Proceedings of the Estonian Academy of Sciences

CONTENTS & ABSTRACTS

In English. Summaries in Estonian

Proceedings of the Estonian Academy of Sciences.

Physics * Mathematics

Volume 53 No. 2 June 2004

Special issue on approximation and regularization methods

Preface; 75–76

Arvet Pedas

On the choice of the regularization parameter for solving self-adjoint ill-posed problems with the approximately given noise level of data; 77–83

Uno Hämarik and Toomas Raus

Abstract. We consider ill-posed problems where the operator has a nonclosed range in the Hilbert space We assume that instead of noisy data are given, with the approximately known noise level The problem is regularized by the (iterated) Lavrentiev method, by iterative methods or by the method of the Cauchy problem. For the choice of the regularization parameter we propose a new a posteriori rule with the property that the regularized solution converges to the exact one in the process provided that the ratio is bounded for The error estimates are given.

Key words: ill-posed problems, regularization methods, Lavrentiev method, iterative method, method of the Cauchy problem, noise level, parameter choice.

Singular and hypersingular integral equations with the Hilbert kernel, delta-function, and method of discrete vortices; 84–91

Nina V. Lebedeva and Ivan K. Lifanov

Abstract. The singular and hypersingular integral equations with the Hilbert kernel, having the delta-function in their right-hand sides, are studied. For these equations a method of discrete vortices type is constructed and justified.

Key words: singular, hypersingular, integral equation, discrete vortex method.

On some properties of piecewise conformal mappings; 92–98

Jüri Lippus

Abstract. We study some properties of a multiresolution-like algorithm for piecewise conformal mapping, based on partitioning the complex plane into convex polygons and using appropriate window functions for these polygons. Some estimates for the nonconformity of the mapping are presented.

Key words: conformal mapping, subdivision, approximation.

Numerical solution of weakly singular Volterra integral equations with change of variables; 99–106

Arvet Pedas and Gennadi Vainikko

Abstract. To construct high-order numerical algorithms for a linear weakly singular Volterra integral equation of the second kind, we first regularize the solution of the integral equation by introducing a suitable new independent variable so that the singularities of the derivatives of the solution will be milder or disappear at all. After that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid.

Key words: weakly singular Volterra integral equation, piecewise polynomial collocation method, regularization of the solution.

Stability analysis of the fast Legendre transform algorithm based on the fast multipole method; 107–115

Reiji Suda

Abstract. The fast Legendre transform algorithm based on the fast multipole method proposed by Suda and Takami (Math. Comput., 2002, 71, 703–715) is discussed. The alpha-beta product is introduced as an indicator of instability, and the effects of the interpolations, splits, and shifts on the alpha-beta products are evaluated. The interpolations are proved to be stable. The instability of the splits and the shifts are evaluated numerically, and the stability is shown to be sufficient for practical use. The fast transform scheme must be applicable to other functions that are stable in the recurrence formula and the Clenshaw summation formula.

Key words: fast Legendre transform, fast multipole method, polynomial interpolation, matrix computation, matrix product, stability analysis, alpha-beta product.

On order optimal regularization under general source conditions; 116–123

Ulrich Tautenhahn

Abstract. We study the problem of solving ill-posed problems with linear operators acting between Hilbert spaces, where instead of exact data noisy data with a known noise level are given. Regularized approximations are obtained by a general regularization scheme. Assuming the unknown solution belongs to some general source set, we prove that the regularized approximations are order optimal on this set provided the regularization parameter is chosen either a priori or a posteriori by the Raus–Gfrerer rule or the monotone error rule. Our results cover the special cases of finitely and infinitely smoothing operators.

Key words: ill-posed problems, regularization, a priori parameter choice, a posteriori rules, order optimal error bounds, general source conditions.

GMRES and discrete approximation of operators; 124–131

Gennadi Vainikko

Abstract. Let Banach spaces and together with connection operators build a discrete convergence framework. Given an equation with we apply GMRES to an approximate (usually finite dimensional) equation with Under certain conditions we establish estimates of the residual and the error of the kth GMRES approximation to Applications to the Galerkin method for singular integral equations are discussed.

Key words: GMRES, optimal reduction factor, discrete convergence, singular integral equations, Galerkin method, fast solvers.

Instructions to authors; 132–134