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Balanced model order reduction for linear random dynamical systems driven by Lévy noise
Computer-assisted proofs for radially symmetric solutions of PDEs
1. | MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, Hungary, H-6720 |
2. | VU Amsterdam, Department of Mathematics, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands |
3. | Université de Montréal, Département de Mathématiques et de Statistique, Pavillon André-Aisenstadt, 2920 chemin de la Tour, Montreal, QC, H3T 1J4, Canada |
4. | Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, Hungary, H-6720 |
5. | McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada |
6. | Simon Fraser University, Department of Mathematics, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada |
We obtain radially symmetric solutions of some nonlinear (geometric) partial differential equations via a rigorous computer-assisted method. We introduce all main ideas through examples, accessible to non-experts. The proofs are obtained by solving for the coefficients of the Taylor series of the solutions in a Banach space of geometrically decaying sequences. The tool that allows us to advance from numerical simulations to mathematical proofs is the Banach contraction theorem.
References:
[1] |
G. Arioli and H. Koch,
Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Ration. Mech. Anal., 197 (2010), 1033-1051.
doi: 10.1007/s00205-010-0309-7. |
[2] |
G. Arioli and H. Koch,
Integration of dissipative partial differential equations: A case study, SIAM J. Appl. Dyn. Syst., 9 (2010), 1119-1133.
doi: 10.1137/10078298X. |
[3] |
G. Arioli and H. Koch, Validated numerical solutions for some semilinear elliptic equations on the disk, 2017. Preprint. Google Scholar |
[4] |
I. Balázs, J.B. van den Berg, J. Courtois, J. Dudás, J.-P. Lessard, A. Vörös-Kiss, J. F. Williams and X. Y. Yin, MATLAB code for "Computer-assisted proofs for radially symmetric solutions of PDEs", 2017, http://www.math.vu.nl/~janbouwe/code/radialpdes/. Google Scholar |
[5] |
A. N. Baltagiannis and K. E. Papadakis, Periodic solutions in the Sun—Jupiter—Trojan Asteroid—Spacecraft system, Planetary and Space Science, 75 (2013), 148-157. Google Scholar |
[6] |
J. F. Barros and E. S. G. Leandro,
The set of degenerate central configurations in the planar restricted four-body problem, SIAM J. Math. Anal., 43 (2011), 634-661.
doi: 10.1137/100789701. |
[7] |
J. F. Barros and E. S. G. Leandro,
Bifurcations and enumeration of classes of relative equilibria in the planar restricted four-body problem, SIAM J. Math. Anal., 46 (2014), 1185-1203.
doi: 10.1137/130911342. |
[8] |
B. Breuer, J. Horák, P. J. McKenna and M. Plum,
A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam, J. Diff. Eq., 224 (2006), 60-97.
doi: 10.1016/j.jde.2005.07.016. |
[9] |
J. Burgos-García and M. Gidea,
Hill's approximation in a restricted four-body problem, Celestial Mech. Dynam. Astronom., 122 (2015), 117-141.
doi: 10.1007/s10569-015-9612-9. |
[10] |
CAPD: Computer assisted proofs in dynamics, a package for rigorous numerics, http://capd.ii.uj.edu.pl/. Google Scholar |
[11] |
A. Castro and E. M. Fischer,
Infinitely many rotationally symmetric solutions to a class of semilinear Laplace-Beltrami equations on spheres, Canad. Math. Bull., 58 (2015), 723-729.
doi: 10.4153/CMB-2015-056-7. |
[12] |
L. Cesari,
Functional analysis and periodic solutions of nonlinear differential equations, Contributions to Differential Equations, 1 (1963), 149-187.
|
[13] |
L. Cesari,
Functional analysis and Galerkin's method, Michigan Math. J., 11 (1964), 385-414.
doi: 10.1307/mmj/1028999194. |
[14] |
J.-L. Figueras, M. Gameiro, J.-P. Lessard and R. de la Llave,
A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations, SIAM J. Appl. Dyn. Syst., 16 (2017), 1070-1088.
doi: 10.1137/16M1073777. |
[15] |
A. Hungria, J.-P. Lessard and J. D. Mireles James,
Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach, Math. Comp., 85 (2016), 1427-1459.
doi: 10.1090/mcom/3046. |
[16] |
H. Koch, A. Schenkel and P. Wittwer,
Computer-assisted proofs in analysis and programming in logic: A case study, SIAM Rev., 38 (1996), 565-604.
doi: 10.1137/S0036144595284180. |
[17] |
O. E. Lanford,
Ⅲ. A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434.
doi: 10.1090/S0273-0979-1982-15008-X. |
[18] |
E. S. G. Leandro,
On the central configurations of the planar restricted four-body problem, J. Diff. Eq., 226 (2006), 323-351.
doi: 10.1016/j.jde.2005.10.015. |
[19] |
P.-L. Lions,
On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.
doi: 10.1137/1024101. |
[20] |
S. McCalla and B. Sandstede,
Snaking of radial solutions of the multi-dimensional Swift-Hohenberg equation: A numerical study, Phys. D, 239 (2010), 1581-1592.
doi: 10.1016/j.physd.2010.04.004. |
[21] |
R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009.
doi: 10.1137/1.9780898717716. |
[22] |
M. T. Nakao,
Numerical verification methods for solutions of ordinary and partial differential equations, Numer. Funct. Anal. Optim., 22 (2001), 321-356.
doi: 10.1081/NFA-100105107. |
[23] |
S. M. Rump, INTLAB - INTerval LABoratory, In Tibor Csendes, editor, Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, 77-104. http://www.ti3.tu-harburg.de/rump/. Google Scholar |
[24] |
S. M. Rump,
Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, 19 (2010), 287-449.
doi: 10.1017/S096249291000005X. |
[25] |
A. Scheel, Radially symmetric patterns of reaction-diffusion systems, Mem. Amer. Math. Soc., 165 (2003), ⅷ+86 pp.
doi: 10.1090/memo/0786. |
[26] |
C. Simó,
Relative equilibrium solutions in the four-body problem, Celestial Mech., 18 (1978), 165-184.
doi: 10.1007/BF01228714. |
[27] |
J. B. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 1977. Google Scholar |
[28] |
W. Tucker,
A rigorous ODE solver and Smale's 14th problem, Foundations of Computational Mathematics, 2 (2002), 53-117.
doi: 10.1007/s002080010018. |
[29] |
W. Tucker, Validated Numerics, Princeton University Press, Princeton, NJ, 2011. A short introduction to rigorous computations. |
[30] |
J. B. van den Berg, A. Deschênes, J.-P. Lessard and J. D. Mireles James,
Stationary coexistence of hexagons and rolls via rigorous computations, SIAM J. Appl. Dyn. Syst., 14 (2015), 942-979.
doi: 10.1137/140984506. |
[31] |
J. B. van den Berg and J.-P. Lessard,
Rigorous numerics in dynamics, Notices Amer. Math. Soc., 62 (2015), 1057-1061.
doi: 10.1090/noti1276. |
[32] |
J. B. van den Berg, J.-P. Lessard and K. Mischaikow,
Global smooth solution curves using rigorous branch following, Math. Comp., 79 (2010), 1565-1584.
doi: 10.1090/S0025-5718-10-02325-2. |
[33] |
M. J. Ward,
Spots, traps, and patches: Asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems, Nonlinearity, 31 (2018), R189-R239.
doi: 10.1088/1361-6544/aabe4b. |
[34] |
N. Yamamoto,
A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM J. Numer. Anal., 35 (1998), 2004-2013.
doi: 10.1137/S0036142996304498. |
[35] |
P. Zgliczyński,
Rigorous numerics for dissipative partial differential equations. Ⅱ. Periodic orbit for the Kuramoto-Sivashinsky PDE—a computer-assisted proof, Found. Comput. Math., 4 (2004), 157-185.
doi: 10.1007/s10208-002-0080-8. |
[36] |
P. Zgliczyński and K. Mischaikow,
Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Found. Comput. Math., 1 (2001), 255-288.
doi: 10.1007/s002080010010. |
show all references
References:
[1] |
G. Arioli and H. Koch,
Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Ration. Mech. Anal., 197 (2010), 1033-1051.
doi: 10.1007/s00205-010-0309-7. |
[2] |
G. Arioli and H. Koch,
Integration of dissipative partial differential equations: A case study, SIAM J. Appl. Dyn. Syst., 9 (2010), 1119-1133.
doi: 10.1137/10078298X. |
[3] |
G. Arioli and H. Koch, Validated numerical solutions for some semilinear elliptic equations on the disk, 2017. Preprint. Google Scholar |
[4] |
I. Balázs, J.B. van den Berg, J. Courtois, J. Dudás, J.-P. Lessard, A. Vörös-Kiss, J. F. Williams and X. Y. Yin, MATLAB code for "Computer-assisted proofs for radially symmetric solutions of PDEs", 2017, http://www.math.vu.nl/~janbouwe/code/radialpdes/. Google Scholar |
[5] |
A. N. Baltagiannis and K. E. Papadakis, Periodic solutions in the Sun—Jupiter—Trojan Asteroid—Spacecraft system, Planetary and Space Science, 75 (2013), 148-157. Google Scholar |
[6] |
J. F. Barros and E. S. G. Leandro,
The set of degenerate central configurations in the planar restricted four-body problem, SIAM J. Math. Anal., 43 (2011), 634-661.
doi: 10.1137/100789701. |
[7] |
J. F. Barros and E. S. G. Leandro,
Bifurcations and enumeration of classes of relative equilibria in the planar restricted four-body problem, SIAM J. Math. Anal., 46 (2014), 1185-1203.
doi: 10.1137/130911342. |
[8] |
B. Breuer, J. Horák, P. J. McKenna and M. Plum,
A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam, J. Diff. Eq., 224 (2006), 60-97.
doi: 10.1016/j.jde.2005.07.016. |
[9] |
J. Burgos-García and M. Gidea,
Hill's approximation in a restricted four-body problem, Celestial Mech. Dynam. Astronom., 122 (2015), 117-141.
doi: 10.1007/s10569-015-9612-9. |
[10] |
CAPD: Computer assisted proofs in dynamics, a package for rigorous numerics, http://capd.ii.uj.edu.pl/. Google Scholar |
[11] |
A. Castro and E. M. Fischer,
Infinitely many rotationally symmetric solutions to a class of semilinear Laplace-Beltrami equations on spheres, Canad. Math. Bull., 58 (2015), 723-729.
doi: 10.4153/CMB-2015-056-7. |
[12] |
L. Cesari,
Functional analysis and periodic solutions of nonlinear differential equations, Contributions to Differential Equations, 1 (1963), 149-187.
|
[13] |
L. Cesari,
Functional analysis and Galerkin's method, Michigan Math. J., 11 (1964), 385-414.
doi: 10.1307/mmj/1028999194. |
[14] |
J.-L. Figueras, M. Gameiro, J.-P. Lessard and R. de la Llave,
A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations, SIAM J. Appl. Dyn. Syst., 16 (2017), 1070-1088.
doi: 10.1137/16M1073777. |
[15] |
A. Hungria, J.-P. Lessard and J. D. Mireles James,
Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach, Math. Comp., 85 (2016), 1427-1459.
doi: 10.1090/mcom/3046. |
[16] |
H. Koch, A. Schenkel and P. Wittwer,
Computer-assisted proofs in analysis and programming in logic: A case study, SIAM Rev., 38 (1996), 565-604.
doi: 10.1137/S0036144595284180. |
[17] |
O. E. Lanford,
Ⅲ. A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434.
doi: 10.1090/S0273-0979-1982-15008-X. |
[18] |
E. S. G. Leandro,
On the central configurations of the planar restricted four-body problem, J. Diff. Eq., 226 (2006), 323-351.
doi: 10.1016/j.jde.2005.10.015. |
[19] |
P.-L. Lions,
On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.
doi: 10.1137/1024101. |
[20] |
S. McCalla and B. Sandstede,
Snaking of radial solutions of the multi-dimensional Swift-Hohenberg equation: A numerical study, Phys. D, 239 (2010), 1581-1592.
doi: 10.1016/j.physd.2010.04.004. |
[21] |
R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009.
doi: 10.1137/1.9780898717716. |
[22] |
M. T. Nakao,
Numerical verification methods for solutions of ordinary and partial differential equations, Numer. Funct. Anal. Optim., 22 (2001), 321-356.
doi: 10.1081/NFA-100105107. |
[23] |
S. M. Rump, INTLAB - INTerval LABoratory, In Tibor Csendes, editor, Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, 77-104. http://www.ti3.tu-harburg.de/rump/. Google Scholar |
[24] |
S. M. Rump,
Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, 19 (2010), 287-449.
doi: 10.1017/S096249291000005X. |
[25] |
A. Scheel, Radially symmetric patterns of reaction-diffusion systems, Mem. Amer. Math. Soc., 165 (2003), ⅷ+86 pp.
doi: 10.1090/memo/0786. |
[26] |
C. Simó,
Relative equilibrium solutions in the four-body problem, Celestial Mech., 18 (1978), 165-184.
doi: 10.1007/BF01228714. |
[27] |
J. B. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 1977. Google Scholar |
[28] |
W. Tucker,
A rigorous ODE solver and Smale's 14th problem, Foundations of Computational Mathematics, 2 (2002), 53-117.
doi: 10.1007/s002080010018. |
[29] |
W. Tucker, Validated Numerics, Princeton University Press, Princeton, NJ, 2011. A short introduction to rigorous computations. |
[30] |
J. B. van den Berg, A. Deschênes, J.-P. Lessard and J. D. Mireles James,
Stationary coexistence of hexagons and rolls via rigorous computations, SIAM J. Appl. Dyn. Syst., 14 (2015), 942-979.
doi: 10.1137/140984506. |
[31] |
J. B. van den Berg and J.-P. Lessard,
Rigorous numerics in dynamics, Notices Amer. Math. Soc., 62 (2015), 1057-1061.
doi: 10.1090/noti1276. |
[32] |
J. B. van den Berg, J.-P. Lessard and K. Mischaikow,
Global smooth solution curves using rigorous branch following, Math. Comp., 79 (2010), 1565-1584.
doi: 10.1090/S0025-5718-10-02325-2. |
[33] |
M. J. Ward,
Spots, traps, and patches: Asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems, Nonlinearity, 31 (2018), R189-R239.
doi: 10.1088/1361-6544/aabe4b. |
[34] |
N. Yamamoto,
A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM J. Numer. Anal., 35 (1998), 2004-2013.
doi: 10.1137/S0036142996304498. |
[35] |
P. Zgliczyński,
Rigorous numerics for dissipative partial differential equations. Ⅱ. Periodic orbit for the Kuramoto-Sivashinsky PDE—a computer-assisted proof, Found. Comput. Math., 4 (2004), 157-185.
doi: 10.1007/s10208-002-0080-8. |
[36] |
P. Zgliczyński and K. Mischaikow,
Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Found. Comput. Math., 1 (2001), 255-288.
doi: 10.1007/s002080010010. |






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