The methods based on splines rank high among the most powerful tools of
computational mathematics. While as early as in the beginning of the 70’s
the word “spline” was sounded just exotic even for many mathematicians,
now it became generally accepted not only among specialists in approximation
theory and computational mathematics but also among engineers related to
solving applied problems on computers. The popularity of splines is explained
mainly by two reasons. First, splines are an extremely powerful and versatile
tool for solving various problems of approximation of functions. Being independently
impotent these problems, lie in the base of many methods of computational
mathematics. Second, algorithms constructed with the aid of splines are
effectively realized on computers.
It should be noted that splines appeared and began developing in response
to calls of practice at that time, when the failure of classical approximation
methods, e.g., by algebraic or trigonometric polynomials, in solving of
the most important applied problems became clear. Splines justified hopes,
setting on them. Now they still remain on the leading edge of applications
of mathematics. Their improvement continues, and their capabilities are
far from being exhausted.
The work on the theory and applications of splines at the Institute of
Mathematics began in the middle of the 60’s under the guidance of doctor
of physical and mathematical sciences, Professor Yu. S. Zavyalov
(1931-1998). He was among pioneers in the development of this new field
of computational mathematics in our country.
The characteristic feature of the spline school created by Yu. S. Zavyalov
is in close connection of theoretical investigations with solving applied
problems. Therefore considerable efforts were devoted first of all to
study of cubic splines that are most often used in computational practice.
The result obtained in this direction are as follows:
there were studied the questions, connected with the algorithms of
construction of interpolating, smoothing, and locally approximating
splines of one and many variables;
there were obtained the estimates (in many cases exact ones) of the
error of approximation by cubic splines;
there were proposed various constructions of generalized cubic splines;
there were developed the method of spline collocation for solution
of boundary value problems.
There were studied the questions connected with the stability of the
algorithms of constructing polynomial splines of arbitrary odd degree.
The problem of asymptotic decomposition of approximation error for periodic
splines of arbitrary degree on a uniform grid was completely solved.
There were the methods of approximation of curves and surfaces by parametric
splines. These methods serve as a basis for mathematical modeling of objects
of complex geometric shape.
The problem of interpolation preserving the properties of monotonicity
and convexity of initial data (isogeometric interpolation) was studied
in detail.
In recent time the considerable attention is paid to developing methods
of approximation functions of many variables, by their values in the arbitrary
scattered points.
Investigations on the theory of splines and its applications are carried
on in the Laboratory of the Theory of Spline Functions. Along with the
studying theoretical questions, the work on development of software for
solving various problems by spline methods is also carried on.
