L2 Projection

The $L^2$ projection of a function $w$ onto a finite element space $V_h=\mathrm{span}\{{\phi }_i\}$ is the best approximation in $L^2(\Omega )$:

$$P_{h}w=\arg \underset{v_h\in V_h}{\min }\Vert w-v_h{\Vert }_{L^2}⟺M\mathbf{z}=\mathbf{b},b_i={\int }_{\Omega }w{\phi }_i\mathrm{d}x,$$

with the Mass Matrix $M$ of $V_h$.

Example: the adjoint gradient. The $L^2$ gradient of the EIT data misfit is the pointwise product $-\nabla u\cdot \nabla \lambda$ (see Functional Derivative of the Data Misfit). For $u,\lambda$ in $P_p$ (or $Q_p$) it is a piecewise polynomial of higher degree ($2(p-1)$ for simplicial $P_p$), and it is discontinuous across element faces. So it generally does not lie in the conductivity space. Projecting it:

• onto piecewise constants $P_0$: exact cell averages, and $M$ is diagonal;
• onto a discontinuous space of sufficient degree: exact;
• onto continuous $P_1/Q_1$: a genuine approximation, which smooths the gradient slightly.

Relation to the discrete gradient. The vector $\mathbf{b}$ with $b_i=-{\int }_{\Omega }\nabla u\cdot \nabla \lambda {\phi }_i$ is exactly the derivative of the discretised objective with respect to the coefficients $z_i$ of $\sigma$. The projection $M^{-1}\mathbf{b}$ is its $L^2$ Riesz representative. Which one to use depends on the metric of the optimiser (see Gradient Representation and the Riesz Map).

In ModularEIT.jl: l2_project, transfer_conductivity.

References

1. S. C. Brenner, L. R. Scott (2008). The Mathematical Theory of Finite Element Methods, 3rd ed. Springer. doi:10.1007/978-0-387-75934-0
2. M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich (2009). Optimization with PDE Constraints. Springer. doi:10.1007/978-1-4020-8839-1