7,056 research outputs found
A class of high-order Runge-Kutta-Chebyshev stability polynomials
The analytic form of a new class of factorized Runge-Kutta-Chebyshev (FRKC)
stability polynomials of arbitrary order is presented. Roots of FRKC
stability polynomials of degree are used to construct explicit schemes
comprising forward Euler stages with internal stability ensured through a
sequencing algorithm which limits the internal amplification factors to . The associated stability domain scales as along the real axis.
Marginally stable real-valued points on the interior of the stability domain
are removed via a prescribed damping procedure.
By construction, FRKC schemes meet all linear order conditions; for nonlinear
problems at orders above 2, complex splitting or Butcher series composition
methods are required. Linear order conditions of the FRKC stability polynomials
are verified at orders 2, 4, and 6 in numerical experiments. Comparative
studies with existing methods show the second-order unsplit FRKC2 scheme and
higher order (4 and 6) split FRKCs schemes are efficient for large moderately
stiff problems.Comment: 24 pages, 5 figures. Accepted for publication in Journal of
Computational Physics, 22 Jul 2015. Revise
A weakly stable algorithm for general Toeplitz systems
We show that a fast algorithm for the QR factorization of a Toeplitz or
Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A.
Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx =
A^Tb, we obtain a weakly stable method for the solution of a nonsingular
Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the
solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further
details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm
Pricing European and American Options under Heston Model using Discontinuous Galerkin Finite Elements
This paper deals with pricing of European and American options, when the
underlying asset price follows Heston model, via the interior penalty
discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM
space discretization with Rannacher smoothing as time integrator with nonsmooth
initial and boundary conditions are illustrated for European vanilla options,
digital call and American put options. The convection dominated Heston model
for vanishing volatility is efficiently solved utilizing the adaptive dGFEM.
For fast solution of the linear complementary problem of the American options,
a projected successive over relaxation (PSOR) method is developed with the norm
preconditioned dGFEM. We show the efficiency and accuracy of dGFEM for option
pricing by conducting comparison analysis with other methods and numerical
experiments
Initial steps towards automatic segmentation of the wire frame of stent grafts in CT data
For the purpose of obtaining a geometrical model of the wire frame of stent grafts, we propose three tracking methods to segment the stent's wire, and compare them in an experiment. A 2D test image was created by obtaining a projection of a 3D volume containing a stent. The image was modified to connect the parts of the stent's frame and thus create a single path. Ten versions of this image were obtained by adding different noise realizations. Each algorithm was started at the start of each of the ten images, after which the traveled paths were compared to the known correct path to determine the performance. Additionally, the algorithms were applied to 3D clinical data and visually inspected. The method based on the minimum cost path algorithm scored excellent in the experiment and showed good results on the 3D data. Future research will focus on establishing a geometrical model by determining the corner points and the crossings from the results of this method.\u
Theory and implementation of -matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels
In this work, we study the accuracy and efficiency of hierarchical matrix
(-matrix) based fast methods for solving dense linear systems
arising from the discretization of the 3D elastodynamic Green's tensors. It is
well known in the literature that standard -matrix based methods,
although very efficient tools for asymptotically smooth kernels, are not
optimal for oscillatory kernels. -matrix and directional
approaches have been proposed to overcome this problem. However the
implementation of such methods is much more involved than the standard
-matrix representation. The central questions we address are
twofold. (i) What is the frequency-range in which the -matrix
format is an efficient representation for 3D elastodynamic problems? (ii) What
can be expected of such an approach to model problems in mechanical
engineering? We show that even though the method is not optimal (in the sense
that more involved representations can lead to faster algorithms) an efficient
solver can be easily developed. The capabilities of the method are illustrated
on numerical examples using the Boundary Element Method
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