Monday, January 14, 2008
Special ``Numerical Analysis Search Candidate'' Scientific Computing Seminar
| Speaker:
|
Johnny Guzman, School of Mathematics, University of Minnesota
|
| Title:
|
Superconvergent Discontinuous Galerkin Methods
|
| Time/Place:
|
12:00 p.m., 182 George Street, Room 110
|
Abstract:
We identify discontinuous Galerkin methods for second-order elliptic
problems having superconvergence properties similar to those of the
Raviart-Thomas and the Brezzi-Douglas-Marini mixed methods. These
methods use polynomials of degree k for both the potential as well as
the flux. We show that the approximate flux converges with the optimal
order of k+1, and that the local averages of the approximate potential
superconverge to the averages of the potential, with order k+2. We also
apply an element-by-element postprocessing of the approximate solution
to obtain a new approximation of the potential. The new approximate
solution of the potential converges with order k+2. We provide numerical
experiments that support our theoretical results.
Tuesday, January 15, 2008
Special ``Numerical Analysis Search Candidate'' Scientific Computing Seminar
| Speaker:
|
Alina Chertock, Department of Mathematics, North Carolina State University
|
| Title:
|
Particle Methods for Nonlinear Time-Dependent PDEs
|
| Time/Place:
|
12:00 p.m., 182 George Street, Room 110
|
Abstract:
In recent years, particle methods have become one of the most useful and
widespread tools for approximating solutions of partial differential equations
in a variety of fields. In these methods, the solution is sought as a linear
combination of Dirac distributions, whose positions and coefficients represent
locations and weights of the particles, respectively. The solution is then
found by following the time evolution of the locations and the weights of the
particles according to a system of ODEs, obtained by considering a weak
formulation of the problem. The main advantage of the particle methods is
their low numerical diffusion that allows to capture a variety of nonlinear
waves with a high resolution. Even though the most ``natural'' application
of the particle methods is linear transport equations, over the years, the
range of these methods has been extended for approximating solutions of
nonlinear equations including degenerate parabolic, convection-diffusion
and dispersive equations.
In this talk, I will review different aspects of a practical implementation of
particle methods such as recovering an approximate solution from the particle
distribution and investigation of various particle redistribution algorithms.
I will also present new numerical techniques for nonlinear PDEs, with
particular reference to problems that admit nonsmooth (discontinuous)
solutions and on problems that involve multiple scales, and therefore, are
difficult to solve numerically by traditional finite-difference methods.
The new techniques are based on the particle methods and their hybridization
with Eulerian (finite-volume) methods.
I will demonstrate the performance of the mew methods in a number of numerical
examples, among which are the Euler-PoincarŽe equation, models of transport of
pollutant in shallow water, reactive Euler equations describing stiff
detonation waves, pressureless gas dynamics, and others.
Thursday, January 17, 2008
Special ``Numerical Analysis Search Candidate'' Scientific Computing Seminar
| Speaker:
|
Mark Tygert, Yale University, New Haven, CT
|
| Title:
|
A fast algorithm for approximating the singular value decomposition of a matrix
|
| Time/Place:
|
12:00 p.m., 182 George Street, Room 110
|
Abstract:
This talk will describe a randomized algorithm for the low-rank
approximation of arbitrary matrices. Constructing a low-rank approximation
is the core step in computing several of the greatest singular values and
corresponding singular vectors of a matrix. The new randomized procedure
is generally significantly faster than the classical pivoted "QR"
decomposition algorithms (such as Gram-Schmidt or Householder), yet
ensures similar or better accuracy.
In order to compute a nearly optimally accurate rank-k approximation to an
n x n matrix, the new algorithm typically requires O(n**2 ln(k) + n k**2)
floating-point operations, whereas pivoted "QR" decomposition algorithms
require at least kn**2 flops. Moreover, the algorithm runs faster than the
classical algorithms in empirical tests on any of several standard PC
microprocessor cores (for almost any k). Furthermore, the scheme
parallelizes naturally. The results will be illustrated via numerical
examples and applications.
This is joint work with Edo Liberty, Vladimir Rokhlin, and Franco Woolfe.
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