January  2014, 1(1): 39-69. doi: 10.3934/jcd.2014.1.39

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

Received  July 2012 Revised  February 2014 Published  April 2014

In this paper we present a novel approach to the computation of convex invariant sets of dynamical systems. Employing a Banach space formalism to describe differences of convex compact subsets of $\mathbb{R}^n$ by directed sets, we are able to formulate the property of a convex, compact set to be invariant as a zero-finding problem in this Banach space. We need either the additional restrictive assumption that the image of sets from a subclass of convex compact sets under the dynamics remains convex, or we have to convexify these images. In both cases we can apply Newton's method in Banach spaces to approximate such invariant sets if an appropriate smoothness of a set-valued map holds. The theoretical foundations for realizing this approach are analyzed, and it is illustrated first by analytical and then by numerical examples.
Citation: R. Baier, M. Dellnitz, M. Hessel-von Molo, S. Sertl, I. G. Kevrekidis. The computation of convex invariant sets via Newton's method. Journal of Computational Dynamics, 2014, 1 (1) : 39-69. doi: 10.3934/jcd.2014.1.39
References:
[1]

J.-P. Aubin, Mutational and Morphological Analysis. Tools for Shape Evolution and Morphogenesis,, Systems & Control: Foundations & Applications, (1999). doi: 10.1007/978-1-4612-1576-9. Google Scholar

[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. doi: 10.1080/10556780600604999. Google Scholar

[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), (2001), 1873. doi: 10.1007/978-1-4613-0279-7_32. Google Scholar

[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, (2008), 18. doi: 10.1090/conm/514/10098. Google Scholar

[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, (2011), 6. doi: 10.1007/978-3-642-29843-1_9. Google Scholar

[6]

R. Baier and G. Perria, Set-valued Hermite interpolation,, J. Approx. Theory, 163 (2011), 1349. doi: 10.1016/j.jat.2010.11.004. Google Scholar

[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. doi: 10.1023/A:1012046027626. Google Scholar

[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. doi: 10.1023/A:1012009529492. Google Scholar

[9]

W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems,, IMA J. Numer. Anal., 10 (1990), 379. doi: 10.1093/imanum/10.3.379. Google Scholar

[10]

F. Blanchini, Set invariance in control,, Automatica, 35 (1999), 1747. doi: 10.1016/S0005-1098(99)00113-2. Google Scholar

[11]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - set oriented numerical methods for dynamical systems,, in Ergodic Theory, (2001), 145. Google Scholar

[12]

M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors,, Numer. Math., 75 (1997), 293. doi: 10.1007/s002110050240. Google Scholar

[13]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM J. Numer. Anal., 36 (1999), 491. doi: 10.1137/S0036142996313002. Google Scholar

[14]

V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis,, vol. 7 of Approximation and Optimization, (1995). Google Scholar

[15]

L. Dieci, J. Lorenz and R. D. Russell, Numerical calculation of invariant tori,, SIAM J. Sci. Statist. Comput., 12 (1991), 607. doi: 10.1137/0912033. Google Scholar

[16]

N. S. Dimitrova and S. M. Markov, Interval methods of Newton type for nonlinear equations,, Pliska Stud. Math. Bulgar., 5 (1983), 105. Google Scholar

[17]

T. Donchev and E. M. Farkhi, Fixed set iterations for relaxed Lipschitz multimaps,, Nonlinear Anal., 53 (2003), 997. doi: 10.1016/S0362-546X(03)00036-1. Google Scholar

[18]

H. Hadwiger, Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt,, Math. Z., 53 (1950), 210. doi: 10.1007/BF01175656. Google Scholar

[19]

E. Hansen, A multidimensional interval Newton method,, Reliab. Comput., 12 (2006), 253. doi: 10.1007/s11155-006-9000-y. Google Scholar

[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. doi: 10.1007/BF00944751. Google Scholar

[21]

C. S. Hsu, Global analysis by cell mapping,, Internat. J. Bifurc. Chaos Appl. Sci. Engrg., 2 (1992), 727. doi: 10.1142/S0218127492000422. Google Scholar

[22]

L. V. Kantorovich, On Newton's method for functional equations,, Dokl. Akad. Nauk SSSR, 59 (1948), 1237. Google Scholar

[23]

L. V. Kantorovich, The majorant principle and Newton's method,, Dokl. Akad. Nauk SSSR, 76 (1951), 17. Google Scholar

[24]

L. V. Kantorovich and G. P. Akilov, Functional Analysis,, 2nd edition, (1982). Google Scholar

[25]

I. G. Kevrekidis, R. Aris, L. D. Schmidt and S. Pelikan, Numerical computation of invariant circles of maps,, Phys. D, 16 (1985), 243. doi: 10.1016/0167-2789(85)90061-2. Google Scholar

[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. doi: 10.1002/aic.10106. Google Scholar

[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. doi: 10.4310/CMS.2003.v1.n4.a5. Google Scholar

[28]

D. Klatte and B. Kummer, Nonsmooth equations in optimization. Regularity, calculus, methods and applications,, vol. 60 of Nonconvex Optimization and its Applications, (2002). Google Scholar

[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. doi: 10.1142/S0218127405012533. Google Scholar

[30]

R. Krawczyk, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken,, Computing (Arch. Elektron. Rechnen), 4 (1969), 187. Google Scholar

[31]

R. Krawczyk, Einschlieş ung von Nullstellen mit Hilfe einer Intervallarithmetik,, Computing (Arch. Elektron. Rechnen), 5 (1970), 356. Google Scholar

[32]

E. Kreuzer, Analysis of chaotic systems using the cell mapping approach,, Arch. Appl. Mech., 55 (1985), 285. doi: 10.1007/BF00538223. Google Scholar

[33]

S. M. Markov, Some applications of extended interval arithmetic to interval iterations,, Comput. Suppl., 2 (1980), 69. Google Scholar

[34]

K. Nickel, Das Auflösungsverhalten von nichtlinearen Fixmengen-Systemen,, in Iterative solution of nonlinear systems of equations (Oberwolfach, (1982), 106. Google Scholar

[35]

D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets, vol. 548 of Mathematics and Its Applications,, Kluwer Academic Publishers, (2002). Google Scholar

[36]

G. Perria, Set-valued Interpolation, vol. 79 of Bayreuth. Math. Schr.,, Department of Mathematics, (2007). Google Scholar

[37]

B. T. Polyak, Convexity of quadratic transformations and its use in control and optimization,, J. Optim. Theory Appl., 99 (1998), 553. doi: 10.1023/A:1021798932766. Google Scholar

[38]

B. T. Polyak, Convexity of nonlinear image of a small ball with applications to optimization,, Set-Valued Anal., 9 (2001), 159. doi: 10.1023/A:1011287523150. Google Scholar

[39]

L. S. Pontryagin, Linear differential games. II,, Sov. Math., 8 (1967), 910. Google Scholar

[40]

R. T. Rockafellar, Convex Analysis,, vol. 28 of Princeton Mathematical Series, (1970). Google Scholar

[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. doi: 10.1080/02331939208843757. Google Scholar

[42]

P. Saint-Pierre, Newton and other continuation methods for multivalued inclusions,, Set-Valued Anal., 3 (1995), 143. doi: 10.1007/BF01038596. Google Scholar

[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. doi: 10.1021/ie021062w. Google Scholar

[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. doi: 10.1073/pnas.97.18.9840. Google Scholar

[45]

L. S. Tuckerman and D. Barkley, Bifurcation analysis for timesteppers,, in Numerical methods for bifurcation problems and large-scale dynamical systems (Minneapolis, (2000), 453. doi: 10.1007/978-1-4612-1208-9_20. Google Scholar

[46]

X. Wang, Convergence of Newton's method and inverse function theorem in Banach space,, Math. Comp., 68 (1999), 169. doi: 10.1090/S0025-5718-99-00999-0. Google Scholar

[47]

T. Yamamoto, A method for finding sharp error bounds for Newton's method under the Kantorovich assumptions,, Numer. Math., 49 (1986), 203. doi: 10.1007/BF01389624. Google Scholar

show all references

References:
[1]

J.-P. Aubin, Mutational and Morphological Analysis. Tools for Shape Evolution and Morphogenesis,, Systems & Control: Foundations & Applications, (1999). doi: 10.1007/978-1-4612-1576-9. Google Scholar

[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. doi: 10.1080/10556780600604999. Google Scholar

[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), (2001), 1873. doi: 10.1007/978-1-4613-0279-7_32. Google Scholar

[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, (2008), 18. doi: 10.1090/conm/514/10098. Google Scholar

[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, (2011), 6. doi: 10.1007/978-3-642-29843-1_9. Google Scholar

[6]

R. Baier and G. Perria, Set-valued Hermite interpolation,, J. Approx. Theory, 163 (2011), 1349. doi: 10.1016/j.jat.2010.11.004. Google Scholar

[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. doi: 10.1023/A:1012046027626. Google Scholar

[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. doi: 10.1023/A:1012009529492. Google Scholar

[9]

W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems,, IMA J. Numer. Anal., 10 (1990), 379. doi: 10.1093/imanum/10.3.379. Google Scholar

[10]

F. Blanchini, Set invariance in control,, Automatica, 35 (1999), 1747. doi: 10.1016/S0005-1098(99)00113-2. Google Scholar

[11]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - set oriented numerical methods for dynamical systems,, in Ergodic Theory, (2001), 145. Google Scholar

[12]

M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors,, Numer. Math., 75 (1997), 293. doi: 10.1007/s002110050240. Google Scholar

[13]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM J. Numer. Anal., 36 (1999), 491. doi: 10.1137/S0036142996313002. Google Scholar

[14]

V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis,, vol. 7 of Approximation and Optimization, (1995). Google Scholar

[15]

L. Dieci, J. Lorenz and R. D. Russell, Numerical calculation of invariant tori,, SIAM J. Sci. Statist. Comput., 12 (1991), 607. doi: 10.1137/0912033. Google Scholar

[16]

N. S. Dimitrova and S. M. Markov, Interval methods of Newton type for nonlinear equations,, Pliska Stud. Math. Bulgar., 5 (1983), 105. Google Scholar

[17]

T. Donchev and E. M. Farkhi, Fixed set iterations for relaxed Lipschitz multimaps,, Nonlinear Anal., 53 (2003), 997. doi: 10.1016/S0362-546X(03)00036-1. Google Scholar

[18]

H. Hadwiger, Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt,, Math. Z., 53 (1950), 210. doi: 10.1007/BF01175656. Google Scholar

[19]

E. Hansen, A multidimensional interval Newton method,, Reliab. Comput., 12 (2006), 253. doi: 10.1007/s11155-006-9000-y. Google Scholar

[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. doi: 10.1007/BF00944751. Google Scholar

[21]

C. S. Hsu, Global analysis by cell mapping,, Internat. J. Bifurc. Chaos Appl. Sci. Engrg., 2 (1992), 727. doi: 10.1142/S0218127492000422. Google Scholar

[22]

L. V. Kantorovich, On Newton's method for functional equations,, Dokl. Akad. Nauk SSSR, 59 (1948), 1237. Google Scholar

[23]

L. V. Kantorovich, The majorant principle and Newton's method,, Dokl. Akad. Nauk SSSR, 76 (1951), 17. Google Scholar

[24]

L. V. Kantorovich and G. P. Akilov, Functional Analysis,, 2nd edition, (1982). Google Scholar

[25]

I. G. Kevrekidis, R. Aris, L. D. Schmidt and S. Pelikan, Numerical computation of invariant circles of maps,, Phys. D, 16 (1985), 243. doi: 10.1016/0167-2789(85)90061-2. Google Scholar

[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. doi: 10.1002/aic.10106. Google Scholar

[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. doi: 10.4310/CMS.2003.v1.n4.a5. Google Scholar

[28]

D. Klatte and B. Kummer, Nonsmooth equations in optimization. Regularity, calculus, methods and applications,, vol. 60 of Nonconvex Optimization and its Applications, (2002). Google Scholar

[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. doi: 10.1142/S0218127405012533. Google Scholar

[30]

R. Krawczyk, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken,, Computing (Arch. Elektron. Rechnen), 4 (1969), 187. Google Scholar

[31]

R. Krawczyk, Einschlieş ung von Nullstellen mit Hilfe einer Intervallarithmetik,, Computing (Arch. Elektron. Rechnen), 5 (1970), 356. Google Scholar

[32]

E. Kreuzer, Analysis of chaotic systems using the cell mapping approach,, Arch. Appl. Mech., 55 (1985), 285. doi: 10.1007/BF00538223. Google Scholar

[33]

S. M. Markov, Some applications of extended interval arithmetic to interval iterations,, Comput. Suppl., 2 (1980), 69. Google Scholar

[34]

K. Nickel, Das Auflösungsverhalten von nichtlinearen Fixmengen-Systemen,, in Iterative solution of nonlinear systems of equations (Oberwolfach, (1982), 106. Google Scholar

[35]

D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets, vol. 548 of Mathematics and Its Applications,, Kluwer Academic Publishers, (2002). Google Scholar

[36]

G. Perria, Set-valued Interpolation, vol. 79 of Bayreuth. Math. Schr.,, Department of Mathematics, (2007). Google Scholar

[37]

B. T. Polyak, Convexity of quadratic transformations and its use in control and optimization,, J. Optim. Theory Appl., 99 (1998), 553. doi: 10.1023/A:1021798932766. Google Scholar

[38]

B. T. Polyak, Convexity of nonlinear image of a small ball with applications to optimization,, Set-Valued Anal., 9 (2001), 159. doi: 10.1023/A:1011287523150. Google Scholar

[39]

L. S. Pontryagin, Linear differential games. II,, Sov. Math., 8 (1967), 910. Google Scholar

[40]

R. T. Rockafellar, Convex Analysis,, vol. 28 of Princeton Mathematical Series, (1970). Google Scholar

[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. doi: 10.1080/02331939208843757. Google Scholar

[42]

P. Saint-Pierre, Newton and other continuation methods for multivalued inclusions,, Set-Valued Anal., 3 (1995), 143. doi: 10.1007/BF01038596. Google Scholar

[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. doi: 10.1021/ie021062w. Google Scholar

[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. doi: 10.1073/pnas.97.18.9840. Google Scholar

[45]

L. S. Tuckerman and D. Barkley, Bifurcation analysis for timesteppers,, in Numerical methods for bifurcation problems and large-scale dynamical systems (Minneapolis, (2000), 453. doi: 10.1007/978-1-4612-1208-9_20. Google Scholar

[46]

X. Wang, Convergence of Newton's method and inverse function theorem in Banach space,, Math. Comp., 68 (1999), 169. doi: 10.1090/S0025-5718-99-00999-0. Google Scholar

[47]

T. Yamamoto, A method for finding sharp error bounds for Newton's method under the Kantorovich assumptions,, Numer. Math., 49 (1986), 203. doi: 10.1007/BF01389624. Google Scholar

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