# Regularization of Inverse Problems (Mathematics and Its Applications) The first is that5 the elements in the various components represent different physical entities. As an example, it might be severely ill-posed w. We will4 use the term multicomponent inverse problem to refer to the inverse problem in5 Definition 5. In order to develop a reasonable mathematical theory for e regularization methods we need of course to restrict the operators T we choose 7 to consider. Following [ 1 4], this is done by introducing a functional which performs this. Then p G SC is a S p -minimizing least squares. We therefore restrict our is description to this case. One can also choose to let the u regularization parameter depend on the iterates.

In particular, for iterated. L with the splitting in 2. One can of course regularize the inverse problem in Definition 5. Now, for fixed f [l] , assume. This method is applicable to. Similarely, consider the inverse problem of rec ponent when -component is assumed to be known. If the mul- ticomponent inverse problem is not ill-posed w.

To make matters interesting, assume that the multicomponent inverse problem is ill-posed w. Under these assumptions, our iterates defined by8 steps S - 13 take the form. In implementations of the algorithm, the value of b is some- 9 times decreased as the number of iterations increases. When we keep the prior w fixed, then the above iterates are a special case of a component-wise reconstruc-.

However, in our method. The result of the invention is a computer. The concretes definition of this transform should be understood by the persons skilled9 in the art for the inverse problem in question.

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This process. It could however also depend on the updated prior and the pre-. It could however also depend on the updated prior and8 the pre-processed data itself. It could however also depend on the updated prior and the pre-7 processed data itself. In scattering theory one usually s wants a potential that fulfills the Rollnick condition which would be the case.

Under the assumptions and approximations. A precise expression of these can e. The actual measured data and forward operator The expression for the actual data measured on a micrograph needs to account for the counting1 stochasticity and the detector usually a slow-scan CCD camera.

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Then, the resulting expression for thes forward operator reads as 7 Definition 5. Wc have a fixed finite set SQ of directions on a smooth curves S C S 2 that defines our data collection geometry.

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Additional difficulties The real experimental situation in ET is unfortu- nately more complicated and a proper formulation of the inverse problem shows that one needs to recover the values of a number of parameters along with the scattering potential. The short exposition here closely follows the one given in .

First, there are parameters that have no corresponding interpretation in the actual real-world ET experiment. Formally, different choices of these parameters can define the an optical setup that has the same imaging properties as the actual setup in the TEM. However, there is a natural way to choose these parameters prior to reconstruction and therefore we considered them as known to sufficient degree of accuracy.

Next, 1 there are parameters, such as detector related parameters e. Some of these, such as the 7 wavenumber k and magnification M, are in most cases known to sufficient degree a of accuracy. The problem of determining2 these parameters is usually referred to as CTF estimation. There are two4 examples of such parameters. Under this assumption, the6 intensity generated by a single electron becomes.

## Geophysical Inverse Theory and Regularization Problems

Bakushinsky and M. Number in Mathematics and Its Applications. Springer Verlag, Cimeticie, F. Dclvaie, M. Jaoua, and F. Solution of the Cauchy problem using iterated Tikhonov regularization. Inverse Problems, , Engl, M.

## [PDF.31ar] Regularization of Inverse Problems (Mathematics and Its Applications)

Hanke, and A. Regularization of inverse problems, volume of Mathematics and its Appkations. Engl and P.

Nonlinear inverse problems: theoretical aspects and some industrial applications. Capasso and J. The European Consortium for Mathematics in Industry, pages Fanelli and O. Electron tomography: A short overview with an emphasis on the absorption potential model for the forward problem. Submitted to Inverse Problems, Three-dimensional electron microscopy of macromolecular assemblies. Oxford University Press, 2nd edition, Golub and V. Separable nonlinear least squares- the variable projection method and its applications.

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## Applications of Tikhonov regularization to inverse problems using reproducing kernels - IOPscience

The inverse problem of emission tomography. Hanke and C. Nonstationary iterated Tikhonov regularization. Journal of Optimization Theory and Applications, 98 l , Hawkes and E. Principles of Electron Optics.

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