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 *k*th
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

Copyright
Transfer Agreement; 135