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Regularization of Nonlinear Inverse Problems

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Inverse Problems

Part of the book series: Lecture Notes in Geosystems Mathematics and Computing ((LNGMC))

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Abstract

Nonlinear inverse problems are much more difficult to solve than linear ones and the corresponding theory is far less developed. Each particular problem may demand a specific regularization. Nevertheless, one can formulate and analyze basic versions of nonlinear Tikhonov regularization and nonlinear iterative methods, which will be done in Sects. 4.1 and 4.6, respectively. These basic versions serve as starting points for regularizations adapted to our nonlinear model problems (Sects. 4.2, 4.5, and 4.6). Nonlinear Tikhonov regularization leads to nonlinear least squares problems. Sections 4.3 and 4.4 present algorithms to solve these numerically.

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Notes

  1. 1.

    If there is no \(\hat{x} \in D\) with \(F(\hat{x}) = y\) due to discretization errors, then hopefully there will at least exist some \(\bar{y} \approx y\) such that a solution \(\hat{x} \in D\) of \(F(x) =\bar{ y}\) exists. If we know δ > 0 such that \(\|\bar{y} - y^{\delta }\|_{2} \leq \delta\), we can tacitly add discretization to measurement errors, interpreting \(\bar{y}\) as “true” right hand side.

  2. 2.

    To arrange multi-indexed components into a vector, a certain index ordering has to be fixed. In case of rectangular index grids, we will always use a row-wise ordering. For example, the indices α ∈ G n for the grid G n defined below, will always be ordered to form the sequence

    $$\displaystyle{(-n,-n),\ldots, (-n,n),\quad \ldots,\quad (n,-n),\ldots, (n,n).}$$
  3. 3.

    This is generally true even if rank(J) = n does not hold, provided only that ∇Z(x i) ≠ 0, see [Bjö96], p. 343.

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  1. Fakultät für Elektrotechnik und Informationstechnik, Universität der Bundeswehr München, Neubiberg, Germany

    Mathias Richter

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  1. Mathias Richter
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Richter, M. (2016). Regularization of Nonlinear Inverse Problems. In: Inverse Problems . Lecture Notes in Geosystems Mathematics and Computing. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-48384-9_4

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