Improved radial basis function methods for multi-dimensional option pricing

In this paper, we have derived a radial basis function (RBF) based method for the pricing of financial contracts by solving the Black–Scholes partial differential equation. As an example of a financial contract that can be priced with this method we have chosen the multi-dimensional European basket call option. We have shown numerically that our scheme is second-order accurate in time and spectrally accurate in space for constant shape parameter. For other non-optimal choices of shape parameter values, the resulting convergence rate is algebraic. We propose an adapted node point placement that improves the accuracy compared with a uniform distribution. Compared with an adaptive finite difference method, the RBF method is 20–40 times faster in one and two space dimensions and has approximately the same memory requirements.     

Error saturation in Gaussian radial basis functions on a finite interval

Radial basis function (RBF) interpolation is a “meshless” strategy with great promise for adaptive approximation. One restriction is “error saturation” which occurs for many types of RBFs including Gaussian RBFs of the form ?(x;α,h)=exp(−α²(x/h)²): in the limit h→0 for fixed α, the error does not converge to zero, but rather to E_S(α). Previous studies have theoretically determined the saturation error for Gaussian RBF on an infinite, uniform interval and for the same with a single point omitted. (The gap enormously increases E_S(α).) We show experimentally that the saturation error on the unit interval, x∈[−1,1], is about 0.06exp(−0.47/α²)‖f‖_∞ — huge compared to the O(2π/α²)exp(−π²/[4α²]) saturation error for a grid with one point omitted. We show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points. The saturation error can be avoided by choosing α?1, the “flat limit”, but the condition number of the interpolation matrix explodes as O(exp(π²/[4α²])). The best strategy is to choose the largest α which yields an acceptably small saturation error: If the user chooses an error tolerance δ, then .     

Algorithms for cardinal interpolation using box splines and radial basis functions

Summary We describe an algorithm for (bivariate) cardinal interpolation which can be applied to translates of basis functions which include box splines or radial basis functions. The algorithm is based on a representation of the Fourier transform of the fundamental interpolant, hence Fast Fourier Transform methods are available. In numerical tests the 4-directional box spline (transformed to the characteristical submodule of ²), the thin plate spline, and the multiquadric case give comparably equal and good results.     

Preconditioning of elliptic problems by approximation in the transform domain

Preconditioned conjugate gradient method is applied for solving linear systemsAx=b where the matrixA is the discretization matrix of second-order elliptic operators. In this paper, we consider the construction of the trnasform based preconditioner from the viewpoint of image compression. Given a smooth image, a major portion of the energy is concentrated in the low frequency regions after image transformation. We can view the matrixA as an image and construct the transform based preconditioner by using the low frequency components of the transformed matrix. It is our hope that the smooth coefficients of the given elliptic operator can be approximated well by the low-rank matrix. Numerical results are reported to show the effectiveness of the preconditioning strategy. Some theoretical results about the properties of our proposed preconditioners and the condition number of the preconditioned matrices are discussed.     

The truncated Stieltjes moment problem solved by using kernel density functions

In this work the problem of the approximate numerical determination of a semi-infinite supported, continuous probability density function (pdf) from a finite number of its moments is addressed. The target space is carefully defined and an approximation theorem is proved, establishing that the set of all convex superpositions of appropriate Kernel Density Functions (KDFs) is dense in this space. A solution algorithm is provided, based on the established approximate representation of the target pdf and the exploitation of some theoretical results concerning moment sequence asymptotics. The solution algorithm also permits us to recover the tail behavior of the target pdf and incorporate this information in our solution. A parsimonious formulation of the proposed solution procedure, based on a novel sequentially adaptive scheme is developed, enabling a very efficient moment data inversion. The whole methodology is fully illustrated by numerical examples.     

Implementation of a modified Marder–Weitzner method for solving nonlinear eigenvalue problems

Our goal is to propose four versions of modified Marder–Weitzner methods and to present the implementation of the new-type methods with incremental unknowns for solving nonlinear eigenvalue problems. By combining with compact schemes and modified Marder–Weitzner methods, six schemes well suited for the calculation of unstable solutions are obtained. We illustrate the efficiency of the new algorithms by using numerical computations and by comparing them with existing methods for some two-dimensional problems.     

Preconditioners for spectral element methods for elliptic and parabolic problems

In this paper we propose preconditioners for spectral element methods for elliptic and parabolic problems. These preconditioners are constructed using separation of variables and are easy to invert. Moreover they are spectrally equivalent to the quadratic forms which they are used to approximate.     

Convergence and optimization of the parallel method of simultaneous directions for the solution of elliptic problems

For the solution of elliptic problems, fractional step methods and in particular alternating directions (ADI) methods are iterative methods where fractional steps are sequential. Therefore, they only accept parallelization at low level. In [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673–708], Lu et al. proposed a method where the fractional steps can be performed in parallel. We can thus speak of parallel fractional step (PFS) methods and, in particular, simultaneous directions (SDI) methods. In this paper, we perform a detailed analysis of the convergence and optimization of PFS and SDI methods, complementing what was done in [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673–708]. We describe the behavior of the method and we specify the good choice of the parameters. We also study the efficiency of the parallelization. Some 2D, 3D and high-dimensional tests confirm our results.     

Some functional equations in the spaces of generalized functions

Summary. Making use of the fundamental solution of the heat equation we find the solutions of some functional equations such as the Cauchy equations, Pexider equations, quadratic functional equations and dAlembert equations in the spaces of Schwartz distributions and Sato hyperfunctions.     

Fourier analysis of schwarz alternating methods for piecewise hermite bicubic orthogonal spline collocation

The rates of convergence of two Schwarz alternating methods are analyzed for the iterative solution of a discrete problem which arises when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet problem for Poisson's equation on a rectangle. In the first method, the rectangle is divided into two overlapping subrectangles, while three overlapping subrectangles are used in the second method. Fourier analysis is used to obtain explicit formulas for the convergence factors by which theH ¹-norm of the errors is reduced in one iteration of the Schwarz methods. It is shown numerically that while these factors depend on the size of overlap, they are independent of the partition stepsize. Results of numerical experiments are presented which confirm the established rates of convergence of the Schwarz methods.This research was supported in part by funds from the National Science Foundation grant CCR-9103451.     

<Emphasis Type=Italic>L</Emphasis><Superscript>1</Superscript>-minimization methods for Hamilton–Jacobi equations: the one-dimensional case

A new approximation technique based on L ¹-minimization is introduced. It is proven that the approximate solution converges to the viscosity solution in the case of one-dimensional stationary Hamilton–Jacobi equation with convex Hamiltonian. This material is based upon work supported by the National Science Foundation grant DMS-0510650. J.-L. Guermond is on leave from LIMSI, UPRR 3251 CNRS, BP 133, 91403 Orsay Cedex, France.     

A posteriori error estimators for the Stokes equations II non-conforming discretizations

Summary We present an a posteriori error estimator for the non-conforming Crouzeix-Raviart discretization of the Stokes equations which is based on the local evaluation of residuals with respect to the strong form of the differential equation. The error estimator yields global upper and local lower bounds for the error of the finite element solution. It can easily be generalized to the stationary, incompressible Navier-Stokes equations and to other non-conforming finite element methods. Numerical examples show the efficiency of the proposed error estimator.     

Discrete mechanics and its application to the solution of the n-body problem

A theory of discrete mechanics is developed based on the results of D. Greenspan. Discrete dynamical equations in an inertial frame, in a coordinate system related to some material point, and in a rotating frame are given and the consistency, stability, and convergence of the methods are studied and some numerical examples presented.     

A parallel implementation of the restarted GMRES iterative algorithm for nonsymmetric systems of linear equations

We describe the parallelisation of the GMRES(c) algorithm and its implementation on distributed-memory architectures, using both networks of transputers and networks of workstations under the PVM message-passing system. The test systems of linear equations considered are those derived from five-point finite-difference discretisations of partial differential equations. A theoretical model of the computation and communication phases is presented which allows us to decide for which values of the parameterc our implementation executes efficiently. The results show that for reasonably large discretisation grids the implementations are effective on a large number of processors.     

Convergence of block iterative methods for linear systems with generalized H-matrices

The paper studies the convergence of some block iterative methods for the solution of linear systems when the coefficient matrices are generalized H$H$-matrices. A truth is found that the class of conjugate generalized H$H$-matrices is a subclass of the class of generalized H$H$-matrices and the convergence results of R. Nabben [R. Nabben, On a class of matrices which arises in the numerical solution of Euler equations, Numer. Math. 63 (1992) 411–431] are then extended to the class of generalized H$H$-matrices. Furthermore, the convergence of the block AOR iterative method for linear systems with generalized H$H$-matrices is established and some properties of special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper.     

A formulation to obtain semi-analytical planetary theories using true anomalies as temporal variables

The main methods used to obtain analytical theories of perturbed motion in celestial mechanics are based on the expansion of the disturbing function in trigonometric series of the mean anomalies (or longitudes). In this paper a new method based on the double Fourier series expansion using the true anomalies (or longitudes) is developed. The method involves a semi-analytical technique to allow the expansion of the inverse of the distance with great accuracy, and a new integration technique using a linear combination of the true anomalies based on an iterative method to integrate each term of the expansion of the Lagrange planetary equations.     

Static condensation,partial orthogonalization of basis functions,and ILU preconditioning in the hp-FEM

Static condensation of internal degrees of freedom, partial orthogonalization of basis functions, and ILU preconditioning are techniques used to facilitate the solution of discrete problems obtained in the hp-FEM. This paper shows that for symmetric linear (not necessarily positive-definite) problems, under mild technical assumptions, these three techniques are completely equivalent. In fact, the same matrices can be obtained by the same arithmetic operations. The study can be extended to nonsymmetric problems naturally.     

A Lie-theoretic setting for the classical interpolation theories

The three classical interpolation theories — Newton-Lagrange, Thiele and Pick-Nevanlinna — are developed within a common Lie-theoretic framework. They essentially involve a recursive process, each step geometrically providing an analytic map from a Riemann surface to a Grassmann manifold. The operation which passes from the (n−1)st to the nth involves the action of what the physicists call a group of gauge transformations. There is also a first-order difference operator which maps the set of solutions of the nth order interpolation to the (n−1)st: This difference operator is, in each case, covariant with respect to the action of the Lie groups involved. For Newton-Lagrange interpolation, this Lie group is the group of affine transformations of the complex plane; for Thiele interpolation the group SL(2, C) of projective transformations; and for Pick-Nevanlinna interpolation the subgroup SU(1, 1) of SL(2, C) which leaves invariant the disk in the complex plane. National Research Council Senior Research Associate at the Ames Research Center (NASA)}.     

A summability method for the arithmetic Fourier transform

The Arithmetic Fourier Transform (AFT) is an algorithm for the computation of Fourier coefficients, which is suitable for parallel processing and in which there are no multiplications by complex exponentials. This is accomplished by the use of the Möbius function and Möbius inversion. However, the algorithm does require the evaluation of the function at an array of irregularly spaced points. In the case that the function has been sampled at regularly spaced points, interpolation is used at the intermediate points of the array. Generally theAFT is most effective when used to calculate the Fourier cosine coefficients of an even function.In this paper a summability method is used to derive a modification of theAFT algorithm. The proof of the modification is quite independent of theAFT itself and involves a summation by primes. One advantage of the new algorithm is that with a suitable sampling scheme low order Fourier coefficients may be calculated without interpolation.