Table 8.
Convergence results for different strategies to precondition the reduced space KKT system. We report results for two preconditioners—our original preconditioner based on the regularization operator (PREG) and the proposed, nested preconditioner (P2L). We solve the reduced space KKT system via a PCG method with a tolerance of 1E−6. We use two different solvers for the latter to invert the preconditioner—a PCG method with a tolerance that is 1E−1 times the tolerance of the PCG method used to solve the reduced space KKT system (i.e., a tolerance of 1E−7) and a CHEB method with a fixed number of 10 iterations. We consider a compressible diffeomorphism with an H2-regularization model. We report results for different images (SMOOTH A, BRAIN, and HAND), for different regularization weights βv, and a varying grid sizes nx (grid convergence; number of unknowns ). We solve the reduced space KKT system at the true solution ; the velocity field corresponds to the test problem SMOOTH A. We consider the RK2A method with a CFL number of 0.2 for the regularization preconditioner and the SL scheme for the nested preconditioner with a CFL number of 5. We report (i) the number of PCG iterations until convergence, (ii) the time spent on the Hessian matvecs (in seconds), (iii) the percentage of that time spent on inverting the preconditioner (if applicable), and (iv) the speedup compared to our original preconditioner (regularization preconditioner in combination with the RK2A scheme).
| SMOOTH A | HAND | BRAIN | |||||||||||||||||
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| n | βv | P | PDE solver | PC solver | run | iter | time | % PC | speedup | run | iter | time | % PC | speedup | run | iter | time | % PC | speedup |
| 8192 | 1E−1 | PREG | RK2A(0.2) | — | #1 | 4 | 2.67 | — | — | #2 | 19 | 7.12 | — | — | #3 | 21 | 8.39 | — | — |
| P2L | SL(5) | PCG(1E−1) | #4 | 2 | 3.08 | 64.04% | 8.66E−1 | #5 | 7 | 1.67E+1 | 90.42% | 4.27E−1 | #6 | 7 | 1.66E+1 | 91.15% | 5.05E−1 | ||
| P2L | SL(5) | CHEB(10) | #7 | 2 | 2.36 | 55.81% | 1.13 | #8 | 6 | 6.19 | 75.56% | 1.15 | #9 | 7 | 6.44 | 75.69% | 1.30 | ||
| 1E−2 | PREG | RK2A(0.2) | — | #10 | 6 | 3.14 | — | — | #11 | 47 | 1.59E+1 | — | — | #12 | 53 | 1.76E+1 | — | — | |
| P2L | SL(5) | PCG(1E−1) | #13 | 2 | 4.31 | 73.56% | 7.29E−1 | #14 | 8 | 4.60E+1 | 95.95% | 3.47E−1 | #15 | 8 | 4.39E+1 | 96.10% | 3.99E−1 | ||
| P2L | SL(5) | CHEB(10) | #16 | 3 | 3.19 | 62.15% | 9.83E−1 | #17 | 8 | 8.26 | 79.30% | 1.93 | #18 | 7 | 6.98 | 76.15% | 2.52 | ||
| 1E−3 | PREG | RK2A(0.2) | — | #19 | 16 | 6.11 | — | — | #20 | 138 | 4.26E+1 | — | — | #21 | 161 | 4.96E+1 | — | — | |
| P2L | SL(5) | PCG(1E−1) | #22 | 2 | 7.77 | 84.55% | 7.86E−1 | #23 | 10 | 1.66E+2 | 98.69% | 2.56E−1 | #24 | 11 | 2.24E+2 | 98.98% | 2.22E−1 | ||
| P2L | SL(5) | CHEB(10) | #25 | 6 | 5.93 | 75.31% | 1.03 | #26 | 24 | 2.16E+1 | 83.93% | 1.97 | #27 | 22 | 2.04E+1 | 82.94% | 2.43 | ||
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| 32 768 | 1E−1 | PREG | RK2A(0.2) | — | #28 | 4 | 1.89E+1 | — | — | #29 | 22 | 4.31E+1 | — | — | #30 | 26 | 5.93E+1 | — | — |
| P2L | SL(5) | PCG(1E−1) | #31 | 2 | 3.53 | 47.19% | 5.37 | #32 | 7 | 2.73E+1 | 85.64% | 1.58 | #33 | 7 | 2.95E+1 | 88.16% | 2.01 | ||
| P2L | SL(5) | CHEB(10) | #34 | 2 | 4.16 | 50.82% | 4.56 | #35 | 6 | 9.70 | 64.92% | 4.44 | #36 | 6 | 9.04 | 71.28% | 6.56 | ||
| 1E−2 | PREG | RK2A(0.2) | — | #37 | 6 | 2.35E+1 | — | — | #38 | 54 | 1.07E+2 | — | — | #39 | 74 | 1.56E+2 | — | — | |
| P2L | SL(5) | PCG(1E−1) | #40 | 3 | 7.14 | 64.44% | 3.29 | #41 | 7 | 6.50E+1 | 94.88% | 1.65 | #42 | 7 | 7.24E+1 | 96.04% | 2.16 | ||
| P2L | SL(5) | CHEB(10) | #43 | 3 | 5.16 | 55.89% | 4.55 | #44 | 9 | 1.26E+1 | 69.26% | 8.51 | #45 | 10 | 1.44E+1 | 69.92% | 1.09E+1 | ||
| 1E−3 | PREG | RK2A(0.2) | — | #46 | 16 | 3.85E+1 | — | — | #47 | 160 | 3.45E+2 | — | — | #48 | 224 | 5.47E+2 | — | — | |
| P2L | SL(5) | PCG(1E−1) | #49 | 3 | 1.27E+1 | 77.88% | 3.04 | #50 | 9 | 2.53E+2 | 98.27% | 1.36 | #51 | 10 | 3.49E+2 | 98.82% | 1.57 | ||
| P2L | SL(5) | CHEB(10) | #52 | 6 | 8.51 | 67.56% | 4.52 | #53 | 27 | 3.92E+1 | 73.91% | 8.80 | #54 | 31 | 4.10E+1 | 76.74% | 1.34E+1 | ||
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| 131072 | 1E−1 | PREG | RK2A(0.2) | — | #55 | 4 | 6.62E+1 | — | — | #56 | 25 | 2.71E+2 | — | — | #57 | 33 | 3.52E+2 | — | — |
| P2L | SL(5) | PCG(1E−1) | #58 | 2 | 1.24E+1 | 31.60% | 5.33 | #59 | 6 | 7.17E+1 | 78.58% | 3.77 | #60 | 6 | 1.02E+2 | 82.62% | 3.46 | ||
| P2L | SL(5) | CHEB(10) | #61 | 2 | 1.33E+1 | 46.49% | 4.97 | #62 | 5 | 2.58E+1 | 53.36% | 1.05E+1 | #63 | 5 | 3.07E+1 | 56.96% | 1.15E+1 | ||
| 1E−2 | PREG | RK2A(0.2) | — | #64 | 6 | 8.38E+1 | — | — | #65 | 63 | 6.61E+2 | — | — | #66 | 92 | 9.42E+2 | — | — | |
| P2L | SL(5) | PCG(1E−1) | #67 | 2 | 1.56E+1 | 39.61% | 5.37 | #68 | 7 | 2.17E+2 | 92.20% | 3.05 | #69 | 7 | 2.51E+2 | 94.07% | 3.76 | ||
| P2L | SL(5) | CHEB(10) | #70 | 3 | 2.06E+1 | 47.60% | 4.06 | #71 | 11 | 5.75E+1 | 57.39% | 1.15E+1 | #72 | 12 | 5.86E+1 | 59.85% | 1.61E+1 | ||
| 1E−3 | PREG | RK2A(0.2) | — | #73 | 16 | 1.66E+2 | — | — | #74 | 188 | 1.74E+3 | — | — | #75 | 279 | 2.29E+3 | — | — | |
| P2L | SL(5) | PCG(1E−1) | #76 | 3 | 3.08E+1 | 75.16% | 5.39 | #77 | 9 | 6.73E+2 | 97.64% | 2.58 | #78 | 8 | 9.54E+2 | 98.19% | 2.40 | ||
| P2L | SL(5) | CHEB(10) | #79 | 6 | 3.04E+1 | 57.77% | 5.46 | #80 | 33 | 1.37E+2 | 65.15% | 1.27E+1 | #81 | 38 | 1.74E+2 | 63.03% | 1.32E+1 | ||
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| 524288 | 1E−1 | PREG | RK2A(0.2) | — | #82 | 4 | 5.40E+2 | — | — | #83 | 25 | 1.97E+3 | — | — | #84 | 37 | 2.68E+3 | — | — |
| P2L | SL(5) | PCG(1E−1) | #85 | 2 | 6.20E+1 | 29.01% | 8.71 | #86 | 5 | 3.54E+2 | 77.07% | 5.58 | #87 | 5 | 4.25E+2 | 83.31% | 6.31 | ||
| P2L | SL(5) | CHEB(10) | #88 | 2 | 7.15E+1 | 35.60% | 7.55 | #89 | 4 | 1.21E+2 | 46.50% | 1.63E+1 | #90 | 5 | 1.60E+2 | 50.11% | 1.68E+1 | ||
| 1E−2 | PREG | RK2A(0.2) | — | #91 | 6 | 6.49E+2 | — | — | #92 | 67 | 4.61E+3 | — | — | #93 | 103 | 7.12E+3 | — | — | |
| P2L | SL(5) | PCG(1E−1) | #94 | 2 | 7.12E+1 | 36.97% | 9.11 | #95 | 6 | 8.72E+2 | 90.68% | 5.29 | #96 | 6 | 1.20E+3 | 93.45% | 5.92 | ||
| P2L | SL(5) | CHEB(10) | #97 | 3 | 8.31E+1 | 38.18% | 7.81 | #98 | 11 | 2.96E+2 | 55.59% | 1.56E+1 | #99 | 14 | 3.51E+2 | 56.00% | 2.03E+1 | ||
| 1E−3 | PREG | RK2A(0.2) | — | #100 | 16 | 1.31E+3 | — | — | #101 | 196 | 1.31E+4 | — | — | #102 | 310 | 2.10E+4 | — | — | |
| P2L | SL(5) | PCG(1E−1) | #103 | 2 | 1.27E+2 | 59.84% | 1.03E+1 | #104 | 7 | 2.93E+3 | 96.85% | 4.47 | #105 | 7 | 4.58E+3 | 97.99% | 4.58 | ||
| P2L | SL(5) | CHEB(10) | #106 | 6 | 1.68E+2 | 49.40% | 7.77 | #107 | 35 | 8.89E+2 | 60.45% | 1.47E+1 | #108 | 46 | 1.17E+3 | 61.08% | 1.80E+1 | ||