December  2019, 8(4): 785-824. doi: 10.3934/eect.2019038

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

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

Received  August 2018 Revised  January 2019 Published  June 2019

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.

Citation: Julius Fergy T. Rabago, Hideyuki Azegami. A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem. Evolution Equations & Control Theory, 2019, 8 (4) : 785-824. doi: 10.3934/eect.2019038
References:
[1]

A. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine. Angew. Math., 325 (1981), 105-144. doi: 10.1515/crll.1981.325.105. Google Scholar

[2]

H. Azegami, Second derivatives of cost functions and $H^1$ Newton method in shape optimization problems, Mathematical Analysis of Continuum Mechanics and Industrial Applications Ⅱ, in Proceedings of the International Conference CoMFoS16 (eds. P. van Meurs, M. Kimura and H. Notsu), vol. 30 of Mathematics for Industry, Springer, Singapore, (2017), 61–72. doi: 10.1007/978-981-10-6283-4_6. Google Scholar

[3]

H. Azegami, Shape Optimization Problems, Morikita Publishing Co., Ltd., Tokyo, 2016 (in Japanese).Google Scholar

[4]

H. Azegami, Solution of shape optimization problem and its application to product design, Mathematical Analysis of Continuum Mechanics and Industrial Applications, in Computer Aided Optimization Design of Structures IV, Structural Optimization (eds. H. Itou, M. Kimura, V. Chalupecký, K. Ohtsuka, D. Tagami and A. Takada), vol. 26 of Mathematics for Industry, Springer, Singapore, (2017), 83–98. doi: 10.1007/978-981-10-2633-1_6. Google Scholar

[5]

H. Azegami, S. Kaizu, M. Shimoda and E. Katamine, Irregularity of shape optimization problems and an improvement technique, in Computer Aided Optimization Design of Structures IV, Structural Optimization (eds. S. Hernandez and C. A. Brebbia), Computational Mechanics Publications, Southampton, (1997), 309–326. doi: 10.2495/OP970301. Google Scholar

[6]

H. Azegami and Z. Q. Wu, Domain optimization analysis in linear elastic problems: Approach using traction method, SME Int. J., Ser. A., 39 (1996), 272-278. doi: 10.1299/jsmea1993.39.2_272. Google Scholar

[7]

H. Azegami, M. Shimoda, E. Katamine and Z. C. Wu, A domain optimization technique for elliptic boundary value problems, in Computer Aided Optimization Design of Structures IV, Structural Optimization (eds. S. Hernandez, M. El-Sayed and C. A. Brebbia), Computational Mechanics Publications, Southampton, (1995), 51–58. doi: 10.2495/OP950071. Google Scholar

[8]

H. Azegami, Solution to domain optimization problems, Trans. Jpn. Soc. Mech. Eng., Ser. A., 60 (1994), 1479–1486 (in Japanese). doi: 10.1299/kikaia.60.1479. Google Scholar

[9]

J. B. Bacani, Methods of Shape Optimization in Free Boundary Problems, Ph.D. Thesis, Karl-Franzens-Universität-Graz, 2013.Google Scholar

[10]

J. B. Bacani and G. Peichl, On the first-order shape derivative of the Kohn-Vogelius cost functional of the Bernoulli problem, Abstr. Appl. Anal., 2013 (2013), Art. ID 384320, 19pp. doi: 10.1155/2013/384320. Google Scholar

[11]

A. Ben AbdaF. BouchonG. PeichlM. Sayeh and R. Touzani, A Dirichlet-Neumann cost functional approach for the Bernoulli problem, J. Eng. Math., 81 (2013), 157-176. doi: 10.1007/s10665-012-9608-3. Google Scholar

[12]

A. BoulkhemairA. Nachaoui and A. Chakib, A shape optimization approach for a class of free boundary problems of Bernoulli type, Appl. Math., 58 (2013), 205-221. doi: 10.1007/s10492-013-0010-x. Google Scholar

[13]

A. BoulkhemairA. Chakib and A. Nachaoui, Uniform trace theorem and application to shape optimization, Appl. Comput. Math., 7 (2008), 192-205. Google Scholar

[14]

A. Boulkhemair and A. Chakib, On the uniform Poincaré inequality, Commun. Partial Differ. Equations, 32 (2007), 1439-1447. doi: 10.1080/03605300600910241. Google Scholar

[15]

D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., 52 (1975), 189-219. doi: 10.1016/0022-247X(75)90091-8. Google Scholar

[16]

M. Dambrine, On variations of the shape Hessian and sufficient conditions for the stability of critical shapes, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A. Mat., 96 (2002), 95-121. Google Scholar

[17]

M. Dambrine and M. Pierre, About stability of equilibrium shapes, Model Math. Anal. Numer., 34 (2000), 811-834. doi: 10.1051/m2an:2000105. Google Scholar

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M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2nd edition, Adv. Des. Control 22, SIAM, Philadelphia, 2011. doi: 10.1137/1.9780898719826. Google Scholar

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[20]

K. Eppler and H. Harbrecht, A regularized Newton method in electrical impedance tomography using shape Hessian information, Control Cybern., 34 (2005), 203-225. Google Scholar

[21]

K. Eppler, Boundary integral representations of second derivatives in shape optimization, Discuss. Math. Differ. Incl. Control. Optim., 20 (2000), 63-78. doi: 10.7151/dmdico.1005. Google Scholar

[22]

K. Eppler, Optimal shape design for elliptic equations via BIE-methods, J. Appl. Math. Comput. Sci., 10 (2000), 487-516. Google Scholar

[23]

A. Fasano, Some free boundary problems with industrial applications, in Shape Optimization and Free Boundaries (eds. M. C. Delfour and G. Sabidussi), vol. 380 of NATO ASI Series (C: Mathematical and Physical Sciences), Springer, Dordrecht, (1992), 113–142. doi: 10.1007/978-94-011-2710-3_3. Google Scholar

[24]

M. Flucher and M. Rumpf, Bernoulli's free-boundary problem, qualitative theory and numerical approximation, J. Reine. Angew. Math., 486 (1997), 165-204. doi: 10.1515/crll.1997.486.165. Google Scholar

[25]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/978-3-642-61798-0. Google Scholar

[26]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Publishing, Marshfield, Massachusetts, 1985. doi: 10.1137/1.9781611972030. Google Scholar

[27]

J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718690. Google Scholar

[28]

J. HaslingerT. KozubekK. Kunisch and G. Peichl, An embedding domain approach for a class of 2-d shape optimization problems: Mathematical analysis, J. Math. Anal. Appl., 290 (2004), 665-685. doi: 10.1016/j.jmaa.2003.10.038. Google Scholar

[29]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265. doi: 10.1515/jnum-2012-0013. Google Scholar

[30]

A. Henrot and M. Pierre, Shape Variation and Optimization: A Geometrical Analysis, Tracts in Mathematics 28, European Mathematical Society, Zürich, 2018. doi: 10.4171/178. Google Scholar

[31]

K. ItoK. Kunisch and G. Peichl, Variational approach to shape derivatives, ESAIM Control Optim. Calc. Var., 14 (2008), 517-539. doi: 10.1051/cocv:2008002. Google Scholar

[32]

T. KashiwabaraC. M. ColciagoL. Dedè and A. Quarteroni, Well-posedness, regularity, and convergence analysis of the finite element approximation of a generalized Robin boundary value problem, SIAM J. Numer. Anal., 53 (2015), 105-126. doi: 10.1137/140954477. Google Scholar

[33]

R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Commun. Pure Appl., 37 (1984), 289-298. doi: 10.1002/cpa.3160370302. Google Scholar

[34]

R. Kress, On Trefftz' integral equation for the Bernoulli free boundary value problem, Numer. Math., 136 (2017), 503-522. doi: 10.1007/s00211-016-0847-5. Google Scholar

[35]

J. Nečas, Direct Methods in the Theory of Elliptic Equations, Corrected 2nd edition, Monographs and Studies in Mathematics, Springer, Berlin, Heidelberg, 2012. doi: 10.1007/978-3-642-10455-8. Google Scholar

[36]

J. W. Neuberger, Sobolev Gradients and Differential Equations, Springer-Verlag, Berlin, 1997. doi: 10.1007/BFb0092831. Google Scholar

[37]

A. Novruzi and M. Pierre, Structure of shape derivatives, J. Evol. Equ., 2 (2002), 365-382. doi: 10.1007/s00028-002-8093-y. Google Scholar

[38]

A. Novruzi and J.-R. Roche, Newton's method in shape optimisation: A three-dimensional case, BIT Numer. Math., 40 (2000), 102-120. doi: 10.1023/A:1022370419231. Google Scholar

[39]

S. Osher and J. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. Google Scholar

[40]

A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, vol. 37 of Texts in Applied Mathematics, 2nd edition, Springer, Berlin, 2007. doi: 10.1007/b98885. Google Scholar

[41]

J. F. T. Rabago and H. Azegami, An improved shape optimization formulation of the Bernoulli problem by tracking the Neumann data, J. Eng. Math., (2019), to appear.Google Scholar

[42]

J. F. T. Rabago and J. B. Bacani, Shape optimization approach to the Bernoulli problem: A Lagrangian formulation, IAENG Int. J. Appl. Math., 47 (2017), 417-424. Google Scholar

[43]

J. F. T. Rabago and J. B. Bacani, Shape optimization approach for solving the Bernoulli problem by tracking the Neumann data: a Lagrangian formulation, Commun. Pur. Appl. Anal., 17 (2018), 2683-2702. doi: 10.3934/cpaa.2018127. Google Scholar

[44]

J. Simon, Second variation for domain optimization problems, in Control and Estimation of Distributed Parameter Systems (eds. F. Kappel, K. Kunisch and W. Schappacher), International Series of Numerical Mathematics, no 91. Birkhäuser, (1989), 361–378. Google Scholar

[45]

J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, in Introduction to Shape Optimization, vol. 16 of Springer Series in Computational Mathematics, Springer, Berlin, Heidelberg, 1992. doi: 10.1007/978-3-642-58106-9_1. Google Scholar

[46]

T. Tiihonen, Shape optimization and trial methods for free boundary problems, RAIRO Modél. Math. Anal. Numér., 31 (1997), 805-825. doi: 10.1051/m2an/1997310708051. Google Scholar

show all references

References:
[1]

A. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine. Angew. Math., 325 (1981), 105-144. doi: 10.1515/crll.1981.325.105. Google Scholar

[2]

H. Azegami, Second derivatives of cost functions and $H^1$ Newton method in shape optimization problems, Mathematical Analysis of Continuum Mechanics and Industrial Applications Ⅱ, in Proceedings of the International Conference CoMFoS16 (eds. P. van Meurs, M. Kimura and H. Notsu), vol. 30 of Mathematics for Industry, Springer, Singapore, (2017), 61–72. doi: 10.1007/978-981-10-6283-4_6. Google Scholar

[3]

H. Azegami, Shape Optimization Problems, Morikita Publishing Co., Ltd., Tokyo, 2016 (in Japanese).Google Scholar

[4]

H. Azegami, Solution of shape optimization problem and its application to product design, Mathematical Analysis of Continuum Mechanics and Industrial Applications, in Computer Aided Optimization Design of Structures IV, Structural Optimization (eds. H. Itou, M. Kimura, V. Chalupecký, K. Ohtsuka, D. Tagami and A. Takada), vol. 26 of Mathematics for Industry, Springer, Singapore, (2017), 83–98. doi: 10.1007/978-981-10-2633-1_6. Google Scholar

[5]

H. Azegami, S. Kaizu, M. Shimoda and E. Katamine, Irregularity of shape optimization problems and an improvement technique, in Computer Aided Optimization Design of Structures IV, Structural Optimization (eds. S. Hernandez and C. A. Brebbia), Computational Mechanics Publications, Southampton, (1997), 309–326. doi: 10.2495/OP970301. Google Scholar

[6]

H. Azegami and Z. Q. Wu, Domain optimization analysis in linear elastic problems: Approach using traction method, SME Int. J., Ser. A., 39 (1996), 272-278. doi: 10.1299/jsmea1993.39.2_272. Google Scholar

[7]

H. Azegami, M. Shimoda, E. Katamine and Z. C. Wu, A domain optimization technique for elliptic boundary value problems, in Computer Aided Optimization Design of Structures IV, Structural Optimization (eds. S. Hernandez, M. El-Sayed and C. A. Brebbia), Computational Mechanics Publications, Southampton, (1995), 51–58. doi: 10.2495/OP950071. Google Scholar

[8]

H. Azegami, Solution to domain optimization problems, Trans. Jpn. Soc. Mech. Eng., Ser. A., 60 (1994), 1479–1486 (in Japanese). doi: 10.1299/kikaia.60.1479. Google Scholar

[9]

J. B. Bacani, Methods of Shape Optimization in Free Boundary Problems, Ph.D. Thesis, Karl-Franzens-Universität-Graz, 2013.Google Scholar

[10]

J. B. Bacani and G. Peichl, On the first-order shape derivative of the Kohn-Vogelius cost functional of the Bernoulli problem, Abstr. Appl. Anal., 2013 (2013), Art. ID 384320, 19pp. doi: 10.1155/2013/384320. Google Scholar

[11]

A. Ben AbdaF. BouchonG. PeichlM. Sayeh and R. Touzani, A Dirichlet-Neumann cost functional approach for the Bernoulli problem, J. Eng. Math., 81 (2013), 157-176. doi: 10.1007/s10665-012-9608-3. Google Scholar

[12]

A. BoulkhemairA. Nachaoui and A. Chakib, A shape optimization approach for a class of free boundary problems of Bernoulli type, Appl. Math., 58 (2013), 205-221. doi: 10.1007/s10492-013-0010-x. Google Scholar

[13]

A. BoulkhemairA. Chakib and A. Nachaoui, Uniform trace theorem and application to shape optimization, Appl. Comput. Math., 7 (2008), 192-205. Google Scholar

[14]

A. Boulkhemair and A. Chakib, On the uniform Poincaré inequality, Commun. Partial Differ. Equations, 32 (2007), 1439-1447. doi: 10.1080/03605300600910241. Google Scholar

[15]

D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., 52 (1975), 189-219. doi: 10.1016/0022-247X(75)90091-8. Google Scholar

[16]

M. Dambrine, On variations of the shape Hessian and sufficient conditions for the stability of critical shapes, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A. Mat., 96 (2002), 95-121. Google Scholar

[17]

M. Dambrine and M. Pierre, About stability of equilibrium shapes, Model Math. Anal. Numer., 34 (2000), 811-834. doi: 10.1051/m2an:2000105. Google Scholar

[18]

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2nd edition, Adv. Des. Control 22, SIAM, Philadelphia, 2011. doi: 10.1137/1.9780898719826. Google Scholar

[19]

K. Eppler and H. Harbrecht, On a Kohn-Vogelius like formulation of free boundary problems, Comput. Optim. App., 52 (2012), 69-85. doi: 10.1007/s10589-010-9345-3. Google Scholar

[20]

K. Eppler and H. Harbrecht, A regularized Newton method in electrical impedance tomography using shape Hessian information, Control Cybern., 34 (2005), 203-225. Google Scholar

[21]

K. Eppler, Boundary integral representations of second derivatives in shape optimization, Discuss. Math. Differ. Incl. Control. Optim., 20 (2000), 63-78. doi: 10.7151/dmdico.1005. Google Scholar

[22]

K. Eppler, Optimal shape design for elliptic equations via BIE-methods, J. Appl. Math. Comput. Sci., 10 (2000), 487-516. Google Scholar

[23]

A. Fasano, Some free boundary problems with industrial applications, in Shape Optimization and Free Boundaries (eds. M. C. Delfour and G. Sabidussi), vol. 380 of NATO ASI Series (C: Mathematical and Physical Sciences), Springer, Dordrecht, (1992), 113–142. doi: 10.1007/978-94-011-2710-3_3. Google Scholar

[24]

M. Flucher and M. Rumpf, Bernoulli's free-boundary problem, qualitative theory and numerical approximation, J. Reine. Angew. Math., 486 (1997), 165-204. doi: 10.1515/crll.1997.486.165. Google Scholar

[25]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/978-3-642-61798-0. Google Scholar

[26]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Publishing, Marshfield, Massachusetts, 1985. doi: 10.1137/1.9781611972030. Google Scholar

[27]

J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718690. Google Scholar

[28]

J. HaslingerT. KozubekK. Kunisch and G. Peichl, An embedding domain approach for a class of 2-d shape optimization problems: Mathematical analysis, J. Math. Anal. Appl., 290 (2004), 665-685. doi: 10.1016/j.jmaa.2003.10.038. Google Scholar

[29]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265. doi: 10.1515/jnum-2012-0013. Google Scholar

[30]

A. Henrot and M. Pierre, Shape Variation and Optimization: A Geometrical Analysis, Tracts in Mathematics 28, European Mathematical Society, Zürich, 2018. doi: 10.4171/178. Google Scholar

[31]

K. ItoK. Kunisch and G. Peichl, Variational approach to shape derivatives, ESAIM Control Optim. Calc. Var., 14 (2008), 517-539. doi: 10.1051/cocv:2008002. Google Scholar

[32]

T. KashiwabaraC. M. ColciagoL. Dedè and A. Quarteroni, Well-posedness, regularity, and convergence analysis of the finite element approximation of a generalized Robin boundary value problem, SIAM J. Numer. Anal., 53 (2015), 105-126. doi: 10.1137/140954477. Google Scholar

[33]

R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Commun. Pure Appl., 37 (1984), 289-298. doi: 10.1002/cpa.3160370302. Google Scholar

[34]

R. Kress, On Trefftz' integral equation for the Bernoulli free boundary value problem, Numer. Math., 136 (2017), 503-522. doi: 10.1007/s00211-016-0847-5. Google Scholar

[35]

J. Nečas, Direct Methods in the Theory of Elliptic Equations, Corrected 2nd edition, Monographs and Studies in Mathematics, Springer, Berlin, Heidelberg, 2012. doi: 10.1007/978-3-642-10455-8. Google Scholar

[36]

J. W. Neuberger, Sobolev Gradients and Differential Equations, Springer-Verlag, Berlin, 1997. doi: 10.1007/BFb0092831. Google Scholar

[37]

A. Novruzi and M. Pierre, Structure of shape derivatives, J. Evol. Equ., 2 (2002), 365-382. doi: 10.1007/s00028-002-8093-y. Google Scholar

[38]

A. Novruzi and J.-R. Roche, Newton's method in shape optimisation: A three-dimensional case, BIT Numer. Math., 40 (2000), 102-120. doi: 10.1023/A:1022370419231. Google Scholar

[39]

S. Osher and J. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. Google Scholar

[40]

A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, vol. 37 of Texts in Applied Mathematics, 2nd edition, Springer, Berlin, 2007. doi: 10.1007/b98885. Google Scholar

[41]

J. F. T. Rabago and H. Azegami, An improved shape optimization formulation of the Bernoulli problem by tracking the Neumann data, J. Eng. Math., (2019), to appear.Google Scholar

[42]

J. F. T. Rabago and J. B. Bacani, Shape optimization approach to the Bernoulli problem: A Lagrangian formulation, IAENG Int. J. Appl. Math., 47 (2017), 417-424. Google Scholar

[43]

J. F. T. Rabago and J. B. Bacani, Shape optimization approach for solving the Bernoulli problem by tracking the Neumann data: a Lagrangian formulation, Commun. Pur. Appl. Anal., 17 (2018), 2683-2702. doi: 10.3934/cpaa.2018127. Google Scholar

[44]

J. Simon, Second variation for domain optimization problems, in Control and Estimation of Distributed Parameter Systems (eds. F. Kappel, K. Kunisch and W. Schappacher), International Series of Numerical Mathematics, no 91. Birkhäuser, (1989), 361–378. Google Scholar

[45]

J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, in Introduction to Shape Optimization, vol. 16 of Springer Series in Computational Mathematics, Springer, Berlin, Heidelberg, 1992. doi: 10.1007/978-3-642-58106-9_1. Google Scholar

[46]

T. Tiihonen, Shape optimization and trial methods for free boundary problems, RAIRO Modél. Math. Anal. Numér., 31 (1997), 805-825. doi: 10.1051/m2an/1997310708051. Google Scholar

Figure 1.  Initial (left) and final (right) free boundaries for Example 1 with $ \alpha = 0.1 $ in (57)
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 $
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
Figure 4.  Results of Example 2 for $ \lambda = -9 $ when $ \eta = 10^{-6} $ in the stopping condition (58) and $ \alpha = 0.99 $ in (57)
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)
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)
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
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
Figure 9.  Results of Example 3 when $ \lambda = -10, -4, -1 $ for both of the proposed and classical formulations
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)
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
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
$ \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
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
$ \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
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
classical $ 0.767890 $ $ 0.000137 $ $ 19 $ $ 55 $ sec
$ -9 $ proposed $ 0.221636 $ $ 6.25 \times 10^{-7} $ $ 13 $ $ 16 $ sec
classical $ 0.738627 $ $ 3.11 \times 10^{-5} $ $ 25 $ $ 96 $ sec
$ -8 $ proposed $ 0.213501 $ $ 7.94 \times 10^{-7} $ $ 12 $ $ 14 $ sec
classical $ 0.702851 $ $ 7.17 \times 10^{-5} $ $ 19 $ $ 57 $ sec
$ -7 $ proposed $ 0.203058 $ $ 1.23 \times 10^{-6} $ $ 10 $ $ 13 $ sec
classical $ 0.658125 $ $ 0.000628 $ $ 12 $ $ 34 $ sec
$ -6 $ proposed $ 0.189163 $ $ 8.27 \times 10^{-7} $ $ 10 $ $ 14 $ sec
classical $ 0.600640 $ $ 0.000190 $ $ 13 $ $ 34 $ sec
$ -5 $ proposed $ 0.169783 $ $ 2.61 \times 10^{-7} $ $ 10 $ $ 16 $ sec
classical $ 0.524113 $ $ 0.000948 $ $ 18 $ $ 54 $ sec
$ -4 $ proposed $ 0.140942 $ $ 2.04 \times 10^{-7} $ $ 10 $ $ 17 $ sec
classical $ 0.417590 $ $ 0.000186 $ $ 8 $ $ 24 $ sec
$ -3 $ proposed $ 0.094111 $ $ 5.68 \times 10^{-7} $ $ 9 $ $ 14 $ sec
classical $ 0.262124 $ $ 4.95 \times 10^{-5} $ $ 11 $ $ 29 $ sec
$ -2 $ proposed $ 0.039805 $ $ 2.37 \times 10^{-7} $ $ 10 $ $ 17 $ sec
classical $ 0.120956 $ $ 6.48 \times 10^{-6} $ $ 10 $ $ 27 $ sec
$ -1 $ proposed $ 0.312388 $ $ 1.08 \times 10^{-6} $ $ 13 $ $ 25 $ sec
classical $ 0.615256 $ $ 5.69 \times 10^{-7} $ $ 9 $ $ 24 $ sec
$ \lambda $ formulation $ t_0 $ cost iteration cpu time
$ -10 $ proposed $ 0.228150 $ $ 1.37 \times 10^{-6} $ $ 14 $ $ 14 $ sec
classical $ 0.767890 $ $ 0.000137 $ $ 19 $ $ 55 $ sec
$ -9 $ proposed $ 0.221636 $ $ 6.25 \times 10^{-7} $ $ 13 $ $ 16 $ sec
classical $ 0.738627 $ $ 3.11 \times 10^{-5} $ $ 25 $ $ 96 $ sec
$ -8 $ proposed $ 0.213501 $ $ 7.94 \times 10^{-7} $ $ 12 $ $ 14 $ sec
classical $ 0.702851 $ $ 7.17 \times 10^{-5} $ $ 19 $ $ 57 $ sec
$ -7 $ proposed $ 0.203058 $ $ 1.23 \times 10^{-6} $ $ 10 $ $ 13 $ sec
classical $ 0.658125 $ $ 0.000628 $ $ 12 $ $ 34 $ sec
$ -6 $ proposed $ 0.189163 $ $ 8.27 \times 10^{-7} $ $ 10 $ $ 14 $ sec
classical $ 0.600640 $ $ 0.000190 $ $ 13 $ $ 34 $ sec
$ -5 $ proposed $ 0.169783 $ $ 2.61 \times 10^{-7} $ $ 10 $ $ 16 $ sec
classical $ 0.524113 $ $ 0.000948 $ $ 18 $ $ 54 $ sec
$ -4 $ proposed $ 0.140942 $ $ 2.04 \times 10^{-7} $ $ 10 $ $ 17 $ sec
classical $ 0.417590 $ $ 0.000186 $ $ 8 $ $ 24 $ sec
$ -3 $ proposed $ 0.094111 $ $ 5.68 \times 10^{-7} $ $ 9 $ $ 14 $ sec
classical $ 0.262124 $ $ 4.95 \times 10^{-5} $ $ 11 $ $ 29 $ sec
$ -2 $ proposed $ 0.039805 $ $ 2.37 \times 10^{-7} $ $ 10 $ $ 17 $ sec
classical $ 0.120956 $ $ 6.48 \times 10^{-6} $ $ 10 $ $ 27 $ sec
$ -1 $ proposed $ 0.312388 $ $ 1.08 \times 10^{-6} $ $ 13 $ $ 25 $ sec
classical $ 0.615256 $ $ 5.69 \times 10^{-7} $ $ 9 $ $ 24 $ sec
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
$ \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
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
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
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
classical $ 0.502658440 $ $ 0.3123875282 $ $ 5.51 \times 10^{-5} $ $ 9 $ $ 24 $ sec
$ \lambda $ formulation $ \alpha $ $ t_0 $ cost iteration cpu time
$ -9 $ proposed $ 0.990000000 $ $ 0.2216361144 $ $ 1.55 \times 10^{-5} $ $ 9 $ $ 10 $ sec
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
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
classical $ 0.502658440 $ $ 0.3123875282 $ $ 5.51 \times 10^{-5} $ $ 9 $ $ 24 $ sec
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
$ \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
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
$ \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
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} $
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 $
formulation iteration cpu time $ \frac{\rm cpu\ time}{\rm iteration} $
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 $
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
$ 0.386646216 $ $ 0.337420581 $ $ 0.001223 $ $ 15 $ $ 24 $ sec
$ -9 $ $ 0.990000000 $ $ 0.331971040 $ $ 5.73 \times 10^{-7} $ $ 13 $ $ 16 $ sec
$ 0.388463867 $ $ 0.331971040 $ $ 0.000473 $ $ 11 $ $ 20 $ sec
$ -8 $ $ 0.990000000 $ $ 0.325160189 $ $ 5.07 \times 10^{-7} $ $ 13 $ $ 14 $ sec
$ 0.390736229 $ $ 0.325160189 $ $ 0.000582 $ $ 10 $ $ 17 $ sec
$ -7 $ $ 0.990000000 $ $ 0.316405255 $ $ 3.36 \times 10^{-7} $ $ 12 $ $ 18 $ sec
$ 0.393657966 $ $ 0.316405255 $ $ 0.000745 $ $ 10 $ $ 16 $ sec
$ -6 $ $ 0.990000000 $ $ 0.304735585 $ $ 3.79 \times 10^{-7} $ $ 11 $ $ 14 $ sec
$ 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
$ 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
$ 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
$ 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
$ 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
$ 0.482819584 $ $ 0.111335863 $ $ 3.54 \times 10^{-5} $ $ 8 $ $ 16 $ sec
$ \lambda $ $ \alpha $ $ t_0 $ cost iter cpu time
$ -10 $ $ 0.990000000 $ $ 0.337420581 $ $ 1.33 \times 10^{-7} $ $ 14 $ $ 14 $ sec
$ 0.386646216 $ $ 0.337420581 $ $ 0.001223 $ $ 15 $ $ 24 $ sec
$ -9 $ $ 0.990000000 $ $ 0.331971040 $ $ 5.73 \times 10^{-7} $ $ 13 $ $ 16 $ sec
$ 0.388463867 $ $ 0.331971040 $ $ 0.000473 $ $ 11 $ $ 20 $ sec
$ -8 $ $ 0.990000000 $ $ 0.325160189 $ $ 5.07 \times 10^{-7} $ $ 13 $ $ 14 $ sec
$ 0.390736229 $ $ 0.325160189 $ $ 0.000582 $ $ 10 $ $ 17 $ sec
$ -7 $ $ 0.990000000 $ $ 0.316405255 $ $ 3.36 \times 10^{-7} $ $ 12 $ $ 18 $ sec
$ 0.393657966 $ $ 0.316405255 $ $ 0.000745 $ $ 10 $ $ 16 $ sec
$ -6 $ $ 0.990000000 $ $ 0.304735585 $ $ 3.79 \times 10^{-7} $ $ 11 $ $ 14 $ sec
$ 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
$ 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
$ 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
$ 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
$ 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
$ 0.482819584 $ $ 0.111335863 $ $ 3.54 \times 10^{-5} $ $ 8 $ $ 16 $ sec
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} $
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 $
formulation iteration cpu time $ \frac{\rm cpu\ time}{\rm iteration} $
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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