Madych and Nelson(1990) proved Multiquadric (MQ) mesh-independent Radial Basis Functions (RBFs) enjoy exponential convergence. The primary disadvantage of the MQ scheme is that it is global, hence the coe cient matrices obtained from this discretization scheme are full. Full matrices tend to become progressively more ill-conditioned as the rank increases. In this paper, we explore several techn...

This paper applies the multiquadric (MQ) as a spatial approximation scheme for solving the nonlinear Burgers' equation. For comparison purposes, a low order explicit nite diierence approximation of the time derivative is employed. By decreasing the time step of the computation , it is shown that the major numerical error is from the time integration instead of the MQ spatial approximation. The ...

The problem of approximating smooth L p-functions from spaces spanned by the integer translates of a radially symmetric function is very well understood. In case the points of translation, , are scattered throughout R d , the approximation problem is only well understood in the \stationary" setting. In this work, we provide lower bounds on the obtainable approximation orders in the \non-station...

An algorithm for use with several meshless schemes is presented based on a local extremum diminishing property. The scheme is applied to the Euler equations in two dimensions. The algorithm is suitable for use with many meshless schemes, three of which are detailed here. First, a method based on Taylor series expansion and least squares is highlighted. Next, a similar least squares method is us...

This article pertains to interpolation of Sobolev functions at shrinking lattices hZ from Lp shift-invariant spaces associated with cardinal functions related to general multiquadrics, φα,c(x) := (|x| + c). The relation between the shift-invariant spaces generated by the cardinal functions and those generated by the multiquadrics themselves is considered. Additionally, Lp error estimates in ter...

A meshfree approximation scheme based on the radial basis function methods is presented for the numerical solution of the options pricing model. This thesis deals with the valuation of the European, Barrier, Asian, American options of a single asset and American options of multi assets. The option prices are modeled by the Black-Scholes equation. The θ-method is used to discretize the equation ...

Shepard interpolation is known to produce flat horizontal spots at and around data points. The phenomenon is caused by Shepard’s use of rational inversedistance weight functions, producing similar values around data points. In this paper, a parametric version of Shepard interpolation is introduced that avoids flat horizontal spots. Because Shepard interpolation or its parametric version does no...

The total variation (TV) denoising method is a PDE-based technique that preserves the edges well but has undesirable staircase effect in some cases, namely, the translation of smooth regions (ramps) into piecewise constant regions (stairs). This paper introduces a novel mesh-free approach using TV (ROFmodel) regularization and radial basis function (RBF) for the numerical approximation of TV-ba...