Minimum Fisher Regularization

Derivation

The MFR (Minimum Fisher Regularization) employs the Fisher information as a objective functional \(O(x)\) introduced in the Generalized Tikhonov Regularization section. The Fisher information is expressed as

\[\begin{equation} O(x) = \int_{\mathbb{R}^3}\frac{\|\nabla x(\mathbf{r})\|_2^2}{x(\mathbf{r})}\mathrm{d}^3\mathbf{r}, \end{equation}\]

where \(x(\mathbf{r})\) is the unknown function parameterized by \(\mathbf{r} \in \mathbb{R}^3\). Using this functional has the advantage of seeking a solution that is smooth and has a localized structure. This non-linear functional can be linearized in the next section.

Definition

The first application of the MFR method to a fusion device was developped to resolve the ill-posedness of the tomography problem for TCV tokamak plasma [1] and later for the ASDEX Upgrade tokamak [2]. In the MFR method, the linearized Fisher information is employed as a objective functional, and the regularization matrix \(\mathbf{H}\) is defined as follows:

\[\begin{split}\begin{align} \mathbf{H} & = \sum_{i, j} \alpha_{ij} \mathbf{D}_i^\mathsf{T} \mathbf{W} \mathbf{D}_j\\ \mathbf{W} & = \rm{diag} \left( \cdots,\frac{1}{\max\left\{\mathbf{x}_\mathit{i}, \epsilon_0\right\}},\cdots \right), \end{align}\end{split}\]

where \(\mathbf{D}_{i,j}\) is derivative matrices along the \(i\) or \(j\) coordinate direction, \(\alpha_{ij}\) is the anisotropic parameter, \(\mathbf{W}\) is the weight matrix, \(\mathbf{x}_\mathit{i}\) is the \(i\)-th element of the unknown solution \(\mathbf{x}\), and \(\epsilon_0\) is a small positive number to avoid division by zero and to push the solution to be positive.

Implementation

The MFR method is the iterative method, and the iteration formula is:

1. Put \(\mathbf{x}^{(0)} = \mathbf{1}\) as the initial guess;

2. Compute \(\mathbf{W}^{(k)}, \mathbf{H}^{(k)}\) with \(\mathbf{x}^{(k)}\);

3. Solve \(\mathbf{x}^{(k+1)}\) optimizing regularization parameter \(\lambda\) by non-iterative inversion methods;

where \(k\) is the iteration number, and the iteration between step 2 and 3 is repeated until the convergence criterion is satisfied or the maximum iteration number is reached.

Several non-iterative inversion methods (e.g. L-curve method) can be used in step 3. This workflow is illustrated in the following figure.

The MFR solution is derived iteratively in the above workflow.

Example

The example shows in Example/MFR tomography.

References

[1]

M Anton, H Weisen, M J Dutch, W Von Der Linden, F Buhlmann, R Chavan, B Marletaz, P Marmillod, and P Paris. X-ray tomography on the TCV tokamak. Plasma Phys. Controlled Fusion, 38(11):1849–1878, 1996. doi:10.1088/0741-3335/38/11/001.

[2]

T Odstrčil, T Pütterich, M Odstrčil, A Gude, V Igochine, and U Stroth. Optimized tomography methods for plasma emissivity reconstruction at the ASDEX upgrade tokamak. Rev. Sci. Instrum., 87(12):123505, 2016. doi:10.1063/1.4971367.