Discrete Fractional Sobolev Norms

The natural spaces for boundary voltages and currents are $H^{1/2}(\partial \Omega )$ and $H^{-1/2}(\partial \Omega )$ (see Sobolev and Trace Spaces). They can be computed discretely by spectral interpolation between $L^2$ and $H^1$ on the boundary.

Solve the generalised eigenproblem with the Boundary Mass and Stiffness Matrices

$$K_{\Gamma }\Phi =M_{\Gamma }\Phi \mathrm{diag}({\lambda }_k),{\Phi }^{\top }{M}_{\Gamma }\Phi =I,{\lambda }_k\ge 0.$$

Clip tiny negative eigenvalues caused by round-off to zero. The columns of $\Phi$ are the discrete Laplace–Beltrami eigenfunctions (on a circle: Fourier modes, with ${\lambda }_k=k^2$). For $s\in [-1,1]$ define

$$A(s)=M_{\Gamma }\Phi \mathrm{diag}((1+{\lambda }_k{)}^s){\Phi }^{\top }{M}_{\Gamma },\Vert \mathbf{f}{\Vert }_{H^s}^2={\mathbf{f}}^{\top }A(s)\mathbf{f}.$$

• $s=0$: $A(0)=M_{\Gamma }$, the $L^2$ norm.
• $s=1$: $A(1)=M_{\Gamma }+K_{\Gamma }$, the $H^1$ norm.
• $s=\pm \frac{1}{2}$: the discrete $H^{\pm 1/2}$ norms. On a circle, $\Vert f{\Vert }_{H^s}^2\simeq {\sum }_k(1+k^2{)}^s|{\hat{f}}_k{|}^2$.

Duality. Since ${\Phi }^{\top }{M}_{\Gamma }\Phi =I$, one gets $A(s{)}^{-1}=\Phi \mathrm{diag}((1+{\lambda }_k{)}^{-s}){\Phi }^{\top }$ and hence

$$A(-s)=M_{\Gamma }A(s{)}^{-1}{M}_{\Gamma }.$$

So the $H^{-1/2}$ matrix is the $L^2$-dual of the $H^{1/2}$ matrix, as in the continuous setting.

Uses in EIT. Weighting the voltage misfit in $H^{1/2}$ or the current misfit in $H^{-1/2}$ matches the mapping properties of the Neumann-to-Dirichlet Map and changes how high-frequency boundary data are weighted relative to low-frequency data (see Data Fidelity Terms).

On uniform rectangle grids the eigenvectors are cosine modes, and the same construction works for σ in the interior at $O(n\log n)$ cost; see Spectral Sobolev Norms on Rectangles.

References

1. M. Arioli, D. Loghin (2009). Discrete Interpolation Norms with Applications. SIAM J. Numer. Anal. 47(4), 2924–2951. doi:10.1137/080729360
2. W. McLean (2000). Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press. ISBN 978-0-521-66375-5