Sobolev Institute of Mathematics
The talk consists of three sections. Section 1 presents a new approach to the problems of tomography (integral geometry) on the plane. This approach treats tomographic problems as boundary-value problems for elliptic equations with operator coefficients. More exactly, we reduce the inverse problem of finding the right-hand side of the stationary one-velocity transport equation to the boundary value problem for an elliptic equation with operator coefficients on the plane. Particular cases of these inverse problems are the problem of inverting the Radon transform in the fan-beam statement and the problem of emission tomography. We present new inversion formulas, numerical algorithms and stability estimates for this problem. The results of this section were obtained partially in collaboration with S.G. Kazantsev and A.A. Bukhgeim. In Section 2 we consider inverse problems for the elliptic equations. Our results generalize well-known theorems obtained by Sylvester, Uhlmann and recent results of Bukhgeim, Uhlmann about recovering a potential from partial Cauchy data. In Section 3 we discuss numerical methods for inverse problems and in particular the stability theory of difference schemes for ill-posed Cauchy problems for partial differential equations. The main techniques employed in sections 2 and 3 are estimates of Carleman type.
Please join us for refreshments before the lecture at 2:30p.m. in room 353 Jabara Hall.
[ Spring 2002]