Numerical solution of differential and integral equations
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
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