MULTIVARIATE INTERPOLATION USING POLYHARMONIC SPLINES

Data measuring and further processing is the fundamental activity in all branches of science and technology. Data interpolation has been an important part of computational mathematics for a long time. In the paper, we are concerned with the interpolation by polyharmonic splines in an arbitrary dimension. We show the connection of this interpolation with the interpolation by radial basis functions and the smooth interpolation by generating functions, which provide means for minimizing the L2 norm of chosen derivatives of the interpolant. This can be useful in 2D and 3D, e.g., in the construction of geographic information systems or computer aided geometric design. We prove the properties of the piecewise polyharmonic spline interpolant and present a simple 1D example to illustratethem

Data approximation using polyharmonic radial basis functions

summary:The paper is concerned with the approximation and interpolation employing polyharmonic splines in multivariate problems. The properties of approximants and interpolants based on these radial basis functions are shown. The methods of such data fitting are applied in practice to treat the problems of, e.g., geographic information systems, signal processing, etc. A simple 1D computational example is presented

Spherical basis function approximation with particular trend functions

summary:The paper is concerned with the measurement of scalar physical quantities at nodes on the $(d-1)$-dimensional unit sphere surface in the \hbox{$d$-dimensional} Euclidean space and the spherical RBF interpolation of the data obtained. In particular, we consider $d=3$. We employ an inverse multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 defined in Cartesian coordinates. We prove the existence of the interpolation formula of the type considered. The formula can be useful in the interpretation of many physical measurements. We show an example concerned with the measurement of anisotropy of magnetic susceptibility having extensive applications in geosciences and present numerical difficulties connected with the high condition number of the matrix of the system defining the interpolation

A comparison of the accuracy of the finite-difference solution to boundary value problems for the Helmholtz equation obtained by direct and iterative methods

summary:The development of iterative methods for solving linear algebraic equations has brought the question of when the employment of these methods is more advantageous than the use of the direct ones. In the paper, a comparison of the direct and iterative methods is attempted. The methods are applied to solving a certain class of boundary-value problems for elliptic partial differential equations which are used for the numerical modeling of electromagnetic fields in geophysics. The numerical experiments performed are studied from the point of view of the time and storage requirements and the achieved accuracy of the solution

Three-dimensional reconstruction from projections

summary:Computerized tomograhphy is a technique for computation and visualization of density (i.e. X- or $\gamma$-ray absorption coefficients) distribution over a cross-sectional anatomic plane from a set of projections. Three-dimensional reconstruction may be obtained by using a system of parallel planes. For the reconstruction of the transverse section it is necessary to choose an appropriate method taking into account the geometry of the data collection, the noise in projection data, the amount of data, the computer power available, the accuracy required etc. In the paper the theory related to the convolution reconstruction methods is reviewed. The principal contribution consists in the exact mathematical treatment of Radon's inverse transform based on the concepts of the regularization of a function and the generalized function. This approach naturally leads to the employment of the generalized Fourier transform. Reconstructions using simulated projection data are presented for both the parallel and divergent-ray collection geometries

On numerical evaluation of integrals involving Bessel functions

summary:The paper is concerned with the efficient evaluation of the integral $\int^\infty_0 f(x)J_n(rx)dx$, where $J_n$ is the Bessel function of index $n$ and $n$ is a nonnegative integer, for a given sequence of values of a real parameter $r$. Two procedures are proposed and compared. One of them consists in a direct generalization of a procedure for the evaluation of of a similar integral with the weight function exp $(irx), which employs the fast Fourier transform. The other approach is based on the construction of a special Gaussian quadrature formula where$J_n$ appears as a weight. The results of the comparison show that the application of the Gaussian formula is much more efficient