Algorithms for magnetic tomography - on the role of a priori knowledge and constraints

Magnetic tomography investigates the reconstruction of currents from their magnetic fields. Here, we will study a number of projection methods in combination with the Tikhonov regularization for stabilization for the solution of the Biot-Savart integral equation Wj = H with the Biot-Savart integral operator W : (L(Ω)) → (L(∂G)) where Ω ⊂ G. In particular, we study the role of a priori knowledge when incorporated into the choice of the projection spaces Xn ⊂ (L(Ω)), n ∈ N, for example the conditions div j = 0 or the use of the full boundary value problem div σgrad φE = 0 in Ω, ν · σgrad φE = g on ∂Ω with some known function g, where j = σgrad φE and σ is an anisotropic matrix valued conductivity. We will discuss and compare these schemes investigating the ill-posedness of each algorithm in terms of the behaviour of the singular values of the corresponding operators both when a priori knowledge is incorporated and when the geometrical setting is modified. Finally, we will numerically evaluate the stability constants in the practical setup of magnetic tomography for fuel cells and, thus, calculate usable error bounds for this important application area. Submitted to: Inverse Problems