Tikhonov Regularization

Tikhonov regularisation penalises a quadratic norm of the conductivity or of its deviation from a reference ${\sigma }^{\ast }$:

$${\mathcal{R}}_{L^2}(\sigma )=\frac{1}{2}\Vert \sigma -{\sigma }^{\ast }{\Vert }_{L^2(\Omega )}^2,{\mathcal{R}}_{H^1}(\sigma )=\frac{1}{2}\Vert \nabla \sigma {\Vert }_{L^2(\Omega )}^2=\frac{1}{2}{\int }_{\Omega }|\nabla \sigma {|}^2\mathrm{d}x.$$

The $L^2$ version prefers small deviations. The $H^1$ seminorm version prefers smooth conductivities and does not penalise constants.

Discretisation. With ${\sigma }_h={\sum }_i{z}_i{\phi }_i$ in a finite element space,

$${\mathcal{R}}_{L^2}(z)=\frac{1}{2}(z-z^{\ast }{)}^{\top }M(z-z^{\ast }),{\mathcal{R}}_{H^1}(z)=\frac{1}{2}{z}^{\top }Kz,$$

with the Mass Matrix $M$ and the Stiffness Matrix $K$. For $\beta {\mathcal{R}}_{H^1}$ the gradient is $\beta Kz$ and the Hessian $\beta K$. The $H^1$ version needs a continuous space ($P_1/Q_1$ or higher); for piecewise constant $\sigma$ one uses a discrete gradient (jumps across faces) instead.

Proximal operator. For use in ADMM (see Proximal Operator):

$$\mathrm{prox}(y)=\arg \underset{z}{\min }\frac{\beta }{2}{z}^{\top }Kz+\frac{\rho }{2}\Vert z-y{\Vert }^2⟺(\beta K+\rho I)z=\rho y.$$

If the proximity term is measured in the $L^2(\Omega )$ norm $\Vert z-y{\Vert }_M^2$, the system becomes $(\beta K+\rho M)z=\rho My$. The matrix is symmetric positive definite, so the problem is strongly convex with a unique minimiser, and CG solves it efficiently.

Properties. It is simple, convex and differentiable, and is the MAP estimate under a Gaussian prior. Its drawback is that edges are blurred. For piecewise constant targets use Total Variation.

In the linearised setting, the Tikhonov solution of $J\delta =r$ is $\delta =(J^{\top }J+\beta L^{\top }L{)}^{-1}{J}^{\top }r$. This is the same system that appears in the Levenberg-Marquardt Method.

In ModularEIT.jl: TikhonovRegularizer.

References

1. A. N. Tikhonov (1963). On the solution of ill-posed problems and the method of regularization. Dokl. Akad. Nauk SSSR 151(3), 501–504 (English transl.: Soviet Math. Dokl. 4, 1035–1038). mathnet.ru/eng/dan28329
2. H. W. Engl, M. Hanke, A. Neubauer (1996). Regularization of Inverse Problems. Kluwer. doi:10.1007/978-94-009-1740-8
3. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344