A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem

Julius Fergy T. Rabago; Hideyuki Azegami

doi:10.3934/eect.2019038

Article Contents

2019, Volume 8, Issue 4: 785-824. Doi: 10.3934/eect.2019038

This issue Previous Article On some nonlinear problem for the thermoplate equations Next Article Optimal energy decay rates for some wave equations with double damping terms

A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem

Julius Fergy T. Rabago^, and
Hideyuki Azegami

Department of Complex Systems Science, Graduate School of Informatics, Nagoya University, A4-2 (780) Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan

^* Corresponding author: J. F. T. Rabago
^* Corresponding author: J. F. T. Rabago

Received: July 31, 2018

Revised: December 31, 2018

Early access: June 2019

Published: June 2019

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Abstract

The solution to a free boundary problem of Bernoulli type, also known as Alt-Caffarelli problem, is studied via shape optimization techniques. In particular, a novel energy-gap cost functional approach with a state constraint consisting of a Robin condition is proposed as a shape optimization reformulation of the problem. Accordingly, the shape derivative of the cost is explicitly determined, and using the gradient information, a Lagrangian-like method is used to formulate an efficient boundary variation algorithm to numerically solve the minimization problem. The second order shape derivative of the cost is also computed, and through its characterization at the solution of the Bernoulli problem, the ill-posedness of the shape optimization formulation is proved. The analysis of the proposed formulation is completed by addressing the existence of optimal solution of the shape optimization problem and is accomplished by proving the continuity of the solution of the state problems with respect to the domain. The feasibility of the newly proposed method and its comparison with the classical energy-gap type cost functional approach is then presented through various numerical results. The numerical exploration issued in the study also includes results from a second-order optimization procedure based on a Newton-type method for resolving such minimization problem. This computational scheme put forward in the paper utilizes the Hessian information at the optimal solution and thus offers a state-of-the-art numerical approach for solving such free boundary problem via shape optimization setting.

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Mathematics Subject Classification: Primary: 35R35, 35N25; Secondary: 49K20, 49Q10.

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Figure 1. Initial (left) and final (right) free boundaries for Example 1 with $ \alpha = 0.1 $ in (57)

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Figure 2. (a)-(b): Respective histories of cost values and Hausdorff distances via first-order method with $ \alpha = 0.1 $ in (57) and varying initial free boundary $ \Sigma_0^i $, $ i = 1, 2, 3 $; (c)-(d): respective histories of cost values and Hausdorff distances via second-order method with $ \alpha = 0.1 $ in (57) and different initial free boundary $ \Sigma_0^i $, $ i = 1, 2, 3 $

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Figure 3. Blue solid lines: Optimal free boundaries for Example 2 when $ \lambda = -10, -9, \ldots, -1 $ (the outermost boundary corresponds to $ \lambda = -1 $ and the innermost boundary to $ \lambda = -10 $); dashed-dot magenta line: initial guess for the free boundary

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Figure 4. Results of Example 2 for $ \lambda = -9 $ when $ \eta = 10^{-6} $ in the stopping condition (58) and $ \alpha = 0.99 $ in (57)

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Figure 5. (a): Histories of descent step sizes for the proposed and classical formulations (with almost equal initial step size $ t_0 $ for the two formulations); (b): optimal free boundaries obtained when $ \lambda = -9, -4, -1 $ in Example 2 using the proposed and classical formulations with $ \eta = 10^{-4} $ in the stopping condition (58)

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Figure 6. (a)-(b): histories of free boundaries obtained through the proposed formulation with initial guess $ \Sigma_0^1 $ and $ \Sigma_0^3 $, respectively, where $ \eta = 10^{-6} $ in (58); (c)-(d): Histories of free boundaries obtained via the classical Kohn-Vogelius formulation with initial guess $ \Sigma_0^1 $ and $ \Sigma_0^3 $, respectively, where $ \eta = 10^{-4} $ in (58)

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Figure 7. (a)-(b): corresponding histories of curvatures of the free boundaries obtained through the proposed formulation with initial guess $ \Sigma_0^1 $ and $ \Sigma_0^3 $, respectively, shown in Figure 6a-6b; (c)-(d): Corresponding histories of curvatures of the free boundaries obtained via the classical Kohn-Vogelius formulation with initial guess $ \Sigma_0^1 $ and $ \Sigma_0^3 $, respectively, shown in Figure 6c-6d

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Figure 8. Blue solid lines: Optimal free boundaries for Example 3 when $ \lambda = -10, -9, \ldots, -1 $ (the outermost boundary corresponds to $ \lambda = -1 $ and the innermost boundary to $ \lambda = -10 $); dashed-dot magenta line: initial guess for the free boundary

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Figure 9. Results of Example 3 when $ \lambda = -10, -4, -1 $ for both of the proposed and classical formulations

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Figure 10. (a)-(b): histories of free boundaries obtained through the proposed formulation with initial guess $ C(\boldsymbol{0}, 0.6) $ and $ \Upsilon $, respectively, where $ \eta = 10^{-6} $ in (58); (c)-(d): histories of free boundaries obtained via the classical Kohn-Vogelius formulation with initial guess $ C(\boldsymbol{0}, 0.6) $ and $ \Upsilon $, respectively, where $ \eta = 10^{-4} $ in (58)

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Figure 11. (a)-(b): Corresponding histories of curvatures of the free boundaries obtained through the proposed formulation with initial guess $ \Sigma_0^1 $ and $ \Sigma_0^3 $ shown in Figure 10a-10b, respectively; (c)-(d): corresponding histories of curvatures of the free boundaries obtained via the classical Kohn-Vogelius formulation with initial guess $ \Sigma_0^1 $ and $ \Sigma_0^3 $ shown in Figure 10c-10d, respectively

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Table 1. Convergence test toward exact solution using the proposed formulation via the modified $ H^1 $-gradient method with initial free boundaries $ \Sigma_0^i $, $ i = 1, 2, 3 $, and $ \alpha = 0.10, 0.50, 0.99 $ in (57)

$ \Sigma_0^i $	$ \alpha $	cost	$ {\rm d}_{\rm H}(\Sigma^\ast, \Sigma^i_f) $	$ \bar{R} $	$ \|R - \bar{R}\|/R $	iter.	cpu time
$ \Sigma_0^1 $	0.10	$ 2.85 \times 10^{-5} $	$ 0.005072 $	$ 0.500888 $	0.001776	72	115 sec
	0.50	$ 2.32 \times 10^{-7} $	$ 0.004983 $	$ 0.500002 $	0.000004	17	26 sec
	0.99	$ 8.55 \times 10^{-8} $	$ 0.004984 $	$ 0.499865 $	0.000270	8	12 sec
$ \Sigma_0^2 $	0.10	$ 1.77 \times 10^{-5} $	$ 0.005044 $	$ 0.499343 $	0.001314	70	103 sec
	0.50	$ 9.26 \times 10^{-7} $	$ 0.005003 $	$ 0.499878 $	0.000244	16	28 sec
	0.99	$ 3.91 \times 10^{-9} $	$ 0.004998 $	$ 0.499956 $	0.000088	7	14 sec
$ \Sigma_0^3 $	0.10	$ 1.65 \times 10^{-5} $	$ 0.005887 $	$ 0.500051 $	0.000102	76	122 sec
	0.50	$ 6.64 \times 10^{-7} $	$ 0.004991 $	$ 0.500002 $	0.000004	19	29 sec
	0.99	$ 8.77 \times 10^{-7} $	$ 0.005001 $	$ 0.499993 $	0.000014	9	13 sec

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Table 2. Convergence test toward exact solution using the proposed formulation via the modified $ H^1 $-Newton method with $ \alpha = 0.1 $ and different initial free boundary $ \Sigma_0^i $, $ i = 1, 2, 3 $

$ \alpha $	$ \Sigma_0^i $	cost	$ {\rm d}_{\rm H}(\Sigma^\ast, \Sigma^i_f) $	$ \bar{R} $	$ \|R - \bar{R}\|/R $	iter.	cpu time
$ 0.1 $	$ \Sigma_0^1 $	$ 6.17 \times 10^{-7} $	$ 0.005007 $	$ 0.500139 $	0.000278	7	25 sec
	$ \Sigma_0^2 $	$ 1.57 \times 10^{-8} $	$ 0.005003 $	$ 0.500013 $	0.000026	8	41 sec
	$ \Sigma_0^3 $	$ 4.47 \times 10^{-6} $	$ 0.005130 $	$ 0.500101 $	0.000202	14	35 sec

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Table 3. Summary of computational results for an L-shaped fixed boundary $ \Gamma = \partial S $ with $ \lambda = -10, -9, \ldots, -1 $ where $ \alpha = 0.99 $ in (57) and $ \eta = 10^{-6} $ in the stopping condition (58)

$ \lambda $	formulation	$ t_0 $	cost	iteration	cpu time
$ -10 $	proposed	$ 0.228150 $	$ 1.37 \times 10^{-6} $	$ 14 $	$ 14 $ sec
$ -10 $	classical	$ 0.767890 $	$ 0.000137 $	$ 19 $	$ 55 $ sec
$ -9 $	proposed	$ 0.221636 $	$ 6.25 \times 10^{-7} $	$ 13 $	$ 16 $ sec
$ -9 $	classical	$ 0.738627 $	$ 3.11 \times 10^{-5} $	$ 25 $	$ 96 $ sec
$ -8 $	proposed	$ 0.213501 $	$ 7.94 \times 10^{-7} $	$ 12 $	$ 14 $ sec
$ -8 $	classical	$ 0.702851 $	$ 7.17 \times 10^{-5} $	$ 19 $	$ 57 $ sec
$ -7 $	proposed	$ 0.203058 $	$ 1.23 \times 10^{-6} $	$ 10 $	$ 13 $ sec
$ -7 $	classical	$ 0.658125 $	$ 0.000628 $	$ 12 $	$ 34 $ sec
$ -6 $	proposed	$ 0.189163 $	$ 8.27 \times 10^{-7} $	$ 10 $	$ 14 $ sec
$ -6 $	classical	$ 0.600640 $	$ 0.000190 $	$ 13 $	$ 34 $ sec
$ -5 $	proposed	$ 0.169783 $	$ 2.61 \times 10^{-7} $	$ 10 $	$ 16 $ sec
$ -5 $	classical	$ 0.524113 $	$ 0.000948 $	$ 18 $	$ 54 $ sec
$ -4 $	proposed	$ 0.140942 $	$ 2.04 \times 10^{-7} $	$ 10 $	$ 17 $ sec
$ -4 $	classical	$ 0.417590 $	$ 0.000186 $	$ 8 $	$ 24 $ sec
$ -3 $	proposed	$ 0.094111 $	$ 5.68 \times 10^{-7} $	$ 9 $	$ 14 $ sec
$ -3 $	classical	$ 0.262124 $	$ 4.95 \times 10^{-5} $	$ 11 $	$ 29 $ sec
$ -2 $	proposed	$ 0.039805 $	$ 2.37 \times 10^{-7} $	$ 10 $	$ 17 $ sec
$ -2 $	classical	$ 0.120956 $	$ 6.48 \times 10^{-6} $	$ 10 $	$ 27 $ sec
$ -1 $	proposed	$ 0.312388 $	$ 1.08 \times 10^{-6} $	$ 13 $	$ 25 $ sec
$ -1 $	classical	$ 0.615256 $	$ 5.69 \times 10^{-7} $	$ 9 $	$ 24 $ sec

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Table 4. Computational results obtained via the classical formulation with $ \eta = 10^{-6} $ in the stopping condition (58) for an L-shaped fixed boundary $ \Gamma = \partial S $ when $ \lambda = -9, -4, -1 $ for different values of $ \alpha $ in (57)

$ \lambda $	$ i $	$ \alpha_i $	$ t_0 $	cost	iteration	cpu time
$ -9 $	$ 1 $	$ 0.2970642705 $	$ 0.2216361137 $	$ 1.34 \times 10^{-5} $	$ 22 $	$ 74 $ sec
	$ 2 $	$ 0.2970642710 $	$ 0.2216361144 $	$ 3.70 \times 10^{-5} $	$ 18 $	$ 102 $ sec
	$ 3 $	$ 0.2970642715 $	$ 0.2216361144 $	$ 6.40 \times 10^{-5} $	$ 17 $	$ 32 $ sec
$ -4 $	$ 1 $	$ 0.334138300 $	$ 0.1409423055 $	$ 1.03 \times 10^{-5} $	$ 11 $	$ 23 $ sec
	$ 2 $	$ 0.334138305 $	$ 0.1409423076 $	$ 4.57 \times 10^{-6} $	$ 12 $	$ 26 $ sec
	$ 3 $	$ 0.334138310 $	$ 0.1409423098 $	$ 4.92 \times 10^{-5} $	$ 14 $	$ 40 $ sec
$ -1 $	$ 1 $	$ 0.502658435 $	$ 0.3123875250 $	$ 1.34 \times 10^{-6} $	$ 14 $	$ 34 $ sec
	$ 2 $	$ 0.502658440 $	$ 0.3123875282 $	$ 1.17 \times 10^{-6} $	$ 14 $	$ 35 $ sec
	$ 3 $	$ 0.502658445 $	$ 0.3123875313 $	$ 1.24 \times 10^{-6} $	$ 14 $	$ 34 $ sec

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Table 5. Comparison of computational results obtained through the proposed and classical formulations with $ \eta = 10^{-4} $ in (58) for an L-shaped fixed boundary $ \Gamma = \partial S $ when $ \lambda = -9, -4, -1 $ with almost the same initial step size $ t_0 $ for both formulations

$ \lambda $	formulation	$ \alpha $	$ t_0 $	cost	iteration	cpu time
$ -9 $	proposed	$ 0.990000000 $	$ 0.2216361144 $	$ 1.55 \times 10^{-5} $	$ 9 $	$ 10 $ sec
$ -9 $	classical	$ 0.297064271 $	$ 0.2216361144 $	$ 14.2 \times 10^{-5} $	$ 12 $	$ 16 $ sec
$ -4 $	proposed	$ 0.990000000 $	$ 0.1409423086 $	$ 6.37 \times 10^{-6} $	$ 7 $	$ 11 $ sec
$ -4 $	classical	$ 0.334138305 $	$ 0.1409423076 $	$ 4.57 \times 10^{-6} $	$ 7 $	$ 11 $ sec
$ -1 $	proposed	$ 0.990000000 $	$ 0.3123875327 $	$ 4.39 \times 10^{-5} $	$ 8 $	$ 17 $ sec
$ -1 $	classical	$ 0.502658440 $	$ 0.3123875282 $	$ 5.51 \times 10^{-5} $	$ 9 $	$ 24 $ sec

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Table 6. Comparison of computational results obtained through the proposed and classical formulations for an L-shaped fixed boundary $ \Gamma = \partial S $ when $ \lambda = -10 $ with almost the same initial step size $ t_0 $ for both formulations

$ \Sigma_0^i $	formulation	$ \eta $	cost	$ {\rm d}_{\rm H}(\Sigma^{\rm in}, \Sigma^i_f) $	iteration	cpu time
$ \Sigma_0^1 $	proposed	$ 10^{-4} $	$ 2.93 \times 10^{-5} $	$ 0.023850 $	$ 9 $	$ 10 $ sec
		$ 10^{-5} $	$ 4.46 \times 10^{-6} $	$ 0.008586 $	$ 12 $	$ 13 $ sec
		$ 10^{-6} $	$ 1.38 \times 10^{-6} $	$ 0.008586 $	$ 14 $	$ 14 $ sec
	classical	$ 10^{-4} $	$ 0.001627 $	$ 0.008435 $	$ 9 $	$ 14 $ sec
		$ 10^{-5} $	$ 0.001627 $	$ 0.010512 $	$ 9 $	$ 14 $ sec
		$ 10^{-6} $	$ 0.000224 $	$ 0.007484 $	$ 15 $	$ 30 $ sec
$ \Sigma_0^3 $	proposed	$ 10^{-4} $	$ 5.69 \times 10^{-5} $	$ 0.026360 $	$ 8 $	$ 9 $ sec
		$ 10^{-5} $	$ 1.22 \times 10^{-5} $	$ 0.008565 $	$ 10 $	$ 10 $ sec
		$ 10^{-6} $	$ 9.47 \times 10^{-7} $	$ 0.007675 $	$ 14 $	$ 14 $ sec
	classical	$ 10^{-4} $	$ 0.000644 $	$ 0.008637 $	$ 9 $	$ 15 $ sec
		$ 10^{-5} $	$ 0.000062 $	$ 0.007394 $	$ 15 $	$ 30 $ sec
		$ 10^{-6} $	$ 0.000027 $	$ 0.007394 $	$ 19 $	$ 114 $ sec

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Table 7. Computational results obtained through the classical formulation with $ \eta = 10^{-4} $ in (58) for an L-shaped fixed boundary $ \Gamma = \partial S $ for $ \lambda = -10, -8, -7, -6, -5, -3, -2 $ with almost the same initial step size $ t_0 $ with respect to that of the proposed formulation shown in Table 3

$ \lambda $	$ \alpha $	$ t_0 $	cost	iteration	cpu time
$ -10 $	$ 0.29414181 $	$ 0.22815001 $	$ 0.001628 $	$ 9 $	$ 14 $ sec
$ -8 $	$ 0.30072761 $	$ 0.21350176 $	$ 0.000119 $	$ 12 $	$ 17 $ sec
$ -7 $	$ 0.30545458 $	$ 0.20305801 $	$ 0.001727 $	$ 10 $	$ 17 $ sec
$ -6 $	$ 0.31178679 $	$ 0.18916314 $	$ 0.000137 $	$ 8 $	$ 13 $ sec
$ -5 $	$ 0.32070413 $	$ 0.16978307 $	$ 0.000925 $	$ 6 $	$ 10 $ sec
$ -3 $	$ 0.35544029 $	$ 0.09411051 $	$ 7.89 \times 10^{-5} $	$ 6 $	$ 14 $ sec
$ -2 $	$ 0.32579718 $	$ 0.03980505 $	$ 2.08 \times 10^{-5} $	$ 7 $	$ 14 $ sec

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Table 8. Means and standard deviations (std) of the number of iterations, computing time and computing time per iteration for the proposed formulation with $ \eta = 10^{-6} $ and classical formulation with $ \eta = 10^{-4} $ in (58)

formulation	iteration		cpu time		$ \frac{\rm cpu\ time}{\rm iteration} $
formulation	mean	std	mean	std	mean	std
proposed	$ \approx 11 (11.1) $	$ \approx 2 (1.73) $	$ 16 $	$ 3.46 $	$ 1.46 $	$ 0.29 $
classical	$ \approx 9 (8.6) $	$ \approx 2 (2.22) $	$ 15 $	$ 3.92 $	$ 1.77 $	$ 0.42 $

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Table 9. Summary of computational results of Example 3 for $ \lambda = -10, -9, \ldots, -1 $ where the highlighted rows correspond to the results due to the proposed formulation

$ \lambda $	$ \alpha $	$ t_0 $	cost	iter	cpu time
$ -10 $	$ 0.990000000 $	$ 0.337420581 $	$ 1.33 \times 10^{-7} $	$ 14 $	$ 14 $ sec
$ -10 $	$ 0.386646216 $	$ 0.337420581 $	$ 0.001223 $	$ 15 $	$ 24 $ sec
$ -9 $	$ 0.990000000 $	$ 0.331971040 $	$ 5.73 \times 10^{-7} $	$ 13 $	$ 16 $ sec
$ -9 $	$ 0.388463867 $	$ 0.331971040 $	$ 0.000473 $	$ 11 $	$ 20 $ sec
$ -8 $	$ 0.990000000 $	$ 0.325160189 $	$ 5.07 \times 10^{-7} $	$ 13 $	$ 14 $ sec
$ -8 $	$ 0.390736229 $	$ 0.325160189 $	$ 0.000582 $	$ 10 $	$ 17 $ sec
$ -7 $	$ 0.990000000 $	$ 0.316405255 $	$ 3.36 \times 10^{-7} $	$ 12 $	$ 18 $ sec
$ -7 $	$ 0.393657966 $	$ 0.316405255 $	$ 0.000745 $	$ 10 $	$ 16 $ sec
$ -6 $	$ 0.990000000 $	$ 0.304735585 $	$ 3.79 \times 10^{-7} $	$ 11 $	$ 14 $ sec
$ -6 $	$ 0.397552887 $	$ 0.304735585 $	$ 9.84 \times 10^{-5} $	$ 12 $	$ 20 $ sec
$ -5 $	$ 0.990000000 $	$ 0.288405794 $	$ 3.33 \times 10^{-7} $	$ 11 $	$ 13 $ sec
$ -5 $	$ 0.403001262 $	$ 0.288405794 $	$ 4.38 \times 10^{-5} $	$ 13 $	$ 18 $ sec
$ -4 $	$ 0.990000000 $	$ 0.263931464 $	$ 2.09 \times 10^{-7} $	$ 11 $	$ 14 $ sec
$ -4 $	$ 0.411150343 $	$ 0.263931464 $	$ 2.25 \times 10^{-5} $	$ 9 $	$ 16 $ sec
$ -3 $	$ 0.990000000 $	$ 0.223215311 $	$ 1.45 \times 10^{-7} $	$ 11 $	$ 13 $ sec
$ -3 $	$ 0.424571657 $	$ 0.223215311 $	$ 3.88 \times 10^{-5} $	$ 7 $	$ 11 $ sec
$ -2 $	$ 0.990000000 $	$ 0.142355886 $	$ 1.70 \times 10^{-7} $	$ 10 $	$ 17 $ sec
$ -2 $	$ 0.448686986 $	$ 0.142355886 $	$ 1.30 \times 10^{-5} $	$ 7 $	$ 13 $ sec
$ -1 $	$ 0.990000000 $	$ 0.111335863 $	$ 9.21 \times 10^{-7} $	$ 12 $	$ 23 $ sec
$ -1 $	$ 0.482819584 $	$ 0.111335863 $	$ 3.54 \times 10^{-5} $	$ 8 $	$ 16 $ sec

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Table 10. Means and standard deviations (std) of the number of iterations, computing time and computing time per iteration of the computational results shown in Table 9

formulation	iteration		cpu time		$ \frac{\rm cpu\ time}{\rm iteration} $
formulation	mean	std	mean	std	mean	std
proposed	$ \approx 12 (11.8) $	$ \approx 1 (1.23) $	$ 16.5 $	$ 3.41 $	$ 1.46 $	$ 0.34 $
classical	$ \approx 10 (10.2) $	$ \approx 3 (2.61) $	$ 17.1 $	$ 3.70 $	$ 1.77 $	$ 0.17 $

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