Discrete Boundary Operator

In practice one never observes the full Neumann-to-Dirichlet Map, only $N$ pairs $(g_i,f_i)$. Collect them as columns, $G=[g_1\cdots g_N]$ and $F=[f_1\cdots f_N]$, with zero-mean columns (currents satisfy $\int g=0$, voltages are grounded).

Least-squares estimate. The matrix $R$ with $RG\approx F$ of minimal norm is

$$\hat{R}=F{G}^{+},$$

where $G^{+}$ is the Moore–Penrose pseudoinverse. If the columns of $G$ are orthonormal, $G^{+}=G^{\top }$. The estimate is only determined on $\mathrm{span}(G)$. Outside the span it acts as zero.

Symmetrisation. The true operator is symmetric positive semidefinite (see Properties of the Boundary Operators). Noise breaks both properties. The closest symmetric matrix is $\frac{1}{2}(\hat{R}+{\hat{R}}^{\top })$. Clipping negative eigenvalues then gives the closest positive semidefinite matrix in the Frobenius norm.

Uses.

• Generating new data pairs, for example optimal, orthogonal ones, through an SVD (see Truncated SVD Regularization).
• Operator-level noise models.
• Operator-level data fidelity: $\Vert {\hat{R}}_{\text{meas}}-R_{\sigma }\Vert$.

The geometry matters: in $L^2(\partial \Omega )$ one should work with $M_{\Gamma }^{1/2}\hat{R}{M}_{\Gamma }^{1/2}$ rather than $\hat{R}$ (see Boundary Mass and Stiffness Matrices).

References

1. N. J. Higham (1988). Computing a nearest symmetric positive semidefinite matrix. Linear Algebra Appl. 103, 103–118. doi:10.1016/0024-3795(88)90223-6
2. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344