# Институт вычислительной математики

и математической геофизики

# The International Conference on Computational Mathematics

ICCM-2004

June, 21-25, 2004
Novosibirsk, Academgorodok

## Abstracts

*Numerical solution of differential and integral equations*

## A numerical solution for the vector tomography problem with usage of polynomial and local bases

**Sobolev Institute of mathematics SD RAS (Novosibirsk)**
The main problem of vector field tomography consists in a
reconstruction of an unknown vector field, given in a bounded
domain, by its known ray transform. This problem can be formulated
as an inverse problem for reconstruction of the right-hand part
(of a special type) of the linear transport equation, as well as
the problem of solving of an integral equation of the first kind.
It is known that the operator of ray transform of a vector field
has non-trivial kernel, so it is possible to reconstruct uniquely
only the solenoidal part of the original vector field.

An approach of direct reconstruction of the solenoidal part of the
original vector field is developed and justified in frameworks of
least squares method. For the purpose of direct reconstruction of
the solenoidal part of the field the procedure of construction of
solenoidal bases of polynomial and local types is developed. The
constructed bases may have some prescribed properties on a
boundary of the domain. The solenoidal bases of vector fields are
constructed either for a medium with rectilinear rule of ray
propagation or for a medium with refraction. In 2D-case the method
of direct reconstruction is realized algorithmically.

*Note. Abstracts are published in author's edition*

© 1996-2000, Siberian Branch of Russian Academy of Sciences, Novosibirsk

Last update: 06-Jul-2012 (11:52:06)