-
Previous Article
Continuation and collapse of homoclinic tangles
- JCD Home
- This Issue
-
Next Article
Global invariant manifolds near a Shilnikov homoclinic bifurcation
The computation of convex invariant sets via Newton's method
1. | Chair of Applied Mathematics, University of Bayreuth, 95440 Bayreuth, Germany |
2. | Chair of Applied Mathematics, University of Paderborn, 33098 Paderborn, Germany, Germany, Germany |
3. | Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544, United States |
References:
[1] |
J.-P. Aubin, Mutational and Morphological Analysis. Tools for Shape Evolution and Morphogenesis, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1999.
doi: 10.1007/978-1-4612-1576-9. |
[2] |
R. Baier, C. Büskens, I. A. Chahma and M. Gerdts, Approximation of reachable sets by direct solution methods of optimal control problems, Optim. Meth. Softw., 22 (2007), 433-452.
doi: 10.1080/10556780600604999. |
[3] |
R. Baier and E. M. Farkhi, Directed derivatives of convex compact-valued mappings, in Advances in Convex Analysis and Global Optimization: Honoring the Memory of C. Caratheodory (1873-1950) (eds. N. Hadjisavvas and P. M. Pardalos), vol. 54 of Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht-Boston-London, (2001), 501-514.
doi: 10.1007/978-1-4613-0279-7_32. |
[4] |
R. Baier and E. M. Farkhi, The directed subdifferential of DC functions, in Nonlinear Analysis and Optimization II: Optimization. A Conference in Celebration of Alex Ioffe's 70th and Simeon Reich's 60th Birthdays, June 18-24, (2008), Haifa, Israel (eds. A. Leizarowitz, B. S. Mordukhovich, I. Shafrir and A. J. Zaslavski), vol. 513 of AMS Contemp. Math., AMS and Bar-Ilan University, (2010), 27-43, http://www.ams.org/bookstore-getitem/item=CONM-514.
doi: 10.1090/conm/514/10098. |
[5] |
R. Baier and M. Hessel-von Molo, Newton's method and secant method for set-valued mappings, in Proceedings on the 8th International Conference on "Large-Scale Scientific Computations'' (LSSC 2011), June 6-10, 2011, Sozopol, Bulgaria (eds. I. Lirkov, S. Margenov and J. Wanśiewski), vol. 7116 of Lecture Notes in Comput. Sci., Springer-Verlag, Berlin-Heidelberg, 2012, 91-98.
doi: 10.1007/978-3-642-29843-1_9. |
[6] |
R. Baier and G. Perria, Set-valued Hermite interpolation, J. Approx. Theory, 163 (2011), 1349-1372.
doi: 10.1016/j.jat.2010.11.004. |
[7] |
R. Baier and E. M. Farkhi, Differences of convex compact sets in the space of directed sets. I. The space of directed sets, Set-Valued Anal., 9 (2001), 217-245.
doi: 10.1023/A:1012046027626. |
[8] |
R. Baier and E. M. Farkhi, Differences of convex compact sets in the space of directed sets. II. Visualization of directed sets, Set-Valued Anal., 9 (2001), 247-272.
doi: 10.1023/A:1012009529492. |
[9] |
W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal., 10 (1990), 379-405, http://imajna.oxfordjournals.org/cgi/content/abstract/10/3/379.
doi: 10.1093/imanum/10.3.379. |
[10] |
F. Blanchini, Set invariance in control, Automatica, 35 (1999), 1747-1767.
doi: 10.1016/S0005-1098(99)00113-2. |
[11] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - set oriented numerical methods for dynamical systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (ed. B. Fiedler), Springer, (2001), 145-174. |
[12] |
M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer. Math., 75 (1997), 293-317.
doi: 10.1007/s002110050240. |
[13] |
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal., 36 (1999), 491-515.
doi: 10.1137/S0036142996313002. |
[14] |
V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, vol. 7 of Approximation and Optimization, Peter Lang, Frankfurt am Main-Berlin-Bern-New York-Paris-Wien, 1995, Updated and revised English edition of V. F. Demyanov, A. M. Rubinov, Foundations of Nonsmooth Analysis, and Quasidifferentiable Calculus, Optimizatsiya i Operatsiĭ 23, Nauka, Moscow, 1990 (in Russian). |
[15] |
L. Dieci, J. Lorenz and R. D. Russell, Numerical calculation of invariant tori, SIAM J. Sci. Statist. Comput., 12 (1991), 607-647, http://link.aip.org/link/?SCE/12/607/1.
doi: 10.1137/0912033. |
[16] |
N. S. Dimitrova and S. M. Markov, Interval methods of Newton type for nonlinear equations, Pliska Stud. Math. Bulgar., 5 (1983), 105-117. |
[17] |
T. Donchev and E. M. Farkhi, Fixed set iterations for relaxed Lipschitz multimaps, Nonlinear Anal., 53 (2003), 997-1015.
doi: 10.1016/S0362-546X(03)00036-1. |
[18] |
H. Hadwiger, Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt, Math. Z., 53 (1950), 210-218.
doi: 10.1007/BF01175656. |
[19] |
E. Hansen, A multidimensional interval Newton method, Reliab. Comput., 12 (2006), 253-272.
doi: 10.1007/s11155-006-9000-y. |
[20] |
A. J. Homburg, H. M. Osinga and G. Vegter, On the computation of invariant manifolds of fixed points, ZAMM Z. Angew. Math. Phys., 46 (1995), 171-187.
doi: 10.1007/BF00944751. |
[21] |
C. S. Hsu, Global analysis by cell mapping, Internat. J. Bifurc. Chaos Appl. Sci. Engrg., 2 (1992), 727-771.
doi: 10.1142/S0218127492000422. |
[22] |
L. V. Kantorovich, On Newton's method for functional equations, Dokl. Akad. Nauk SSSR, 59 (1948), 1237-1240. |
[23] |
L. V. Kantorovich, The majorant principle and Newton's method, Dokl. Akad. Nauk SSSR, 76 (1951), 17-20. |
[24] |
L. V. Kantorovich and G. P. Akilov, Functional Analysis, 2nd edition, Pergamon Press, Oxford, 1982, Translated from the Russian by H. L. Silcock. |
[25] |
I. G. Kevrekidis, R. Aris, L. D. Schmidt and S. Pelikan, Numerical computation of invariant circles of maps, Phys. D, 16 (1985), 243-251, http://www.sciencedirect.com/science/article/B6TVK-46MNK8Y-2T /2/48df9752ceb87d5170b0eabe206cbfb9.
doi: 10.1016/0167-2789(85)90061-2. |
[26] |
I. G. Kevrekidis, C. W. Gear and G. Hummer, Equation-free: The computer-aided analysis of complex multiscale systems, AIChE Journ., 50 (2004), 1346-1354.
doi: 10.1002/aic.10106. |
[27] |
I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg and C. Theodoropoulos, Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis, Commun. Math. Sci., 1 (2003), 715-762, http://projecteuclid.org/getRecord?id=euclid.cms/1119655353.
doi: 10.4310/CMS.2003.v1.n4.a5. |
[28] |
D. Klatte and B. Kummer, Nonsmooth equations in optimization. Regularity, calculus, methods and applications, vol. 60 of Nonconvex Optimization and its Applications, Kluwer Academic Publishers, Dordrecht, 2002. |
[29] |
B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791.
doi: 10.1142/S0218127405012533. |
[30] |
R. Krawczyk, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken, Computing (Arch. Elektron. Rechnen), 4 (1969), 187-201. |
[31] |
R. Krawczyk, Einschlieş ung von Nullstellen mit Hilfe einer Intervallarithmetik, Computing (Arch. Elektron. Rechnen), 5 (1970), 356-370. |
[32] |
E. Kreuzer, Analysis of chaotic systems using the cell mapping approach, Arch. Appl. Mech., 55 (1985), 285-294.
doi: 10.1007/BF00538223. |
[33] |
S. M. Markov, Some applications of extended interval arithmetic to interval iterations, Comput. Suppl., 2 (1980), 69-84. |
[34] |
K. Nickel, Das Auflösungsverhalten von nichtlinearen Fixmengen-Systemen, in Iterative solution of nonlinear systems of equations (Oberwolfach, 1982), vol. 953 of Lecture Notes in Math., Springer, Berlin, (1982), 106-124. |
[35] |
D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets, vol. 548 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 2002. |
[36] |
G. Perria, Set-valued Interpolation, vol. 79 of Bayreuth. Math. Schr., Department of Mathematics, University of Bayreuth, Germany, 2007. |
[37] |
B. T. Polyak, Convexity of quadratic transformations and its use in control and optimization, J. Optim. Theory Appl., 99 (1998), 553-583.
doi: 10.1023/A:1021798932766. |
[38] |
B. T. Polyak, Convexity of nonlinear image of a small ball with applications to optimization, Set-Valued Anal., 9 (2001), 159-168, Special issue on Wellposedness in Optimization and Related Topics (Gargnano, 1999).
doi: 10.1023/A:1011287523150. |
[39] |
L. S. Pontryagin, Linear differential games. II, Sov. Math., Dokl., 8 (1967), 910-912. |
[40] |
R. T. Rockafellar, Convex Analysis, vol. 28 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1970. |
[41] |
A. M. Rubinov and I. S. Akhundov, Difference of compact sets in the sense of Demyanov and its application to non-smooth analysis, Optimization, 23 (1992), 179-188.
doi: 10.1080/02331939208843757. |
[42] |
P. Saint-Pierre, Newton and other continuation methods for multivalued inclusions, Set-Valued Anal., 3 (1995), 143-156.
doi: 10.1007/BF01038596. |
[43] |
C. I. Siettos, C. C. Pantelides and I. G. Kevrekidis, Enabling dynamic process simulators to perform alternative tasks: A time-stepper-based toolkit for computer-aided analysis, Ind. Engrg. Chem. Res., 42 (2003), 6795-6801.
doi: 10.1021/ie021062w. |
[44] |
C. Theodoropoulos, Y.-H. Qian and I. G. Kevrekidis, "Coarse'' stability and bifurcation analysis using time-steppers: A reaction-diffusion example, Proc. Natl. Acad. Sci., 97 (2000), 9840-9843, http://www.pnas.org/content/97/18/9840.abstract.
doi: 10.1073/pnas.97.18.9840. |
[45] |
L. S. Tuckerman and D. Barkley, Bifurcation analysis for timesteppers, in Numerical methods for bifurcation problems and large-scale dynamical systems (Minneapolis, MN, 1997), vol. 119 of IMA Vol. Math. Appl., Springer, New York, (2000), 453-466.
doi: 10.1007/978-1-4612-1208-9_20. |
[46] |
X. Wang, Convergence of Newton's method and inverse function theorem in Banach space, Math. Comp., 68 (1999), 169-186.
doi: 10.1090/S0025-5718-99-00999-0. |
[47] |
T. Yamamoto, A method for finding sharp error bounds for Newton's method under the Kantorovich assumptions, Numer. Math., 49 (1986), 203-220.
doi: 10.1007/BF01389624. |
show all references
References:
[1] |
J.-P. Aubin, Mutational and Morphological Analysis. Tools for Shape Evolution and Morphogenesis, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1999.
doi: 10.1007/978-1-4612-1576-9. |
[2] |
R. Baier, C. Büskens, I. A. Chahma and M. Gerdts, Approximation of reachable sets by direct solution methods of optimal control problems, Optim. Meth. Softw., 22 (2007), 433-452.
doi: 10.1080/10556780600604999. |
[3] |
R. Baier and E. M. Farkhi, Directed derivatives of convex compact-valued mappings, in Advances in Convex Analysis and Global Optimization: Honoring the Memory of C. Caratheodory (1873-1950) (eds. N. Hadjisavvas and P. M. Pardalos), vol. 54 of Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht-Boston-London, (2001), 501-514.
doi: 10.1007/978-1-4613-0279-7_32. |
[4] |
R. Baier and E. M. Farkhi, The directed subdifferential of DC functions, in Nonlinear Analysis and Optimization II: Optimization. A Conference in Celebration of Alex Ioffe's 70th and Simeon Reich's 60th Birthdays, June 18-24, (2008), Haifa, Israel (eds. A. Leizarowitz, B. S. Mordukhovich, I. Shafrir and A. J. Zaslavski), vol. 513 of AMS Contemp. Math., AMS and Bar-Ilan University, (2010), 27-43, http://www.ams.org/bookstore-getitem/item=CONM-514.
doi: 10.1090/conm/514/10098. |
[5] |
R. Baier and M. Hessel-von Molo, Newton's method and secant method for set-valued mappings, in Proceedings on the 8th International Conference on "Large-Scale Scientific Computations'' (LSSC 2011), June 6-10, 2011, Sozopol, Bulgaria (eds. I. Lirkov, S. Margenov and J. Wanśiewski), vol. 7116 of Lecture Notes in Comput. Sci., Springer-Verlag, Berlin-Heidelberg, 2012, 91-98.
doi: 10.1007/978-3-642-29843-1_9. |
[6] |
R. Baier and G. Perria, Set-valued Hermite interpolation, J. Approx. Theory, 163 (2011), 1349-1372.
doi: 10.1016/j.jat.2010.11.004. |
[7] |
R. Baier and E. M. Farkhi, Differences of convex compact sets in the space of directed sets. I. The space of directed sets, Set-Valued Anal., 9 (2001), 217-245.
doi: 10.1023/A:1012046027626. |
[8] |
R. Baier and E. M. Farkhi, Differences of convex compact sets in the space of directed sets. II. Visualization of directed sets, Set-Valued Anal., 9 (2001), 247-272.
doi: 10.1023/A:1012009529492. |
[9] |
W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal., 10 (1990), 379-405, http://imajna.oxfordjournals.org/cgi/content/abstract/10/3/379.
doi: 10.1093/imanum/10.3.379. |
[10] |
F. Blanchini, Set invariance in control, Automatica, 35 (1999), 1747-1767.
doi: 10.1016/S0005-1098(99)00113-2. |
[11] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - set oriented numerical methods for dynamical systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (ed. B. Fiedler), Springer, (2001), 145-174. |
[12] |
M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer. Math., 75 (1997), 293-317.
doi: 10.1007/s002110050240. |
[13] |
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal., 36 (1999), 491-515.
doi: 10.1137/S0036142996313002. |
[14] |
V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, vol. 7 of Approximation and Optimization, Peter Lang, Frankfurt am Main-Berlin-Bern-New York-Paris-Wien, 1995, Updated and revised English edition of V. F. Demyanov, A. M. Rubinov, Foundations of Nonsmooth Analysis, and Quasidifferentiable Calculus, Optimizatsiya i Operatsiĭ 23, Nauka, Moscow, 1990 (in Russian). |
[15] |
L. Dieci, J. Lorenz and R. D. Russell, Numerical calculation of invariant tori, SIAM J. Sci. Statist. Comput., 12 (1991), 607-647, http://link.aip.org/link/?SCE/12/607/1.
doi: 10.1137/0912033. |
[16] |
N. S. Dimitrova and S. M. Markov, Interval methods of Newton type for nonlinear equations, Pliska Stud. Math. Bulgar., 5 (1983), 105-117. |
[17] |
T. Donchev and E. M. Farkhi, Fixed set iterations for relaxed Lipschitz multimaps, Nonlinear Anal., 53 (2003), 997-1015.
doi: 10.1016/S0362-546X(03)00036-1. |
[18] |
H. Hadwiger, Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt, Math. Z., 53 (1950), 210-218.
doi: 10.1007/BF01175656. |
[19] |
E. Hansen, A multidimensional interval Newton method, Reliab. Comput., 12 (2006), 253-272.
doi: 10.1007/s11155-006-9000-y. |
[20] |
A. J. Homburg, H. M. Osinga and G. Vegter, On the computation of invariant manifolds of fixed points, ZAMM Z. Angew. Math. Phys., 46 (1995), 171-187.
doi: 10.1007/BF00944751. |
[21] |
C. S. Hsu, Global analysis by cell mapping, Internat. J. Bifurc. Chaos Appl. Sci. Engrg., 2 (1992), 727-771.
doi: 10.1142/S0218127492000422. |
[22] |
L. V. Kantorovich, On Newton's method for functional equations, Dokl. Akad. Nauk SSSR, 59 (1948), 1237-1240. |
[23] |
L. V. Kantorovich, The majorant principle and Newton's method, Dokl. Akad. Nauk SSSR, 76 (1951), 17-20. |
[24] |
L. V. Kantorovich and G. P. Akilov, Functional Analysis, 2nd edition, Pergamon Press, Oxford, 1982, Translated from the Russian by H. L. Silcock. |
[25] |
I. G. Kevrekidis, R. Aris, L. D. Schmidt and S. Pelikan, Numerical computation of invariant circles of maps, Phys. D, 16 (1985), 243-251, http://www.sciencedirect.com/science/article/B6TVK-46MNK8Y-2T /2/48df9752ceb87d5170b0eabe206cbfb9.
doi: 10.1016/0167-2789(85)90061-2. |
[26] |
I. G. Kevrekidis, C. W. Gear and G. Hummer, Equation-free: The computer-aided analysis of complex multiscale systems, AIChE Journ., 50 (2004), 1346-1354.
doi: 10.1002/aic.10106. |
[27] |
I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg and C. Theodoropoulos, Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis, Commun. Math. Sci., 1 (2003), 715-762, http://projecteuclid.org/getRecord?id=euclid.cms/1119655353.
doi: 10.4310/CMS.2003.v1.n4.a5. |
[28] |
D. Klatte and B. Kummer, Nonsmooth equations in optimization. Regularity, calculus, methods and applications, vol. 60 of Nonconvex Optimization and its Applications, Kluwer Academic Publishers, Dordrecht, 2002. |
[29] |
B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791.
doi: 10.1142/S0218127405012533. |
[30] |
R. Krawczyk, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken, Computing (Arch. Elektron. Rechnen), 4 (1969), 187-201. |
[31] |
R. Krawczyk, Einschlieş ung von Nullstellen mit Hilfe einer Intervallarithmetik, Computing (Arch. Elektron. Rechnen), 5 (1970), 356-370. |
[32] |
E. Kreuzer, Analysis of chaotic systems using the cell mapping approach, Arch. Appl. Mech., 55 (1985), 285-294.
doi: 10.1007/BF00538223. |
[33] |
S. M. Markov, Some applications of extended interval arithmetic to interval iterations, Comput. Suppl., 2 (1980), 69-84. |
[34] |
K. Nickel, Das Auflösungsverhalten von nichtlinearen Fixmengen-Systemen, in Iterative solution of nonlinear systems of equations (Oberwolfach, 1982), vol. 953 of Lecture Notes in Math., Springer, Berlin, (1982), 106-124. |
[35] |
D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets, vol. 548 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 2002. |
[36] |
G. Perria, Set-valued Interpolation, vol. 79 of Bayreuth. Math. Schr., Department of Mathematics, University of Bayreuth, Germany, 2007. |
[37] |
B. T. Polyak, Convexity of quadratic transformations and its use in control and optimization, J. Optim. Theory Appl., 99 (1998), 553-583.
doi: 10.1023/A:1021798932766. |
[38] |
B. T. Polyak, Convexity of nonlinear image of a small ball with applications to optimization, Set-Valued Anal., 9 (2001), 159-168, Special issue on Wellposedness in Optimization and Related Topics (Gargnano, 1999).
doi: 10.1023/A:1011287523150. |
[39] |
L. S. Pontryagin, Linear differential games. II, Sov. Math., Dokl., 8 (1967), 910-912. |
[40] |
R. T. Rockafellar, Convex Analysis, vol. 28 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1970. |
[41] |
A. M. Rubinov and I. S. Akhundov, Difference of compact sets in the sense of Demyanov and its application to non-smooth analysis, Optimization, 23 (1992), 179-188.
doi: 10.1080/02331939208843757. |
[42] |
P. Saint-Pierre, Newton and other continuation methods for multivalued inclusions, Set-Valued Anal., 3 (1995), 143-156.
doi: 10.1007/BF01038596. |
[43] |
C. I. Siettos, C. C. Pantelides and I. G. Kevrekidis, Enabling dynamic process simulators to perform alternative tasks: A time-stepper-based toolkit for computer-aided analysis, Ind. Engrg. Chem. Res., 42 (2003), 6795-6801.
doi: 10.1021/ie021062w. |
[44] |
C. Theodoropoulos, Y.-H. Qian and I. G. Kevrekidis, "Coarse'' stability and bifurcation analysis using time-steppers: A reaction-diffusion example, Proc. Natl. Acad. Sci., 97 (2000), 9840-9843, http://www.pnas.org/content/97/18/9840.abstract.
doi: 10.1073/pnas.97.18.9840. |
[45] |
L. S. Tuckerman and D. Barkley, Bifurcation analysis for timesteppers, in Numerical methods for bifurcation problems and large-scale dynamical systems (Minneapolis, MN, 1997), vol. 119 of IMA Vol. Math. Appl., Springer, New York, (2000), 453-466.
doi: 10.1007/978-1-4612-1208-9_20. |
[46] |
X. Wang, Convergence of Newton's method and inverse function theorem in Banach space, Math. Comp., 68 (1999), 169-186.
doi: 10.1090/S0025-5718-99-00999-0. |
[47] |
T. Yamamoto, A method for finding sharp error bounds for Newton's method under the Kantorovich assumptions, Numer. Math., 49 (1986), 203-220.
doi: 10.1007/BF01389624. |
[1] |
Liqun Qi, Zheng yan, Hongxia Yin. Semismooth reformulation and Newton's method for the security region problem of power systems. Journal of Industrial and Management Optimization, 2008, 4 (1) : 143-153. doi: 10.3934/jimo.2008.4.143 |
[2] |
Matthias Gerdts, Martin Kunkel. A nonsmooth Newton's method for discretized optimal control problems with state and control constraints. Journal of Industrial and Management Optimization, 2008, 4 (2) : 247-270. doi: 10.3934/jimo.2008.4.247 |
[3] |
Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015 |
[4] |
Andy M. Yip, Wei Zhu. A fast modified Newton's method for curvature based denoising of 1D signals. Inverse Problems and Imaging, 2013, 7 (3) : 1075-1097. doi: 10.3934/ipi.2013.7.1075 |
[5] |
Xing Wang, Nan-Jing Huang. Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces. Journal of Industrial and Management Optimization, 2013, 9 (1) : 57-74. doi: 10.3934/jimo.2013.9.57 |
[6] |
Zhiang Zhou, Xinmin Yang, Kequan Zhao. $E$-super efficiency of set-valued optimization problems involving improvement sets. Journal of Industrial and Management Optimization, 2016, 12 (3) : 1031-1039. doi: 10.3934/jimo.2016.12.1031 |
[7] |
Bernd Hofmann, Barbara Kaltenbacher, Elena Resmerita. Lavrentiev's regularization method in Hilbert spaces revisited. Inverse Problems and Imaging, 2016, 10 (3) : 741-764. doi: 10.3934/ipi.2016019 |
[8] |
T. Tachim Medjo. On the Newton method in robust control of fluid flow. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1201-1222. doi: 10.3934/dcds.2003.9.1201 |
[9] |
Xiaojiao Tong, Felix F. Wu, Yongping Zhang, Zheng Yan, Yixin Ni. A semismooth Newton method for solving optimal power flow. Journal of Industrial and Management Optimization, 2007, 3 (3) : 553-567. doi: 10.3934/jimo.2007.3.553 |
[10] |
Zhi-Feng Pang, Yu-Fei Yang. Semismooth Newton method for minimization of the LLT model. Inverse Problems and Imaging, 2009, 3 (4) : 677-691. doi: 10.3934/ipi.2009.3.677 |
[11] |
Matthias Gerdts, Stefan Horn, Sven-Joachim Kimmerle. Line search globalization of a semismooth Newton method for operator equations in Hilbert spaces with applications in optimal control. Journal of Industrial and Management Optimization, 2017, 13 (1) : 47-62. doi: 10.3934/jimo.2016003 |
[12] |
Robert Denk, Leonid Volevich. A new class of parabolic problems connected with Newton's polygon. Conference Publications, 2007, 2007 (Special) : 294-303. doi: 10.3934/proc.2007.2007.294 |
[13] |
Juhi Jang, Ian Tice. Passive scalars, moving boundaries, and Newton's law of cooling. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1383-1413. doi: 10.3934/dcds.2016.36.1383 |
[14] |
Anatoli Babin, Alexander Figotin. Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1685-1718. doi: 10.3934/cpaa.2014.13.1685 |
[15] |
Honglan Zhu, Qin Ni, Meilan Zeng. A quasi-Newton trust region method based on a new fractional model. Numerical Algebra, Control and Optimization, 2015, 5 (3) : 237-249. doi: 10.3934/naco.2015.5.237 |
[16] |
Xiaojiao Tong, Shuzi Zhou. A smoothing projected Newton-type method for semismooth equations with bound constraints. Journal of Industrial and Management Optimization, 2005, 1 (2) : 235-250. doi: 10.3934/jimo.2005.1.235 |
[17] |
Saeed Ketabchi, Hossein Moosaei, M. Parandegan, Hamidreza Navidi. Computing minimum norm solution of linear systems of equations by the generalized Newton method. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 113-119. doi: 10.3934/naco.2017008 |
[18] |
Hans J. Wolters. A Newton-type method for computing best segment approximations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 133-149 . doi: 10.3934/cpaa.2004.3.133 |
[19] |
Shuang Chen, Li-Ping Pang, Dan Li. An inexact semismooth Newton method for variational inequality with symmetric cone constraints. Journal of Industrial and Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733 |
[20] |
Hong-Yi Miao, Li Wang. Preconditioned inexact Newton-like method for large nonsymmetric eigenvalue problems. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 677-685. doi: 10.3934/naco.2021012 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]