Table 4.

Quantitative analysis of the convergence of the Picard, N-PCG, and GN-PCG methods. The test problem is the sinusoidal images (see section 5.1 for 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; H¹-regularization; bottom block). 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. 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 at least 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). The regularization parameter is empirically set to β = 1E–3. 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) and the average number of the required line search steps ᾱ are reported. Note that we introduced a memory for the step size into the Picard method (preconditioned gradient descent) to stabilize the optimization (see section 4.3.5 and the description of the results). The definitions for the reported measures are summarized in Table 2.

		$n_x^i$	k★	nPDE	${\Vert m_R^h-m_1^h\Vert }_{2,\text{rel}}$	$\delta {\mathcal{J}}_{\text{rel}}^h$	||gh||∞,rel	ᾱ
H2-regularization	Picard	64	30	420	4.78E–3	6.05E–2	7.86E–3	1.72
		128	34	414	4.69E–3	6.04E–2	6.48E–3	1.82
		256	33	414	4.68E–3	6.03E–2	7.42E–3	1.75
	GN-PCG	64	5	78	4.63E–3	6.05E–2	5.57E–5	1.00
		128	4	45	4.59E–3	6.04E–2	5.60E–4	1.00
		256	4	45	4.58E–3	6.03E–2	5.10E–4	1.00
	N-PCG	64	5	63	4.63E–3	6.05E–2	1.94E–4	1.00
		128	5	61	4.57E–3	6.04E–2	2.83E–4	1.00
		256	5	61	4.57E–3	6.03E–2	2.86E–4	1.00
Stokes regularization	Picard	64	57	269	6.61E–4	1.56E–2	2.55E–3	1.68
		128	52	250	5.79E–4	1.55E–2	3.55E–3	1.65
		256	50	233	5.67E–4	1.55E–2	2.31E–3	1.71
	GN-PCG	64	5	88	5.86E–4	1.56E–2	7.30E–5	1.00
		128	5	86	4.87E–4	1.55E–2	8.09E–5	1.00
		256	5	86	4.86E–4	1.55E–2	8.33E–5	1.00
	N-PCG	64	5	75	5.86E–4	1.56E–2	2.60E–4	1.00
		128	5	75	4.87E–4	1.55E–2	2.69E–4	1.00
		256	5	75	4.87E–4	1.55E–2	2.51E–4	1.00