Table 8.

Comparison of the inversion results for the GN-PCG method for a varying number of the spatial coefficient fields ${\mathit{v}}_l^h:{\mathbf{R}}^d\to {\mathbf{R}}^d$, l = 1, …, n_c (i.e., we change the number of the unknowns in time). We report results for plain H²-regularization. We consider the hand images (n_x = (128, 128)^⊤; top block) and the brain images (n_x = (200, 200)^⊤; bottom block). We consider the full set of stopping conditions in (4.8) with τ_𝒥 = 1E–3, as we no longer study grid convergence and/or compare different optimization methods. We report the number of the (outer) iterations (k^★), the number of the hyperbolic PDE solves (n_PDE) and the relative change of (i) the L²-distance ( ${\Vert m_R^h-m_1^h\Vert }_{2,\mathit{rel}}$), (ii) the objective ( $\delta {\mathcal{J}}_{\mathit{rel}}^h$), and (iii) the (reduced) gradient (||g^h||_∞,rel) as well as the minimal and maximal values of the determinant of the deformation gradient. The definitions of these measures can be found in Table 2. The number of the coefficient fields n_c used to solve the individual registration problems is chosen to be in {1, 2, 4, 8, 16}.

	nc	k	nPDE	${\Vert m_R^h-m_1^h\Vert }_{2,\text{rel}}$	$\delta {\mathcal{J}}_{\text{rel}}^h$	||gh||∞,rel	$min(det({\mathit{F}}_1^h))$	$max(det({\mathit{F}}_1^h))$
Hand images	1	7	279	6.65E–2	8.52E–2	2.27E–2	2.14E–1	6.60
	2	7	279	6.60E–2	8.48E–2	2.29E–2	2.15E–1	6.44
	4	7	283	6.41E–2	8.24E–2	2.29E–2	2.07E–1	6.49
	8	7	277	6.41E–2	8.24E–2	2.51E–2	2.08E–1	6.46
	16	7	281	6.41E–2	8.24E–2	2.56E–2	2.08E–1	6.45
Brain images	1	20	669	5.49E–1	6.68E–1	3.71E–2	4.93E–2	6.47
	2	20	667	5.47E–1	6.66E–1	3.72E–2	4.96E–2	6.45
	4	21	710	5.31E–1	6.51E–1	3.66E–2	4.55E–2	7.33
	8	21	710	5.30E–1	6.51E–1	3.52E–2	4.53E–2	7.37
	16	21	708	5.30E–1	6.51E–1	3.54E–2	4.53E–2	7.37