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Report CopyRight/DMCA Form For : A Fast Iterative Method For Inverse Problems With Inexact

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Introduction,1 Introduction,2 Sequential subspace optimization. SESOP for linear inverse problems,RESESOP for inexact forward operators. RESESOP Kaczmarz for linear inverse problems,3 Application Dynamic computerized tomography. 4 Numerical results,5 Conclusion and outlook, A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 2 24. Introduction,Inverse problems with inexact forward operator.

We consider an inverse problem,A f g A D A X Y, where the given data g are subject to noise with noise level. In addition we assume that only an inexact version A of A is given with inexactness. 0 such that,kA f A f k for all f B 0 D A, include information on inexactness in reconstruction. A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 3 24. Introduction,Inexact forward operators two examples. Dynamic Computerized Tomography Talk by Bernadette Hahn Tuesday 9 40. A slight motion of the object during the scan affects the data. g s R f0 s f0 x s xT dx,We use the static model with inexactness. g s Rf0 s s f0 x s xT dx s,Magnetic Particle Imaging.

The high complexity of the physical model for the system function s suggests to use an. inexact model,u t c x s x t s x t dx c x s x t dx t. A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 4 24. Sequential subspace optimization,1 Introduction,2 Sequential subspace optimization. SESOP for linear inverse problems,RESESOP for inexact forward operators. RESESOP Kaczmarz for linear inverse problems,3 Application Dynamic computerized tomography. 4 Numerical results,5 Conclusion and outlook, A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 5 24.

Sequential subspace optimization SESOP for linear inverse problems. Notation and definitions, Let X Y be real Hilbert spaces and MA f g f X A f g the solution set. Hyperplanes halfspaces and stripes,Let u X 0 and R 0 We define the affine hyperplane. H u f X hu f i,the halfspace,H u f X hu f i,and the stripe. H u f X hu f i, A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 6 24. Sequential subspace optimization SESOP for linear inverse problems. Sequential subspace optimization SESOP for linear inverse problems in. Hilbert spaces,fn 1 fn tn i A wn i, In a finite index set wn i Y for all i In and the parameters tn tn i i In minimize.

hn t fn ti A wn i ti hwn i gi,Lemma Scho pfer Schuster Louis 2008. The minimization of hn t is equivalent to computing the metric projection. fn 1 PHn fn Hn Hn i,onto the intersection of hyperplanes. Hn i f X hA wn i f i hwn i gi, A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 7 24. Sequential subspace optimization SESOP for linear inverse problems. Adaptions of SESOP,Linear inverse problems with noisy data. Use stripes with width,Hn i f X A wn i,f wn i k MAf g.

Nonlinear inverse problems with noisy data,f X A0 fi wn i. k ctc kRi k,kwn i MA f g, A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 8 24. Sequential subspace optimization RESESOP for inexact forward operators. RESESOP for noisy data and inexact forward operator. kA f A f k for all f B 0 D A, Linear inverse problems with noisy data and inexact forward operator. f X A wn i,Hn i f wn i k MAf g, Nonlinear inverse problems with noisy data and inexact forward operator. f X A 0 fi wn i,kwn i k ctc kwn i k MA f g, A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 9 24.

Sequential subspace optimization RESESOP Kaczmarz for linear inverse problems. Semi discrete setting,Let Ak X Yk be linear bounded operators and. Ak f gk kgk gk k kAk Ak k,for all k 1 K as well as. n In for all n N wn i A n fi g n,for all i In n N,where n n mod K. Time dependent inverse problems, The index k may refer to different time points tk 0 T. Changes in the physical setting may be incorporated in k tk. Periodic motion in dynamic CT calls for a periodic function t. Reference state at tk tk 0, Rising temperature in MPI scanner during the scan increasing t.

A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 10 24. Sequential subspace optimization RESESOP Kaczmarz for linear inverse problems. RESESOP Kaczmarz algorithm, Choose a starting value f0 f0 B 0 X and constants k 1 k 1 K. As long as the discrepancy principle is not yet fulfilled i e. A n fn g n n n n,calculate the new iterate as,PH fn Hn H u. fn 1 n i n i n i,Here we choose In 0 1 n such that n In and. n i wn i g i,n i i i wn i, A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 11 24. Sequential subspace optimization RESESOP Kaczmarz for linear inverse problems. Convergence of the SESOP Kaczmarz algorithm,Now let 0 and k 0 for all k 1 K.

Theorem Blanke Hahn W 2019, Let fn n N be the sequence generated by the SESOP Kaczmarz algorithm with initial. value f0 and,In n N 1 n N N N 0 fixed,wi wn i A i fi g i for all i In. for all n N If there is an upper bound for the set of optimization parameters tn i i e. tn i t for all i In n N then fn n N converges strongly to a solution f of. Ak f gk k 1 K, A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 12 24. Sequential subspace optimization RESESOP Kaczmarz for linear inverse problems. The RESESOP Kaczmarz algorithm as a regularization method. Let n In n N 1 n N and wn i A i fi g i,for all i In in the. RESESOP Kaczmarz algorithm,Theorem Blanke Hahn W 2019.

a The RESESOP Kaczmarz algorithm yields a finite stopping index n. b We have fn fn for 0 0 where fn n N is the sequence of iterates. generated by the SESOP Kaczmarz algorithm,c If fn n N converges strongly then. f MAf g B 0, The previous statements also hold for the sequences frK r N and frK r N of full. A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 13 24. Application Dynamic computerized tomography,1 Introduction. 2 Sequential subspace optimization,SESOP for linear inverse problems. RESESOP for inexact forward operators,RESESOP Kaczmarz for linear inverse problems.

3 Application Dynamic computerized tomography,4 Numerical results. 5 Conclusion and outlook, A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 14 24. Application Dynamic computerized tomography,Dynamic computerized tomography. Motion in computerized tomography,The object undergoes motion during the scan. Typical examples from medical CT imaging, Breathing the entire body expands periodically global motion.

Heartbeat a local periodical motion, Problem If the motion is not taken into account this causes severe artefacts in the. reconstruction and details are no longer visible,Motion compensation. In general the motion is unknown or hard to estimate. interpret the motion of the object as a model inexactness. A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 15 24. 2 Sequential subspace optimization SESOP for linear inverse problems RESESOP for inexact forward operators RESESOP Kaczmarz for linear inverse problems 3 Application Dynamic computerized tomography 4 Numerical results 5 Conclusion and outlook A Wald Saarland University Fast iteration for inexact inverse problems August 8 2019 2 24

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