Table 11.

Convergence results for the inversion using our formulation for a near-incompressible diffeomorphism (linear Stokes regularization). We consider the HAND images in Fig. 3. We report results for our original preconditioner (P_REG; RK2A scheme with a CFL number of 0.2) and the proposed preconditioner (P_2L; SL scheme with a CFL number of 5; CHEB method with a fixed number of 10 iterations). We terminate the inversion if the change in the ℓ^∞-norm of the reduced gradient ${\mathit{g}}_k^h$, k = 1, 2, …, is at least two orders of magnitude or if the ℓ^∞-norm of ${\mathit{g}}_k^h$ is smaller or equal to 1E−5. We report (i) the relative change of the reduced gradient ‖g^★‖_rel, (ii) the relative change of the residual ‖r‖_rel (L²-distance between m_R and m₁), (iii) the number of outer iterations, (iv) the number of Hessian matvecs, (v) the time to solution, and (vi) the speedup compared to our original scheme.

βv	βw	run	P	PDE solver	PC solver	‖g★‖rel	‖r‖rel	iter	matvecs	time	speedup
1.00E−1	1.00E−3	#1	PREG	RK2A(0.2)	—	9.20E−3	1.55E−1	10	119	1.76E+2	—
		#2	P2L	SL(5)	CHEB(10)	5.55E−3	1.57E−1	10	29	2.40E+1	7.32
	1.00E−4	#3	PREG	RK2A(0.2)	—	9.49E−3	1.48E−1	9	99	1.63E+2	—
		#4	P2L	SL(5)	CHEB(10)	9.02E−3	1.50E−1	9	25	1.82E+1	8.98
1.00E−2	1.00E−3	#5	PREG	RK2A(0.2)	—	9.14E−3	6.56E−2	14	731	1.50E+3	—
		#6	P2L	SL(5)	CHEB(10)	9.57E−3	6.60E−2	13	60	7.05E+1	2.13E+1
	1.00E−4	#7	PREG	RK2A(0.2)	—	8.52E−3	5.31E−2	13	513	1.09E+3	—
		#8	P2L	SL(5)	CHEB(10)	9.58E−3	5.37E−2	13	60	6.28E+1	1.74E+1