Table 6.

Quantitative analysis of the convergence for the Picard and the GN-PCG method. The test problem is the UT images (see section 5.1 for more details on the construction of this synthetic registration problem). We compare convergence results for plain H²-regularization (γ = 0; top block) and the Stokes regularization scheme (γ = 1; bottom block; H¹-regularization) for empirically chosen regularization parameters β ∈ {1E–2, 1E–3}. We report results for different grid sizes ${\mathit{n}}_x={(n_x^1,n_x^2)}^{\top },n_x^i\in \{64,128,256\}$, i = 1, 2, n_t = 4 max(n_x). We invert for a stationary velocity field (i.e., n_c = 1). We terminate the optimization if the relative change of the ℓ^∞-norm of the reduced gradient g^h is larger than or equal to three orders of magnitude or if the change in 𝒥^h between 10 successive iterations is below or equal to 1E–6 (i.e., the algorithm stagnates). 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 average number of the line search steps ᾱ. Note that we introduced a memory for the step size into the Picard method to stabilize the optimization (see section 4.3.5 and the description of the results). The definitions for the reported measures can be found in Table 2. This study directly relates to the results for the smooth registration problem (see section 5.3.2, in particular Table 4).

			$n_x^i$	k★	nPDE	${\Vert m_R^h-m_1^h\Vert }_{2,\text{rel}}$	$\delta {\mathcal{J}}_{\text{rel}}^h$	||gh||∞,rel	ᾱ
H2-regularization	β = 1E–2	Picard	64	130	752	1.99E–2	1.18E–1	2.06E–2	1.68
			128	231	1589	8.29E–3	8.47E–2	4.03E–2	1.72
			256	388	3022	5.05E–3	7.92E–2	9.86E–2	1.68
		GN-PCG	64	7	282	1.94E–2	1.16E–1	3.30E–4	1.00
			128	9	450	8.09E–3	8.37E–2	6.07E–4	1.00
			256	13	789	4.87E–3	7.85E–2	7.28E–4	1.00
	β = 1E–3	Picard	64	339	4410	1.84E–3	1.54E–2	2.46E–2	1.56
			128	466	9671	7.19E–4	9.84E–3	5.98E–2	1.64
			256	632	8690	5.35E–4	8.81E–3	1.08E–1	1.64
		GN-PCG	64	7	670	1.44E–3	1.50E–2	2.59E–4	1.00
			128	8	929	4.47E–4	9.60E–3	4.76E–4	1.00
			256	13	1744	2.45E–4	8.55E–3	4.04E–4	1.00
Stokes regularization	β = 1E–2	Picard	64	92	448	2.73E–3	3.62E–2	7.54E–3	1.64
			128	136	958	8.92E–4	2.38E–2	1.63E–2	1.72
			256	143	1004	1.09E–3	2.53E–2	1.75E–2	1.68
		GN-PCG	64	7	313	2.72E–3	3.59E–2	4.70E–4	1.00
			128	9	437	8.88E–4	2.37E–2	5.59E–4	1.00
			256	10	514	1.09E–3	2.52E–2	5.49E–4	1.00
	β = 1E–3	Picard	64	216	2823	9.99E–4	4.69E–3	1.34E–2	1.56
			128	179	5465	3.57E–4	2.77E–3	1.79E–2	1.69
			256	175	5569	5.27E–4	3.07E–3	2.22E–2	1.68
		GN-PCG	64	8	1162	7.57E–4	4.55E–3	8.23E–4	1.00
			128	9	1069	2.15E–4	2.65E–3	5.67E–4	1.00
			256	9	769	3.43E–4	2.92E–3	6.44E–4	1.00