Search results for "Numerical Analysis"
showing 10 items of 883 documents
Hermite interpolation: The barycentric approach
1991
The barycentric formulas for polynomial and rational Hermite interpolation are derived; an efficient algorithm for the computation of these interpolants is developed. Some new interpolation principles based on rational interpolation are discussed.
Efficient numerical methods for pricing American options under stochastic volatility
2007
Five numerical methods for pricing American put options under Heston's stochastic volatility model are described and compared. The option prices are obtained as the solution of a two-dimensional parabolic partial differential inequality. A finite difference discretization on nonuniform grids leading to linear complementarity problems with M-matrices is proposed. The projected SOR, a projected multigrid method, an operator splitting method, a penalty method, and a componentwise splitting method are considered. The last one is a direct method while all other methods are iterative. The resulting systems of linear equations in the operator splitting method and in the penalty method are solved u…
Reconstructions that combine interpolation with least squares fitting
2015
We develop a reconstruction that combines interpolation and least squares fitting for point values in the context of multiresolution a la Harten. We study the smoothness properties of the reconstruction as well as its approximation order. We analyze how different adaptive techniques (ENO, SR and WENO) can be used within this reconstruction. We present some numerical examples where we compare the results obtained with the classical interpolation and the interpolation combined with least-squares approximation. We develop a reconstruction that combines interpolation and least squares fitting.We study the smoothness properties of the reconstruction and its approximation order.We present some nu…
A fast dual boundary element method for 3D anisotropic crack problems
2009
In the present paper a fast solver for dual boundary element analysis of 3D anisotropic crack problems is formulated, implemented and tested. The fast solver is based on the use of hierarchical matrices for the representation of the collocation matrix. The admissible low rank blocks are computed by adaptive cross approximation (ACA). The performance of ACA against the accuracy of the adopted computational scheme for the evaluation of the anisotropic kernels is investigated, focusing on the balance between the kernel representation accuracy and the accuracy required for ACA. The system solution is computed by a preconditioned GMRES and the preconditioner is built exploiting the hierarchical …
Numerically stable computation of step-sizes for descent methods. The nonconvex case
1977
The computation of step-sizes which guarantee convergence in unconstrained minimization by descent methods is considered. The use of a “control” or “range” function is highly attractive for this purpose because of its simplicity. Since the Armijo-Goldstein test may fail prematurely due to numerical instability near the minimizer, we consider a range function based on gradient values alone as has been done forg convex in [8]. Numerical algorithms are given for the computation of step-sizes whose behaviour under roundoff is shown to be benign in the sense of F. L. Bauer [5].
Numerical Study of Two Sparse AMG-methods
2003
A sparse algebraic multigrid method is studied as a cheap and accurate way to compute approximations of Schur complements of matrices arising from the discretization of some symmetric and positive definite partial differential operators. The construction of such a multigrid is discussed and numerical experiments are used to verify the properties of the method.
Multiresolution based on weighted averages of the hat function I: Linear reconstruction techniques
1998
In this paper we analyze a particular example of the general framework developed in [A. Harten, {\it SIAM J. Numer. Anal}., 33 (1996) pp. 1205--1256], the case in which the discretization operator is obtained by taking local averages with respect to the hat function. We consider a class of reconstruction procedures which are appropriate for this multiresolution setting and describe the associated prediction operators that allow us to climb up the ladder from coarse to finer levels of resolution. In Part I we use data-independent (linear) reconstruction techniques as our approximation tool. We show how to obtain multiresolution transforms in bounded domains and analyze their stability with r…
The Hu-Washizu variational principle for the identification of imperfections in beams
2008
This paper presents a procedure for the identification of imperfections of structural parameters based on displacement measurements by static tests. The proposed procedure is based on the well-known Hu–Washizu variational principle, suitably modified to account for the response measurements, which is able to provide closed-form solutions to some inverse problems for the identification of structural parameter imperfections in beams. Copyright © 2008 John Wiley & Sons, Ltd.
Well-balanced bicharacteristic-based scheme for multilayer shallow water flows including wet/dry fronts
2013
The aim of this paper is to present a new well-balanced finite volume scheme for two-dimensional multilayer shallow water flows including wet/dry fronts. The ideas, presented here for the two-layer model, can be generalized to a multilayer case in a straightforward way. The method developed here is constructed in the framework of the Finite Volume Evolution Galerkin (FVEG) schemes. The FVEG methods couple a finite volume formulation with evolution operators. The latter are constructed using the bicharacteristics of multidimensional hyperbolic systems. However, in the case of multilayer shallow water flows the required eigenstructure of the underlying equations is not readily available. Thus…
Discrete multiresolution based on hermite interpolation: computing derivatives
2004
Abstract Harten’s framework for multiresolution representation of data has been extended by Warming and Beam in [SIAM J. Sci. Comp. 22 (2000) 1269] to include Hermite interpolation. It needs the point-values of the derivative, which are usually unavailable, so they have to be approximated. In this work we show that the way in which the derivatives are approximated is crucial for the success of the method, and we present a new way to compute them that makes the scheme adequate for non-smooth data.