*Background:*
My research interests are primarily concerned with the theory of splines
and their applications to computer aided geometric design (CAGD).
The design of curves and surfaces plays an important role not only in the
construction of different products such as car bodies, ship hulls,
airplane fuselages and wings, etc., but also in the description of
geological, physical and even medical phenomena. New areas of CAGD
applications include computer vision and inspection of manufactured
parts, medical research (software for digital diagnostic equipment),
image analysis, high resolution TV systems, cartography, the film
industry, etc.

*Exact Solutions of Finite-Difference Schemes:*
At the first stage of my scientific career I was involved in the numerical
solution of problems of mathematical physics in several variables.
It was the time when the method of fractional steps, better known as
the method of splitting, gained popularity in many areas of scientific
computing. I constructed high-order accuracy finite-difference schemes for
Laplace and biharmonic equations.
Using the idea of fractional steps, I suggested efficient
iterative algorithms for the numerical treatment of the corresponding
finite-difference schemes. In the case of a difference analogue of
the Dirichlet problem for Laplace and biharmonic equations on a 2-D square
domain and uniform mesh with N knots in both directions one obtains
the exact finite-difference solution after N iterations.
This is a much more efficient approach than the traditional successive
over-relaxation technique.
Later these results were extended by different authors to the equations
of elasticity and plasticity and to multiply connected domains.

*Optimal Error Bounds for Spline Approximation:*
In the year 1980 I completed a book on
Methods of Spline Functions in collaboration with Professor
Yuri S. Zav'yalov and Dr. Valeri L. Miroshnichenko (Sobolev Institute of
Mathematics in Novosibirsk, Russia), published by Nauka in Moscow.
This book is a comprehensive survey of different efficient algorithms for
computing 1-D and 2-D splines. Our own main results relate to a new technique
which we developed to obtain optimal error bounds for polynomial spline
interpolation. This method is based on an integral representation
of the error estimate. In many cases
it gives minimal values of constants in error bounds for interpolation
splines. The same approach is also used to find optimal error bounds for
local spline approximation methods. This book has become a standard textbook
for students, researchers and engineers in Russia and over eleven thousands
copies were sold.

*Shape-Preserving Spline Interpolation:*
Standard methods of spline functions do not preserve shape properties of the
data. By introducing shape control parameters into the spline structure, one
can preserve various characteristics of the initial data including positivity,
monotonicity, convexity, linear and planar sections. Based on interpolating
splines, methods with shape control are usually called
methods of shape-preserving spline interpolation.
Here the main challenge is to develop algorithms that choose shape control
parameters automatically.
The majority of such algorithms solve the problem only for some special data.
To solve the problem in a general setting, I gave a classification of the
initial data and reduced the problem of shape-preserving interpolation to the
problem of Hermite interpolation with constraints of inequality type.
The solution is a C2 local generalized tension spline with additional knots.
This allows for development of a local algorithm of shape-preserving spline
interpolation where shape control parameters are selected automatically
to meet the monotonicity and convexity constraints for the data.
Its application makes it possible to give a complete solution to the
shape-preserving interpolation problem for arbitrary data and isolate the
sections of linearity, the angles, etc.

*Tension GB-Splines:*
In my opinion, my most significant contribution
to the theory of splines, involves the development of
"direct methods"
for constructing explicit formulae for tension
generalized basis splines (GB-splines for short) and finding
recursive algorithms for the calculation
of GB-splines. This approach has yielded new local bases for various tension
splines including among others rational, exponential, hyperbolic,
variable order splines, and splines with additional knots.
These results were expanded to GB-splines of arbitrary
order. I investigated the main properties of GB-splines and their series
such as shape-preserving properties, invariance with respect to affine
transformations, etc. I have shown that the GB-spline
series is a variation diminishing function and the systems of GB-splines
are weak Chebyshev systems.

*Shape-Preserving Local Spline Approximation:*
The method of local spline approximation due to Lyche and Schumaker [J. Aprox. Theory 15(1975)],
combined with recursive algorithms for computing polynomial splines,
was found to be efficient in practical applications. However it gives
a curve which only approximates the data, but changes the data shape
substantially. For data with specified tolerances, I developed algorithms
of shape-preserving local spline approximation based on GB-splines with
automatic selection of the shape control parameters.
These parameters are chosen to satisfy the given tolerances and
the monotonicity and convexity conditions for the initial data.
Such methods are successfully used in the shape-preserving approximation
of multivalued surfaces where the initial data is assumed to be given as
a set of pointwise-assigned non-intersecting curvilinear sections of a
3-D solid. Additionally, my algorithms of shape-preserving parametrization
may be used to improve the quality of representation of the shape-preserving
spline curves and surfaces. This technique is described in full detail in
my book
Methods of Shape-Preserving Spline Approximation.
World Scientific Publ. Co. Pte. Ltd., Singapore, 2000.

*Research plans:*
My current research interests include shape-preserving spline approximation,
difference methods for constructing splines, surface reconstruction from
cross-sections, and scientific visualization.

*Shape-Preserving Curve and Surface
Approximation:*
A fundamental requirement of all CAGD software is to be able to approximate
data in a manner which produces either a curve or surface which has the same
"shape" as the data. The need for such methods is essential for a wide range
of applications in industrial design, and in the visualization of scientific
data. The known algorithms of shape-preserving approximation with automatic
selection of shape control parameters are mainly comonotone,
i.e. the spline on the i-th data interval is
increasing or decreasing with the data on that interval. Such splines have
the disadvantage that they must have slope zero at a point where the
neighboring secant lines have a sign change in their slope; hence, any local
extrema of the underlying approximation are assumed to be in the data sample.
I am working on developing algorithms of shape-preserving approximation with
automatic choice of shape control parameters which are free of this
disadvantage and generalize comonotone approximations.

*Difference Methods for Constructing Splines:*
Spline theory is mainly
grounded on two approaches: the algebraic one (where splines are understood
as smooth piecewise functions) and the variational one (where splines are
obtained via minimization of quadratic functionals with equality and/or
inequality constraints). Although less common, a third approach, where
splines are defined as the solutions of differential multipoint boundary
value problems (DMBVP for short), has been considered. Even though some of
the important classes of splines can be obtained from all three schemes,
specific features sometimes make the last one an important tool in practical
settings. I want to develop difference methods for the construction of
shape-preserving splines. In the 1-D case the basic DMBVP gives hyperbolic
splines. In the 2-D case, I plan to consider a generalization of thin plate
splines. This approach permits to avoid evaluation of hyperbolic and other
special functions. It can be generalized to smoothing splines and to the
approximation of scattered data in a straightforward manner.

*Surface Reconstruction From Cross-Sections:*
This application is widely
used in automobile and aircraft manufacturing, shipbuilding, and in Computer
Aided Tomography. The aim is to provide a
method for reconstruction a complete surface from a sequence of
cross-sectional images or contours. In tomography one has the problem of
matching appropriate points on adjacent contours, and when branching
occurs within the object, i.e. a single contour at one level branches into
two or more at the next. Research is in progress to devise algorithms to
perform reconstruction automatically with local shape control.

*Scientific Visualization:*
Visualization involves the use of computer
graphics tools to explore data resulting from measurements, or data produced
by simulation models. Visualization of complex data is an excellent way to gain
a better understanding of the system under study, and it is also valuable as
a means of communication. This is an exciting area which combines many
skills: graphics, knowledge of the system under study, interface design, and
imagination. Visualisation has widespread applications whenever a complex
system is being investigated. The methods of shape-preserving spline
approximation of curves and surfaces described in my above mentioned book
were realized as a package of computer programs which enable one to construct
and render complex multivariate surfaces. I plan to improve this package
for purposes of scientific visualization and aim to include it in standard
software Matlab.

Page Last Updated on February 12, 2008