Search results for "Computational Mathematic"
showing 10 items of 987 documents
Padé approximants and the prediction of non-perturbative parameters in particle physics
2010
Conference on Approximation and extrapolation of Convergent and Divergent Sequences and Series Luminy, FRANCE, SEP 28-OCT 02, 2009
GPU accelerated Monte Carlo simulation of the 2D and 3D Ising model
2009
The compute unified device architecture (CUDA) is a programming approach for performing scientific calculations on a graphics processing unit (GPU) as a data-parallel computing device. The programming interface allows to implement algorithms using extensions to standard C language. With continuously increased number of cores in combination with a high memory bandwidth, a recent GPU offers incredible resources for general purpose computing. First, we apply this new technology to Monte Carlo simulations of the two dimensional ferromagnetic square lattice Ising model. By implementing a variant of the checkerboard algorithm, results are obtained up to 60 times faster on the GPU than on a curren…
An iterative method for pricing American options under jump-diffusion models
2011
We propose an iterative method for pricing American options under jump-diffusion models. A finite difference discretization is performed on the partial integro-differential equation, and the American option pricing problem is formulated as a linear complementarity problem (LCP). Jump-diffusion models include an integral term, which causes the resulting system to be dense. We propose an iteration to solve the LCPs efficiently and prove its convergence. Numerical examples with Kou@?s and Merton@?s jump-diffusion models show that the resulting iteration converges rapidly.
Order optimal preconditioners for fully implicit Runge-Kutta schemes applied to the bidomain equations
2010
The partial differential equation part of the bidomain equations is discretized in time with fully implicit Runge–Kutta methods, and the resulting block systems are preconditioned with a block diagonal preconditioner. By studying the time-stepping operator in the proper Sobolev spaces, we show that the preconditioned systems have bounded condition numbers given that the Runge–Kutta scheme is A-stable and irreducible with an invertible coefficient matrix. A new proof of order optimality of the preconditioners for the one-leg discretization in time of the bidomain equations is also presented. The theoretical results are verified by numerical experiments. Additionally, the concept of weakly po…
Controllability method for the Helmholtz equation with higher-order discretizations
2007
We consider a controllability technique for the numerical solution of the Helmholtz equation. The original time-harmonic equation is represented as an exact controllability problem for the time-dependent wave equation. This problem is then formulated as a least-squares optimization problem, which is solved by the conjugate gradient method. Such an approach was first suggested and developed in the 1990s by French researchers and we introduce some improvements to its practical realization. We use higher-order spectral elements for spatial discretization, which leads to high accuracy and lumped mass matrices. Higher-order approximation reduces the pollution effect associated with finite elemen…
Monotonic solution of heterogeneous anisotropic diffusion problems
2013
Anisotropic problems arise in various areas of science and engineering, for example groundwater transport and petroleum reservoir simulations. The pure diffusive anisotropic time-dependent transport problem is solved on a finite number of nodes, that are selected inside and on the boundary of the given domain, along with possible internal boundaries connecting some of the nodes. An unstructured triangular mesh, that attains the Generalized Anisotropic Delaunay condition for all the triangle sides, is automatically generated by properly connecting all the nodes, starting from an arbitrary initial one. The control volume of each node is the closed polygon given by the union of the midpoint of…
Efficient numerical method for simulating static and dynamic properties of superfluid helium
2004
Density functional theory (DFT) offers computationally affordable way of describing static and dynamic properties of superfluid 4He. In general, the DFT models yield single particle-like Schrodinger equations with a nonlinear potential term that accounts for all the many-body interactions. The resulting equations can be solved for small amplitude plane wave excitations in the bulk whereas fully numerical solution must be sought in more complicated cases. In this paper we propose a numerical method that can be used in solving the time-dependent nonlinear Schrodinger equation in both real and imaginary times. The method is based on operator splitting technique where each component operator is…
The MAST-edge centred lumped scheme for the flow simulation in variably saturated heterogeneous porous media
2012
A novel methodology is proposed for the solution of the flow equation in a variably saturated heterogeneous porous medium. The computational domain is descretized using triangular meshes and the governing PDEs are discretized using a lumped in the edge centres numerical technique. The dependent unknown variable of the problem is the piezometric head. A fractional time step methodology is applied for the solution of the original system, solving consecutively a prediction and a correction problem. A scalar potential of the flow field exists and in the prediction step a MArching in Space and Time (MAST) formulation is applied for the sequential solution of the Ordinary Differential Equation of…
Time-harmonic elasticity with controllability and higher-order discretization methods
2008
The time-harmonic solution of the linear elastic wave equation is needed for a variety of applications. The typical procedure for solving the time-harmonic elastic wave equation leads to difficulties solving large-scale indefinite linear systems. To avoid these difficulties, we consider the original time dependent equation with a method based on an exact controllability formulation. The main idea of this approach is to find initial conditions such that after one time-period, the solution and its time derivative coincide with the initial conditions.The wave equation is discretized in the space domain with spectral elements. The degrees of freedom associated with the basis functions are situa…
Solution of time-independent Schrödinger equation by the imaginary time propagation method
2007
Numerical solution of eigenvalues and eigenvectors of large matrices originating from discretization of linear and non-linear Schrodinger equations using the imaginary time propagation (ITP) method is described. Convergence properties and accuracy of 2nd and 4th order operator-splitting methods for the ITP method are studied using numerical examples. The natural convergence of the method is further accelerated with a new dynamic time step adjustment method. The results show that the ITP method has better scaling with respect to matrix size as compared to the implicitly restarted Lanczos method. An efficient parallel implementation of the ITP method for shared memory computers is also demons…