In English. Summaries in Estonian

Proceedings of the Estonian Academy of Sciences.

Physics * Mathematics


Volume 53 No. 3 September 2004


On the linear spline collocation for pseudodifferential equations on the torus; 139–147

Juha Anttila, Jyri Hämäläinen, and Jukka Saranen

Abstract. We examine the spline collocation method for a class of pseudodifferential equations on a two-dimensional torus. In the analysis, we assume nonuniform mesh, continuous piecewise linear splines, and nodal point collocation. By employing the “Arnold–Wendland trick”; we are able to carry out the stability and convergence analysis. The results show quasioptimal order estimates for the convergence of the collocation solution.

Key words: boundary elements, collocation.

On a method of the construction of smoothing histosplines; 148–155

Natalia Budkina

Abstract. The problem of the approximation of a given histogram by a function from Sobolev space under inequality constraints for area matching conditions is considered. The smoothing problem is reduced to the problem of linear programming with some nonlinear restrictions.

Key words: smoothing problem, histospline simplex method.

Pseudodifferential calculus on the 2-sphere; 156–164

Ville Turunen

Abstract. We show how pseudodifferential equations on the unit sphere of the 3-dimensional Euclidean space can be studied using the spherical harmonic Fourier series on the symmetry group of the sphere.

Key words: pseudodifferential operations, symbol calculus, asymtotic expansions, spherical harmonics rotation group.

A convergence theorem for approximate methods of tangent hyperbolas; 165–176

Indrek Kaldo and Otu Vaarmann

Abstract. For solving an operator equation F(x) = 0, where F is a nonlinear operator from a Banach space X into another Banach space Y, approximate variants of the method of tangent hyperbolas are developed, provided F is twice Frechet-differentiable and its first derivative has the uniformly bounded inverse. A local convergence theorem is provided for the methods under consideration and their computational aspects are briefly discussed.

Key words: nonlinear equation, Banach space, Newton’s method, cubically convergent method, approximate variants of methods, midpoint method.

Optimization problems with points of discontinuity and discrete arguments; 177–185

Ants Tauts

Abstract. Minimization of such functions is considered, where some arguments are related to the final function by intermediate functions with discontinuity points, but other arguments have only 0 and 1 for the allowed values, although the theoretical generalization allows also intermediate values. Both of the circumstances create difficulties in the use of the gradient method. We solve the first problem by approximation, primarily by a square polynomial obtained using the integral form of the least squares method, and later by the partial sums of orthogonal series of the wave function treated with the logarithmic averages method. The second problem can be solved with the help of the planes, which have been taken in the n-dimensional space in such a way that any allowed point on the side of the space relative to this plane is better than all the points on the other side.

Key words: optimization, discontinuity points, discrete values, least squares method, wave func­tions, logarithmic average.

On the description of stochastic systems; 186–200

Jaak Heinloo

Abstract. A set-up of the systemic description of a (stochastic) system, particular states of which form (on the lowest determination level of the system state) a population of random events, is presented. Elementary and hierarchic stochastic systems are considered and the structure of their description is formulated. As an application of the conception, the set-up of the description of liquid media motion is considered.

Key words: probability, system, liquid media.