SPLINE ALGORITHMS FOR DATA PROCESSING AND SOLVING SOME INVERSE PROBLEMS
Grebennikov A.I.
SRCC of the Moscow Lomonosov State University
Vorobjovy Gory, Moscow 119899, Russia.
e-mail: aig@srcc.msu.su, fax: (095) 938 21 36.

A class of incorrect problems similar on unstability with a problem of
differentiation, such as problems of numerical calculation of function's
derivatives, solving the Fredholm (with logarithmic singularity in a
kernel), Abel,Volterra type integral equations of the 1-st kind is
considered. We assume that the input data are given approximately and
discretely by a set of functionals.
The author suggested and theoretically justified in
operator form a stable spline-approximation method for solving a considered
class of problems. The essence of a method consists of two steps. The first
one is a special processing of initial data for construction spline by consecutive
smoothings. The smoothings are carried out consecutively some
times, but does not require the solution of linear algebraic equations
systems, as it is based on the explicit formular, namely on a local
approximating splines. Therefore its realization is very effective. At the
second step the derivatives are calculating, or
the value of the inverse operator (if it is given by known formular)
on the smoothed data is calculating.
Otherwise the approximate solution is searched in a form of
spline with unknown coefficients, which are found from a collocation
conditions for a smoothed right-hand side of equation.
At certain dependence a number of smoothings from parametr of discretization
and level of an error regularization takes place.
At the present time the author suggested, theoretically and numerically justified
the algorithms of solving coefficients invers problems for ordinary and partial
differential equations so as of solving two-dimensional integral equations
of the 1-st kind with singularity in a kernel. Some applications and
numerical results are discused.