# Rigorous Numerics for Infinite Dimensional Nonlinear Dynamics (17w5141)

Arriving in Banff, Alberta Sunday, May 7 and departing Friday May 12, 2017

## Organizers

Jean-Philippe Lessard (Université Laval)

Jan Bouwe van den Berg (Vrije Universiteit Amsterdam)

Konstantin Mischaikow (Rutgers University, USA)

Michael Plum (Karlsruhe Institute of Technology)

JF Williams (Simon Fraser University)

## Objectives

The method of rigorously verified computing, developed in the past decade, has primarily been applied to a range of ordinary differential equations and maps (e.g. [1, 11, 16, 18, 26] to name just a few) and one-variable periodic patterns in PDEs and delay equations [28, 10]. Additionally, several multi-dimensional stationary PDE problems have been studied, e.g. in the area of elliptic PDEs [27, 23, 14] and in settings with periodic boundary conditions [7]. Furthermore, eigenvalue problems have been considered [2, 24], parameter continuation has been implemented [20], and parametrization methods for invariant manifolds enable the study of connecting orbits in ODEs [21]. Also, rigorously verified computations were combined with Morse-Conley theory to study global dynamics of gradient PDEs [5].

The list of examples given in the short overview above illustrates that we stand at the start of the emerging field of computer-assisted proofs for infinite dimensional nonlinear dynamics generated by PDEs, integral equations, delay equations, as well as infinite dimensional maps.

Rigorous verification goes well beyond an a posteriori analysis of numerical computations. In a nutshell, verification methods are mathematical theorems formulated in such a way that the assumptions can be rigorously verified on a computer. Indeed, it requires an a priori setup that allows analysis and numerics to go hand in hand: the choice of function spaces, the choice of the basis functions/elements and Galerkin projections, the analytic estimates, and the computational parameters must all work together to bound the errors due to approximation, rounding and truncation sufficiently tightly for the verification proof to go through.

On top of that, for high and infinite dimensional problems additional aspects arise. On the analysis side, we need to deal with much subtler and more involved truncation estimates and, for connecting orbits, with high or infinite dimensional invariant manifolds. On the algorithm side, we must find suitable pre-conditioners and develop efficient interval-arithmetic based algorithms. Finally we need to understand how to tie these computational results to the geometric and topological ideas of global nonlinear analysis that form the framework for our understanding of nonlinear dynamics.

This leads us to the following questions, which will be central to the workshop:

• How should we proceed to improve our algorithms to adapt them to suit a variety of infinite dimensional problems, including computational proofs of connecting orbits in PDEs, in delay equations, and between periodic orbits, etc.?

• How to deal with the computational cost for such problems? Which are the ingredients from scientific computing and numerical linear algebra that are needed for these large scale problems?

• Which types of infinite dimensional dynamical systems can be analyzed with rigorously verified computing in the near future? For which types is there not much hope at present, and why not? Which tools are missing?

To answer these questions we plan to bring together experts in scientific computing, numerical analysis with result verification, applied PDE analysis, and nonlinear dynamical systems. The aim of the workshop is to foster collaboration in order to develop a flexible analytic strategy tied with corresponding fast and robust algorithms.

As indicated above, rigorously verified computations require a variety of technical tools ranging from specific estimates to theorems in dynamics to specialized algorithms. To spur parallel progress in this direction we will identify a select set of challenging infinite dimensional problems around which the group can focus its efforts.

A few examples of the type of challenges that we strive for are:

1. Connecting orbits in the Navier-Stokes equation for fluids. Recently, numerical evidence has been found for trajectories in the Navier-Stokes equations (with suitably chosen boundary conditions) that are homoclinic to periodic orbits. If we can rigorously validate the existence of such orbits, this would imply, through forcing results, the first mathematically rigorous proof of chaotic flow in fluids (as described by the Navier-Stokes equations in 3D).

2. Connecting orbits in ill-posed PDEs. Ill-posed PDEs (with no suitable initial value problem) that come with a variational structure allow for the construction of a Floer homology. Connecting orbits are essential ingredients of this construction. If we can rigorously compute such connecting orbits, they yield specific “local” information, which when combined with generic global analytic result, will lead to powerful forcing results.

3. State dependent delay. Feedback loops often lead to delayed responses. Incorporating such phenomena in mathematical models results in delay equations, which essentially moves the dynamics from a finite to an infinite dimensional phase space. When the time span of the delay is not fixed a priori, but depends crucially on the state of the system, analysis of the equations becomes especially elusive. We aim at extending the rigorous numerical methods to these ill-understood systems, which appear in a variety of applications.

In order to stimulate the dynamics and interaction at the workshop, we plan to have a limited number of lectures combined with plenty of time to collaborate on identified open problems. We envision a few overview lectures on the state of the art, as well as some lectures that have the aim to introduce types of infinite dimensional problems that may serve as pivots for further development. We will discuss open problems and split up into groups to collaborate. We think that such an informal setting provides excellent opportunities for junior participants to start new research directions. Indeed, we aim for a good mix of junior and senior mathematicians. All junior participants will have the opportunity (and be encouraged) to present themselves and their research interest during a dedicated session.

REFERENCES

[1] B. Breuer, J. Horak, P. J. McKenna, and M. Plum. A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly sup- ported beam. J. Differential Equations, 224(1):60–97, 2006.

[2] B. M. Brown, D. K. R. McCormack, and A. Zettl. On a computer assisted proof of the existence of eigenvalues below the essential spectrum of the Sturm-Liouville problem. J. Comput. Appl. Math., 125(1-2):385–393, 2000. Numerical analysis 2000, Vol. VI, Ordinary differential equations and integral equations.

[3] R. Castelli, M. Gameiro and J.-P. Lessard. Rigorous numerics for ill-posed partial differential equations: periodic orbits in the Boussinesq equation. Preprint, 2015.

[4] J. Cyranka and P. Zgliczynski. Existence of globally attracting solutions for one-dimensional viscous Burgers equation with nonautonomous forcing - a computer assisted proof. SIAM J. Appl. Dyn. Syst., 14(2): 787–821, 2015.

[5] S. Day, Y. Hiraoka, K. Mischaikow, and T. Ogawa. Rigorous numerics for global dynamics: a study of the Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst., 4(1):1–31 (electronic), 2005.

[6] F. Pacella, M. Plum, D. Rutters. A computer-assisted existence proof for Emden’s equation on an unbounded L-shaped domain. Preprint, 2015.

[7] M. Gameiro and J.-P. Lessard. Rigorous computation of smooth branches of equilibria for the three dimensional Cahn-Hilliard equation. Numer. Math., 117(4):753–778, 2011.

[8] M. Gameiro and J.-P. Lessard. A posteriori verification of invariant objects of evolution equations: periodic orbits in the Kuramoto-Sivashinsky PDE. Preprint, 2015.

[9] O.E. Lanford, III. A computer-assisted proof of the Feigenbaum conjectures. Bull. Amer. Math. Soc. (N.S.), 6(3):427–434, 1982.

[10] J.-P. Lessard. Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright’s equation. J. Differential Equations, 248(5):992–1016, 2010.

[11] H.E. Lomeli and J.D. Meiss. Quadratic volume-preserving maps. Nonlinearity, 11(3):557–574, 1998.

[12] R. de la Llave, and J.D. Mireles James. Connecting orbits for compact infinite dimensional maps: computer assisted proofs of existence. In preparation, 2015.

[13] J.D. Mireles James, and C. Reinhardt. Rigorous numerics for (un)stable manifolds in infinite dimensions. In preparation, 2015.

[14] M. Plum. Computer-assisted proofs for semilinear elliptic boundary value problems. Japan J. Indust. Appl. Math., 26(2-3):419–442, 2009.

[15] S.M. Rump. INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages 77–104. Kluwer Academic Publishers, Dordrecht, 1999. http://www.ti3.tu-harburg.de/rump/.

[16] W. Tucker. A rigorous ODE solver and Smale’s 14th problem. Found. Comput. Math., 2(1):53–117, 2002.

[17] J.B. van den Berg, A. Deschênes, J.D. Mireles James, and J.-P. Lessard. Stationary coexistence of hexagons and rolls via rigorous computations. SIAM J. Appl. Dyn. Syst., 14(2):942–979, 2015.

[18] J.B. van den Berg and J.-P. Lessard. Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst., 7(3):988–1031, 2008.

[19] J.B. van den Berg and J.-P. Lessard. Rigorous numerics in dynamics. Notices Amer. Math. Soc., 62(9):1057–1061, October 2015.

[20] J.B. van den Berg, J.-P. Lessard, and K. Mischaikow. Global smooth solution curves using rigorous branch following. Math. Comp., 79(271):1565–1584, 2010.

[21] J.B. van den Berg, J.D. Mireles James, J.-P. Lessard, and K. Mischaikow. Rigorous numerics for symmetric connecting orbits: even homoclinics of the Gray-Scott equation. SIAM J. Math. Anal., 43(4):1557–1594, 2011.

[22] J.B. van den Berg and J.F. Williams. Rigorous numerics for the Ohta-Kawasaki problem. In preparation, 2015.

[23] Y. Watanabe and M.T. Nakao. Numerical verifications of solutions for nonlinear elliptic equations. Japan J. Indust. Appl. Math., 10(1):165–178, 1993.

[24] Y. Watanabe, M. Plum, and M.T. Nakao. A computer-assisted instability proof for the Orr-Sommerfeld problem with Poiseuille flow. ZAMM Z. Angew. Math. Mech., 89(1):5–18, 2009.

[25] Y. Watanabe, K. Nagatou, M. Plum, M.T. Nakao. Verified computations of eigenvalue exclosures for eigenvalue problems in Hilbert spaces. SIAM J. Numer. Anal., 52(2), 975–992, 2014.

[26] D. Wilczak and P. Zgliczynski. Heteroclinic connections between periodic orbits in planar restricted circular three-body problem—a computer assisted proof. Comm. Math. Phys., 234(1):37–75, 2003.

[27] N. Yamamoto and M.T. Nakao. Numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element. J. Comput. Appl. Math., 60(1-2):271–279, 1995. Linear/nonlinear iterative methods and verification of solution (Matsuyama, 1993).

[28] P. Zgliczynski and K. Mischaikow. Rigorous numerics for partial differential equations: the Kuramoto-Sivashinsky equation. Found. Comput. Math., 1(3):255–288, 2001.

The list of examples given in the short overview above illustrates that we stand at the start of the emerging field of computer-assisted proofs for infinite dimensional nonlinear dynamics generated by PDEs, integral equations, delay equations, as well as infinite dimensional maps.

Rigorous verification goes well beyond an a posteriori analysis of numerical computations. In a nutshell, verification methods are mathematical theorems formulated in such a way that the assumptions can be rigorously verified on a computer. Indeed, it requires an a priori setup that allows analysis and numerics to go hand in hand: the choice of function spaces, the choice of the basis functions/elements and Galerkin projections, the analytic estimates, and the computational parameters must all work together to bound the errors due to approximation, rounding and truncation sufficiently tightly for the verification proof to go through.

On top of that, for high and infinite dimensional problems additional aspects arise. On the analysis side, we need to deal with much subtler and more involved truncation estimates and, for connecting orbits, with high or infinite dimensional invariant manifolds. On the algorithm side, we must find suitable pre-conditioners and develop efficient interval-arithmetic based algorithms. Finally we need to understand how to tie these computational results to the geometric and topological ideas of global nonlinear analysis that form the framework for our understanding of nonlinear dynamics.

This leads us to the following questions, which will be central to the workshop:

• How should we proceed to improve our algorithms to adapt them to suit a variety of infinite dimensional problems, including computational proofs of connecting orbits in PDEs, in delay equations, and between periodic orbits, etc.?

• How to deal with the computational cost for such problems? Which are the ingredients from scientific computing and numerical linear algebra that are needed for these large scale problems?

• Which types of infinite dimensional dynamical systems can be analyzed with rigorously verified computing in the near future? For which types is there not much hope at present, and why not? Which tools are missing?

To answer these questions we plan to bring together experts in scientific computing, numerical analysis with result verification, applied PDE analysis, and nonlinear dynamical systems. The aim of the workshop is to foster collaboration in order to develop a flexible analytic strategy tied with corresponding fast and robust algorithms.

As indicated above, rigorously verified computations require a variety of technical tools ranging from specific estimates to theorems in dynamics to specialized algorithms. To spur parallel progress in this direction we will identify a select set of challenging infinite dimensional problems around which the group can focus its efforts.

A few examples of the type of challenges that we strive for are:

1. Connecting orbits in the Navier-Stokes equation for fluids. Recently, numerical evidence has been found for trajectories in the Navier-Stokes equations (with suitably chosen boundary conditions) that are homoclinic to periodic orbits. If we can rigorously validate the existence of such orbits, this would imply, through forcing results, the first mathematically rigorous proof of chaotic flow in fluids (as described by the Navier-Stokes equations in 3D).

2. Connecting orbits in ill-posed PDEs. Ill-posed PDEs (with no suitable initial value problem) that come with a variational structure allow for the construction of a Floer homology. Connecting orbits are essential ingredients of this construction. If we can rigorously compute such connecting orbits, they yield specific “local” information, which when combined with generic global analytic result, will lead to powerful forcing results.

3. State dependent delay. Feedback loops often lead to delayed responses. Incorporating such phenomena in mathematical models results in delay equations, which essentially moves the dynamics from a finite to an infinite dimensional phase space. When the time span of the delay is not fixed a priori, but depends crucially on the state of the system, analysis of the equations becomes especially elusive. We aim at extending the rigorous numerical methods to these ill-understood systems, which appear in a variety of applications.

In order to stimulate the dynamics and interaction at the workshop, we plan to have a limited number of lectures combined with plenty of time to collaborate on identified open problems. We envision a few overview lectures on the state of the art, as well as some lectures that have the aim to introduce types of infinite dimensional problems that may serve as pivots for further development. We will discuss open problems and split up into groups to collaborate. We think that such an informal setting provides excellent opportunities for junior participants to start new research directions. Indeed, we aim for a good mix of junior and senior mathematicians. All junior participants will have the opportunity (and be encouraged) to present themselves and their research interest during a dedicated session.

REFERENCES

[1] B. Breuer, J. Horak, P. J. McKenna, and M. Plum. A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly sup- ported beam. J. Differential Equations, 224(1):60–97, 2006.

[2] B. M. Brown, D. K. R. McCormack, and A. Zettl. On a computer assisted proof of the existence of eigenvalues below the essential spectrum of the Sturm-Liouville problem. J. Comput. Appl. Math., 125(1-2):385–393, 2000. Numerical analysis 2000, Vol. VI, Ordinary differential equations and integral equations.

[3] R. Castelli, M. Gameiro and J.-P. Lessard. Rigorous numerics for ill-posed partial differential equations: periodic orbits in the Boussinesq equation. Preprint, 2015.

[4] J. Cyranka and P. Zgliczynski. Existence of globally attracting solutions for one-dimensional viscous Burgers equation with nonautonomous forcing - a computer assisted proof. SIAM J. Appl. Dyn. Syst., 14(2): 787–821, 2015.

[5] S. Day, Y. Hiraoka, K. Mischaikow, and T. Ogawa. Rigorous numerics for global dynamics: a study of the Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst., 4(1):1–31 (electronic), 2005.

[6] F. Pacella, M. Plum, D. Rutters. A computer-assisted existence proof for Emden’s equation on an unbounded L-shaped domain. Preprint, 2015.

[7] M. Gameiro and J.-P. Lessard. Rigorous computation of smooth branches of equilibria for the three dimensional Cahn-Hilliard equation. Numer. Math., 117(4):753–778, 2011.

[8] M. Gameiro and J.-P. Lessard. A posteriori verification of invariant objects of evolution equations: periodic orbits in the Kuramoto-Sivashinsky PDE. Preprint, 2015.

[9] O.E. Lanford, III. A computer-assisted proof of the Feigenbaum conjectures. Bull. Amer. Math. Soc. (N.S.), 6(3):427–434, 1982.

[10] J.-P. Lessard. Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright’s equation. J. Differential Equations, 248(5):992–1016, 2010.

[11] H.E. Lomeli and J.D. Meiss. Quadratic volume-preserving maps. Nonlinearity, 11(3):557–574, 1998.

[12] R. de la Llave, and J.D. Mireles James. Connecting orbits for compact infinite dimensional maps: computer assisted proofs of existence. In preparation, 2015.

[13] J.D. Mireles James, and C. Reinhardt. Rigorous numerics for (un)stable manifolds in infinite dimensions. In preparation, 2015.

[14] M. Plum. Computer-assisted proofs for semilinear elliptic boundary value problems. Japan J. Indust. Appl. Math., 26(2-3):419–442, 2009.

[15] S.M. Rump. INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages 77–104. Kluwer Academic Publishers, Dordrecht, 1999. http://www.ti3.tu-harburg.de/rump/.

[16] W. Tucker. A rigorous ODE solver and Smale’s 14th problem. Found. Comput. Math., 2(1):53–117, 2002.

[17] J.B. van den Berg, A. Deschênes, J.D. Mireles James, and J.-P. Lessard. Stationary coexistence of hexagons and rolls via rigorous computations. SIAM J. Appl. Dyn. Syst., 14(2):942–979, 2015.

[18] J.B. van den Berg and J.-P. Lessard. Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst., 7(3):988–1031, 2008.

[19] J.B. van den Berg and J.-P. Lessard. Rigorous numerics in dynamics. Notices Amer. Math. Soc., 62(9):1057–1061, October 2015.

[20] J.B. van den Berg, J.-P. Lessard, and K. Mischaikow. Global smooth solution curves using rigorous branch following. Math. Comp., 79(271):1565–1584, 2010.

[21] J.B. van den Berg, J.D. Mireles James, J.-P. Lessard, and K. Mischaikow. Rigorous numerics for symmetric connecting orbits: even homoclinics of the Gray-Scott equation. SIAM J. Math. Anal., 43(4):1557–1594, 2011.

[22] J.B. van den Berg and J.F. Williams. Rigorous numerics for the Ohta-Kawasaki problem. In preparation, 2015.

[23] Y. Watanabe and M.T. Nakao. Numerical verifications of solutions for nonlinear elliptic equations. Japan J. Indust. Appl. Math., 10(1):165–178, 1993.

[24] Y. Watanabe, M. Plum, and M.T. Nakao. A computer-assisted instability proof for the Orr-Sommerfeld problem with Poiseuille flow. ZAMM Z. Angew. Math. Mech., 89(1):5–18, 2009.

[25] Y. Watanabe, K. Nagatou, M. Plum, M.T. Nakao. Verified computations of eigenvalue exclosures for eigenvalue problems in Hilbert spaces. SIAM J. Numer. Anal., 52(2), 975–992, 2014.

[26] D. Wilczak and P. Zgliczynski. Heteroclinic connections between periodic orbits in planar restricted circular three-body problem—a computer assisted proof. Comm. Math. Phys., 234(1):37–75, 2003.

[27] N. Yamamoto and M.T. Nakao. Numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element. J. Comput. Appl. Math., 60(1-2):271–279, 1995. Linear/nonlinear iterative methods and verification of solution (Matsuyama, 1993).

[28] P. Zgliczynski and K. Mischaikow. Rigorous numerics for partial differential equations: the Kuramoto-Sivashinsky equation. Found. Comput. Math., 1(3):255–288, 2001.