Incompressible TwoFluid Model. 
Title  Finite Difference Weighted Essentially NonOscillatory Schemes with Constrained Transport for Ideal MHD 

Speaker  Prof Andrew J. Christlieb,Department of Mathematics, Michigan State University 
Date  1/14/2014 
Comments 
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. 8wave formulation; 3. divergence clean approach; 4. constrained transport methods (staggered and nonstaggered). In parrell setting, the uniform meshes offer speed and scalability advantages over nonuniform meshes. 
Reference 
Title  Free Boundary Problems: When Partial Differential Equations meet Calculus of Variations 

Speaker  Prof Danielli Donatella , Purdue University 
Date  10/28/2013 
Comments  Wellposedness: (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)}{xy^{\alpha}} , for some 0<\alpha<1. Classical obstacle problems, e.g. Signorini problem. 
Reference  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 

Speaker  Prof Mark Lyon,University of New Hampshire 
Date  9/27/2013 
Comments  
Reference  Highorder unconditionally stable FCAD solvers for general smooth domains 
Title  A High Order AStable Wave Propagation Method 

Speaker  Prof Andrew J. Christlieb,Department of Mathematics, Michigan State University 
Date  9/20/2013 
Comments  
Reference  Method of Lines Transpose: A Fast Implicit Wave Propagator 
Title  A Minimum Sobolev Norm Technique for the Numerical Solution of PDEs 

Speaker  Prof Shivkumar Chandrasekaran ,University of California, Santa Barbara 
Date  9/13/2013 
Comments  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. 
Reference  Minimum Sobolev norm interpolation with trigonometric polynomials on the torus 
Title  AggregationBased Multilevel Methods for the Graph Laplacian 

Speaker  Prof James Brannick , Penn State University 
Date  8/23/2013 
Comments  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 offdiagonal 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. 
Reference  Algebraic multilevel preconditioners for the graph Laplacian based on matching in graphs 
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 + nonperiodical boundary conditions => Polynomial interpolation; However, analytic function + equispaced points + nonperiodical BC leads to Runge phenomenon (wild oscillations near the endpoints). Several ways to get exponential convergence rate for equispaced interpolation: (i) MockChebyshev 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 interiorpoint 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'  codebreaking 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 nonconvex functions in Euclidean space could be convex in some kind of curved space. (x) Geometry: Euclidean > Riemannian > Lorentzian > multicellular. 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}{1m(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}{1m(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  Leastsquares 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 muiltidimensional 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 convectiondominante problem
\varepsilon \Delta u + \beta \nabla u + \sigma u =f: \uu_h\_{H^1} \leq C \varepsilon^{1}h^2 u_{H^2}, \quad h \geq \varepsilon, p=1. (ii) hperror estimate for Helmholtz equation \Delta u + ku = f: uu_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 UrbanaChampaign 
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 Ngon among all Ngons ??? (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 FastSlow 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 hpadaptivity; (iii) builtin stabilization mechanism, not degrade highorder 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  HighOrder Factorization of Source Singularities and HighOrder Methods for PointSource 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 Kernelfree 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 d1; (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 (kernelfree boundary integral method): just need the matrixvector multiplication, suitable for variable coefficients. 
Reference  A kernelfree 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 biconjugate function h^{**} is itself, h = h^{**}; If h is not convex, then the biconjugate 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 nonlinear NavierStokes 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. 
'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