Internationale conference ”Inverse and Ill-Posed Problems of Mathematical Physics”, dedicated to Professor M.M.Lavrent’ev in occasion of his 75-th birthday, August 20-25, 2007, Novosibirsk, Russia

On stability and approximate solution of inverse and computerized tomography problems

D.K.Ugulava and D.N.Zarnadze

(Niko Muskhelishvili Institute of Computing Mathematics, 0193 Tbilisi, Georgia)

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Let H be a Hilbert space and K:HH be compact selfadjoint one-to-one operator, having everywhere dense range. An ill-posed problem Ku=f in a Hilbert space H with the operator K having positive eigenvalues is considered. It is assumed that the solution existence and uniqueness conditions are satisfied, but the stability is not [1]. This means that the inverse operator K-1 is not continuous. Similarly A.N.Tikhinov [2], this equation is transfered in the Ftechet space D(K-), where K- : D(K-)D(K-) is defined by equality

K-(x)={ K-1x, K-2x,..., K-nx,...}.

Due to this notation, the space D(K-) acquire a new meaning that differs from classical case, where D(K-) was whole symbol, where K-, if taken separately, meant nothing.

The Frechet space D(K-), as a set is the part of the Hilbert space H and the restriction K of the operator K on the space D(K-) maps the space D(K-) isomorpically onto. More exactly, restriction of K on D(K-), taking into account the topology, coincides with the restriction of AN from the Frechet-Hilbert space HN to D(A) and is selfadjoint operator on Frechet space D(K-), which isomorphically maps D(K-) onto [3]. Therefore, the considered equation Ku=f has in the Frechet space D(K-) unique and stable solution. For approximate solution of this operator equation in the energetic Frechet space of the operator K Ritz’s extended method is applied [3].

These results are also applied to operators, mapping a separable Hilbert space into the same space and admitting a singular decomposition. In particular, the well-known Radon transformation admitts the singular decomposition [4] and therefore the Ritz’s extended method is applied to the computerized tomography problem.

It will be also noted, that according to Schwartz theorem Radon transformation is one-to-one operator from the Frechet-Schwartz space in the same space. The classicalmethod of least squares is generalized for the equations in a Frechet spaces [5].This method can be also used for the approximate solution of equation containing Radon transformation and acting from a Frechet-Schwartz space in the same space.

References

  1. D.K.Ugulava and D.N.Zarnadze. On the application of Ritz’s extended method for some ill-posed problem. Reports of enlarged session of I.N.Vekua Inst. of applid Math. 2006, v.24, N3.
  2. A.N.Tikhinov. On stability of inverse problems. Dokl. AN SSSR, v.39, 5(1943), 195-198 (in Russian).
  3. D.N.Zarnadze and S.A.Tsotniashvili. Selfadjoint operators and generalized central algorithms in Frechet spaces. Georgian Math. Journal, v.13(2006), n.2, 324-343.
  4. F. Natterer. The mathematics of Computerized Tomography. Stuttgart.

5. D.N. Zarnadze. A generalization of the method of least squares for operator equations in some Frechet spaces. Izv. Akad. Nauk Russia. Ser.Mat. 59 (1995). 59-72; English transl. in Russian Acad. Sci Izv. Math. 59:5 (1995), 935-948.