Table 5.

Results for the analytic tumor/analytic velocity with non-zero diffusion (ATAV-DIF) test case; ground truth: (ρ _f = 10, ρ_w = 1, ρ_g = 0, k _f = 1.00E−2, k_w = 1, k_g = 0, p = p^⋆, v = −v^⋆). We report values for the (summed) norm of the residual between the respective probability maps for the different brain tissue classes ${\mu }_{B,L^2}$ and tumor ${\mu }_{T,L^2}$ in patient space, the mean Dice coefficient for brain tissue DICE_B and tumor DICE_T , respectively, as well as the relative norm of the gradient ${\Vert g\Vert }_{\text{rel}}$ for the global coupled problem (3). We report results for different values of ρ _f ∈ {5, 10, 15} used in the inversion (ρ _f = 10 is the ground truth). We report the time spent per iteration (in seconds; top run) or in total (in seconds; bottom runs) for the entire Picard inversion, and the amount of that time spent in the tumor inversion and image registration (in percent (top run); in seconds (bottom runs)), respectively. Note that the latter sums up to less than 100% as we do not explicitly measure time spent in additional coupling functionality and forward solvers. These runs are performed using 64 MPI tasks on three nodes of HazelHen (see Section 5.1 for details). The top block shows the course of the inversion with respect to the Picard iteration index for the correct parameters (ground truth) for ρ _f and k _f . The four rows on the bottom show the final result for our Picard scheme for different parameter and model combinations.

iterations	βv	${\mu }_{B,L^2}$	DICEB	${\mu }_{T,L^2}$	DICET	${\Vert g\Vert }_{\text{rel}}$	$T^{it}[s]$	$T_{inv}^{tu}[\%]$	$T_{inv}^{reg}[\%]$
non-zero diffusion with ground truth parameters ρf = 10,kf = 1E−2
ref	−	1.00	7.14E−1	1.00	0.00	1.00
1	1	9.32E−1	7.32E−1	1.00	0.00	9.94E−1	2.11E+3	99.7	0.2
2	1E−1	6.52E−1	8.01E−1	2.08E−1	8.56E−1	7.45E−2	2.88E+2	96.0	2.1
3	1E−2	3.79E−1	8.88E−1	1.47E−1	9.12E−1	3.87E−2	3.94E+2	91.8	6.9
4	1E−3	1.75E−1	9.39E−1	8.08E−2	9.63E−I	2.45E−2	3.98E+2	72.7	26.1
5	1E−4	1.60E−1	9.48E−1	5.93E−2	9.78E−1	1.77E−2	2.00E+2	35.4	62.1
6	1E−4	1.60E−1	9.48E−1	5.77E−2	9.77E−1	1.72E−2	1.34E+2	46.6	49.6
7	1E−4	1.60E−1	9.47E−1	5.63E−2	9.76E−1	1.73E−2	7.10E+1	0.0	93.0
varying ρ f ∈ {5,10,15},kf = 1E−2, ground truth ρf =10,kf = 1E−2							T otal [s]	$T_{inv}^{tu}[\text{s}]$	$T_{inv}^{reg}[\text{s}]$
ρf = 5	1E−4	1.61E−1	9.46E−1	6.48E−2	9.67E−1	2.21E−2	4.37E+3	3.95E+3	3.97E+2
ρf = 10	1E−4	1.60E−1	9.47E−1	5.63E−2	9.76E−1	1.73E−2	3.60E+3	3.17E+3	4.01E+2
ρf = 15	1E−4	1.61E−1	9.48E−1	6.39E−2	9.72E−1	1.41E−2	3.96E+3	3.53E+3	4.00E+2
ρf = 10, kf = 0, ground truth pf =10,kf =1E−2							Ttotal [s]	$T_{inv}^{tu}[\text{s}]$	$T_{inv}^{reg}[\text{s}]$
ρf = 10	1E−4	1.60E−1	9.48E−1	5.95E−2	9.71E−1	1.74E−2	4.25E+2	2.25E+1	4.01E+2