Table 3.

Results for the sinusoidal analytic tumor / analytic velocity (SIN) test case; ground truth: (ρ _f = 15, ρ_w = 1, ρ_g = 0, k _f = 0, p = p^⋆, v = −v^⋆). We report convergence of the Picard iteration scheme for the SIN test case for three runs with increasingly refined mesh. For the registration, we increase the number of inversion variables by a factor of eight starting from N_i = 64 points per dimension, to N_i = 128 and N_i = 256; for the tumor parameters, we choose n _p to be 8,64, and 512 respectively. To study the convergence of our scheme, we perform 20 Picard iterations after termination of the parameter-continuation scheme for the regularization parameter of the registration solver. The relative objective function value ${\Vert \mathcal{J}\Vert }_{\text{rel}}$ and the relative gradient norm ${\Vert g\Vert }_{\text{rel}}$ of the coupled problem in (3) are reported for every iteration. Further, for each iteration we report the relative norm of the update $e_c{}_{0,L^2,rel}$ and $e_{\mathit{v}}{}_{,L^2,rel}$ of the inversion variables p and v respectively.

		Ni = 64,np = 8				Ni = 128,np = 64				Ni = 256,np = 512
It	${\beta }_{\upsilon }$	${\Vert \mathcal{J}\Vert }_{\text{rel}}$	${\Vert g\Vert }_{\text{rel}}$	$e_{c0,L^2,rel}$	$e_{\upsilon ,L^2,rel}$	${\Vert \mathcal{J}\Vert }_{\text{rel}}$	${\Vert g\Vert }_{\text{rel}}$	$e_{c0,L^2,rel}$	$e_{\upsilon ,L^2,rel}$	${\Vert \mathcal{J}\Vert }_{\text{rel}}$	${\Vert g\Vert }_{\text{rel}}$	$e_{c0,L^2,rel}$	$e_{\upsilon ,L^2,rel}$
init	-	1.00	1.00	–	–	1.00	1.00	–	–	1.00	1.00	–	–
1	1	6.67E−1	9.42E−3	–	–	6.73E−1	2.48E−3	–	–	6.67E−1	8.76E−4	–	–
2	1E−1	5.84E−1	5.37E−4	–	–	5.88E−1	2.27E−4	–	–	5.84E−1	7.58E−5	–	–
3	1E−2	3.08E−1	2.42E−4	8.08E−3	1.01E+1	3.08E−1	1.88E−4	2.66E−2	6.02	3.08E−1	6.97E−5	2.28E−2	4.97
4	1E−2	3.03E−1	1.77E−4	2.48E−3	1.51	3.04E−1	1.01E−4	2.35E−2	2.48	3.03E−1	4.37E−5	2.55E−2	2.94
5	1E−2	3.02E−1	1.44E−4	1.34E−3	2.60E−2	3.00E−1	6.20E−5	7.16E−3	1.33E−1	3.02E−1	2.62E−5	1.00E−2	1.48E−1
6	1E−2	3.01E−1	1.18E−4	8.72E−4	1.43E−2	3.02E−1	4.66E−5	2.51E−3	7.70E−2	3.01E−1	1.81E−5	3.61E−3	9.79E−2
7	1E−2	3.01E−1	9.64E−5	5.61E−4	7.25E−3	3.02E−1	3.78E−5	1.29E−3	4.61E−2	3.01E−1	1.41E−5	1.63E−3	6.43E−2
8	1E−2	3.00E−1	7.91E−5	3.62E−4	4.37E−3	3.02E−1	3.12E−5	8.04E−4	2.70E−2	3.00E−1	1.15E−5	9.29E−4	3.96E−2
9	1E−2	3.00E−1	6.50E−5	2.38E−4	3.30E−3	3.02E−1	2.59E−5	5.29E−4	1.61E−2	3.00E−1	9.51E−6	5.89E−4	2.48E−2
10	1E−2	3.00E−1	5.35E−5	1.57E−4	2.71E−3	3.02E−1	2.15E−5	3.54E−4	9.98E−3	3.00E−1	7.89E−6	3.89E−4	1.58E−2
11	1E−2	3.00E−1	4.40E−5	1.05E−4	2.31E−3	3.02E−1	1.79E−5	2.39E−4	6.42E−3	3.00E−1	6.56E−6	2.62E−4	1.02E−2
12	1E−2	3.00E−1	3.63E−5	6.99E−5	1.90E−3	3.02E−1	1.49E−5	1.63E−4	4.22E−3	3.00E−1	5.47E−6	1.78E−4	6.75E−3
13	1E−2	3.00E−1	2.99E−5	4.69E−5	1.58E−3	3.02E−1	1.24E−5	1.12E−4	2.09E−3	3.00E−1	4.55E−6	1.21E−4	4.54E−3
14	1E−2	3.00E−1	2.46E−5	3.16E−5	1.30E−3	3.02E−1	1.04E−5	7.63E−5	3.09E−3	3.00E−1	3.50E−6	8.33E−5	3.10E−3
15	1E−2	3.00E−1	2.03E−5	2.13E−5	1.07E−3	3.02E−1	8.64E−6	5.26E−5	1.35E−3	3.00E−1	3.17E−6	5.74E−5	2.19E−3
16	1E−2	3.00E−1	1.67E−5	1.44E−5	8.85E−4	3.02E−1	7.21E−6	3.63E−5	1.71E−3	3.00E−1	2.64E−6	3.96E−5	1.60E−3
17	1E−2	3.00E−1	1.38E−5	9.73E−6	7.29E−4	3.02E−1	6.05E−6	2.51E−5	9.31E−4	3.00E−1	2.21E−6	2.74E−5	1.20E−3
18	1E−2	3.00E−1	1.14E−5	6.59E−6	6.01E−4	3.02E−1	5.08E−6	1.59E−5	7.94E−4	3.00E−1	1.86E−6	1.89E−5	8.04E−4
19	1E−2	3.00E−1	9.37E−6	4.47E−6	4.95E−4	3.02E−1	4.30E−6	1.12E−5	6.62E−4	3.00E−1	1.57E−6	1.16E−5	6.50E−4
20	1E−2	3.00E−1	7.73E−6	3.03E−6	4.08E−4	3.02E−1	3.64E−6	7.61E−6	5.22E−4	3.00E−1	1.34E−6	8.04E−6	5.29E−4
21	1E−2	3.00E−1	6.37E−6	2.06E−6	3.36E−4	3.02E−1	3.09E−6	5.23E−6	4.27E−4	3.00E−1	1.14E−6	5.69E−6	4.42E−4
22	1E−2	3.00E−1	5.26E−6	1.40E−6	2.77E−4	3.02E−1	2.62E−6	3.71E−6	3.59E−4	3.00E−1	9.63E−7	4.08E−6	3.74E−4
23	1E−2	3.00E−1	4.35E−6	6.45E−7	2.29E−4	3.02E−1	2.23E−6	1.91E−6	3.05E−4	3.00E−1	8.17E−7	2.10E−6	3.17E−4