CRM CAMP in Nonlinear Analysis

Computer-Assisted Mathematical Proofs in Nonlinear Analysis
CRM CAMP in Nonlinear Analysis Seminar Series Participants Videos Open Problems Series Other Scientific Activities About Scientific Coordinators Resources

CRM CAMP in Nonlinear Analysis

The main goal of the CRM CAMP project is to bring together the worldwide community of researchers in the area of computer-assisted methods of proof, especially those working in the areas of dynamical systems theory and nonlinear analysis. This community has enjoyed dramatic growth over the last three decades, and has developed methods to resolve a number of important unsolved problems in mathematics. Yet participating researchers are scattered around the globe, and there is a growing need for a regular forum for discussion and dissemination of results. This is especially important in current time of unprecedented travel interruption.

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Scientific Activities

December 1, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

FUTURE DIRECTIONS SERIES: Defect and front dynamics: analysis and computation

Seminar presented by Arnd Scheel (University of Minnesota, USA)

This will be a personal selection of results and related open problems in dissipative spatially extended systems. I will focus on simple, sometimes universal models such as the complex Ginzburg-Landau, the Swift-Hohenberg, and extended KPP equations and attempts to describe their dynamics based on coherent structures. I will present "conceptual' analytical results, and describe gaps that rigorous computations may be able to close. Topics include invasion fronts, defects in one- and two-dimensional oscillatory media, and point defects in striped phases. 

November 24, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Symmetry breaking and Hopf bifurcations for the planar Navier-Stokes equation

Seminar presented by Gianni Arioli (Politecnico di Milano, Italy)

We consider the Navier-Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. The uniqueness of stationary solutions is studied in dependence of the kinematic viscosity. For some particular forcing, it is shown that uniqueness persists on some continuous branch of stationary solutions, when the viscosity becomes arbitrarily small. On the other hand, for a different forcing, a branch of symmetric solutions is shown to bifurcate, giving rise to a secondary branch of nonsymmetric stationary solutions. Furthermore, as the kinematic viscosity is varied, the branch of symmetric stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.

Slides for the activity

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Scientific coordinators

Jean-Philippe Lessard (McGill University, Canada)

Jason D. Mireles James (Florida Atlantic University, USA)

Jan Bouwe van den Berg (VU Amsterdam, Netherlands)

Seminar Series

Weekly Seminar Series: every Tuesdays of the summer at 10am (Montreal/Miami time).

To have access to the Zoom meeting link, please register to the Seminar Series. Registering once gives you access to every seminar.

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Upcoming seminars
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December 1, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

FUTURE DIRECTIONS SERIES: Defect and front dynamics: analysis and computation

Seminar presented by Arnd Scheel (University of Minnesota, USA)

This will be a personal selection of results and related open problems in dissipative spatially extended systems. I will focus on simple, sometimes universal models such as the complex Ginzburg-Landau, the Swift-Hohenberg, and extended KPP equations and attempts to describe their dynamics based on coherent structures. I will present "conceptual' analytical results, and describe gaps that rigorous computations may be able to close. Topics include invasion fronts, defects in one- and two-dimensional oscillatory media, and point defects in striped phases. 

December 8, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

CRM-CAMP COLLOQUIUM: A complete proof of the Feigenbaum conjectures

Seminar presented by Jean-Pierre Eckmann (University of Geneva, Switzerland)

In the late 1970s, Mitchell Feigenbaum discovered the universality of bifurcations in one-parameter families of maps. This universality was explained with a fixed point equation, and a flow in the space of all one-parameter families of maps. With Peter Wittwer, I showed in 1987 a rigorous proof of this phenomenon, using a "computer assisted" proof of the Feigenbaum conjectures. I will explain what are the somehow unconventional issues of this computer assisted proof, and how they are solved. Please keep in mind that this is quite old stuff, and a more modern implementation of the ideas would be much easier now than it was over 30 years ago.

January 19, 2021 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Details of the seminar of N. Brisebarre to come

Seminar presented by Nicolas Brisebarre (ENS Lyon, France)

Start time Title Speaker
2020-12-01 10:00 FUTURE DIRECTIONS SERIES: Defect and front dynamics: analysis and computation Arnd Scheel (University of Minnesota, USA)
2020-12-08 10:00 CRM-CAMP COLLOQUIUM: A complete proof of the Feigenbaum conjectures Jean-Pierre Eckmann (University of Geneva, Switzerland)
2021-01-19 10:00 Details of the seminar of N. Brisebarre to come Nicolas Brisebarre (ENS Lyon, France)

Past seminars
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November 24, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Symmetry breaking and Hopf bifurcations for the planar Navier-Stokes equation

Seminar presented by Gianni Arioli (Politecnico di Milano, Italy)

We consider the Navier-Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. The uniqueness of stationary solutions is studied in dependence of the kinematic viscosity. For some particular forcing, it is shown that uniqueness persists on some continuous branch of stationary solutions, when the viscosity becomes arbitrarily small. On the other hand, for a different forcing, a branch of symmetric solutions is shown to bifurcate, giving rise to a secondary branch of nonsymmetric stationary solutions. Furthermore, as the kinematic viscosity is varied, the branch of symmetric stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.

Slides for the activity

November 17, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Stable periodic orbits for the Mackey-Glass equation

Seminar presented by Ferenc Bartha (University of Szeged, Hungary)

We consider the classical Mackey-Glass delay differential equation. By letting n go to infinity in the part encapsulating the delayed term, we obtain a limiting hybrid delay equation. We investigate how periodic solutions of this limiting equation are related to periodic solutions of the original MG equation for large n. Then, we establish a procedure for constructing such periodic solutions via forward time integration. Finally, we use rigorous numerics to establish the existence of stable periodic orbits for various parameters of the MG equation. We note that some of these solutions exhibit seemingly complicated dynamics, yet they are stable periodic orbits. 

Stable periodic orbits for the Mackey-Glass equation

November 10, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Recent progress in proving stability of traveling waves in the 1D Navier-Stokes equations using rigorous computations

Seminar presented by Blake Barker (Brigham Young University, USA)

We discuss recent progress developing and applying rigorous computation to prove stability of traveling waves in the 1D Navier-Stokes equation. In particular, we talk about rigorous computation of the Evans function, an analytic function whose zeros correspond to eigenvalues of the linearized PDE problem. Nonlinear stability results by Zumbrun and collaborators show that the underlying traveling waves are stable if there are no eigenvalues in the right half of the complex plane. Thus one may use rigorous computation of the Evans function to prove nonlinear-orbital stability of traveling waves.

Recent progress in proving stability of traveling waves in the 1D Navier-Stokes equations using rigorous computations

November 3, 2020 from 16:00 to 17:00 (Montreal/Miami time) Zoom meeting

Stability and approximation of statistical limit laws

Seminar presented by Gary Froyland (UNSW Sydney, Australia)

The unpredictability of chaotic nonlinear dynamics leads naturally to statistical descriptions, including probabilistic limit laws such as the central limit theorem and large deviation principle. A key tool in the Nagaev-Guivarc'h spectral method for establishing statistical limit theorems is a "twisted" transfer operator. We prove stability of the variance in the central limit theorem and the rate function from a large deviation principle with respect to deterministic and stochastic perturbations of the dynamics and perturbations induced by numerical schemes. We then apply these results to piecewise expanding maps in one and multiple dimensions. This theory can be extended to uniformly hyperbolic maps and in this setting we develop two new Fourier-analytic methods to provide the first rigorous estimates of the variance and rate function for Anosov maps.  This is joint work with Harry Crimmins.

Stability and approximation of statistical limit laws

October 27, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Equilibrium validation in models for pattern formation based on Sobolev embeddings

Seminar presented by Evelyn Sander (George Mason University, USA)

In this talk, I describe a method of computer-assisted proof focused on continuation of solutions depending on a parameter. These techniques are applied to the Ohta-Kawasaki model for the dynamics of diblock copolymers in dimensions one, two, and three. The functional analytic approach and techniques can be generalized to other parabolic partial differential equations. This is joint work with Thomas Wanner (George Mason University).

Equilibrium validation in models for pattern formation based on Sobolev embeddings

October 20, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Formally verified computer-assisted mathematical proofs

Seminar presented by Assia Mahboubi (Inria, France & VU Amsterdam, Netherlands)

Proof assistants are pieces of software designed for defining formally mathematical objects, statement and proofs. In particular, such a formalization reduces the verification of proofs to a purely mechanical well-formedness check. Since the early 70s, proof assistants have been extensively used for applications in program verification, notably for security-related issues. They have also been used to verify landmark results in mathematics, including theorems with a computational proof, like the Four Colour Theorem (Appel and Haken, 1977) or Hales and Ferguson's proof of the Kepler conjecture (2005). This talk will discuss what are formalized mathematics and formal proofs, and sketch the architecture of modern proof assistants. It will also showcase a few applications in formally verified rigorous computation.

The slides of the talk are available here

October 13, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Rigorous computation of periodic solutions and Floquet multipliers in delay differential equations with time-forced discontinuities

Seminar presented by Kevin Church (McGill University, Canada)

I will present some recent work on rigorous computation of periodic solutions for delay differential equations with impulse effects. At fixed moments in time, the state of such a system is reset and solutions become discontinuous. Once a periodic solution of such a system has been computed, its Floquet spectrum can be rigorously computed by discretization of the monodromy operator (period map) and some technical error estimates. As an application, we compute a branch of periodic solutions in the pulse-harvested Hutchinson equation and examine its stability.

Rigorous computation of periodic solutions and Floquet multipliers in DDEs with time-forced discontinuities

October 6, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Computer-assisted proofs for finding the monodromy of hypergeometric differential equations

Seminar presented by Akitoshi Takayasu (University of Tsukuba, Japan)

In this talk, we introduce a numerical method for rigorously finding the monodromy matrix of hypergeometric differential equations. From a base point defined by fundamental solutions, we analytically continue the solution on a contour around a singular point of the differential equation using a rigorous integrator. Depending on the contour we obtain the monodromy representation of fundamental solutions, which represents the fundamental group of the equation. As an application of this method, we consider a Picard-Fuchs type hypergeometric differential equation arising from a polarized K3 surface. The monodromy matrix shows a deformation of homologically independent 2-cycles for the surface along the contour, which is regarded as a change of characterization for the K3 surface. This is joint work with Naoya Inoue (University of Tsukuba) and Toshimasa Ishige (Chiba University).

Computer-assisted proofs for finding the monodromy of hypergeometric differential equations

September 29, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

A computer-assisted proof of Kazhdan’s property (T) for automorphism groups of free groups

Seminar presented by Piotr Nowak (Polish Academy of Sciences, Poland)

Property (T) was introduced in 1967 by Kazhdan and is an important rigidity property of groups. The most elementary way to define it is through a fixed point property: a group G has property (T) if every action of G by affine isometries on a Hilbert space has a fixed point. Property (T) has numerous applications in the form of rigidity of actions and operator algebras associated to the group, constructions of expander graphs or constructions of counterexamples to Baum-Connes-type conjectures. 

In this talk I will give a brief introduction to property (T) and explain the necessary group-theoretic background in order to present a computer-assisted approach to proving property (T) by showing that the Laplacian on the group has a spectral gap. This approach allowed us show that Aut(F_n), the group of automorphisms of the free group F_n on n generators, has property (T) when n is at least 5: the case n=5 is joint work with Marek Kaluba and Narutaka Ozawa, and the case of n at least 6 is joint work with Kaluba and Dawid Kielak. Important aspects of our methods include passing from a computational result to a rigorous proof and later obtaining the result for an infinite family of groups using a single computation. I will present an overview of these arguments.

A computer-assisted proof of Kazhdan’s property (T) for automorphism groups of free groups

September 22, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Computing and validating collisions, ejections, and homoclinics for the three body problem

Seminar presented by Shane Kepley (Rutgers University, USA)

Understanding connecting and collision/ejection orbits is central to the study of transport in Celestial Mechanics. The atlas algorithm combines the parameterization method with rigorous numerical techniques for solving initial value problems in order to find and validate connecting orbits. However, difficulties arise when parameterizing orbits passing near a singularity such as “near miss” homoclinics or ejection/collision orbits. In this talk we present a method of overcoming this obstacle based on rigorous Levi-Civita regularization which desingularizes the vector field near the primaries. This regularization is performed dynamically allowing invariant manifolds to be parameterized globally, even near singularities.

Computing and validating collisions, ejections, and homoclinics for the three body problem

September 15, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Validating Hopf bifurcations in the Kuramoto-Sivashinsky PDE

Seminar presented by Elena Queirolo (Rutgers University, USA)

We prove the existence of a Hopf bifurcation in the Kuramoto–Sivashinsky PDE. For this, we rewrite the Kuramoto–Sivashinsky equation into a desingularized formulation near the Hopf point via a blow-up approach and we apply the radii polynomial approach to validate a solution branch of periodic solutions. Then this solution branch includes the Hopf bifurcation point.

Validating Hopf bifurcations in the Kuramoto-Sivashinsky PDE

September 8, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

A proof of Noise Induced Order in the BZ map, and some remarks on the phenomenon

Seminar presented by Isaia Nisoli (Universidade Federal do Rio de Janeiro, Brazil)

In this talk I will present a Computer Aided Proof of Noise Induced Order (NIO) in a model associated with the Belousov-Zhabotinsky reaction: when studying the random dynamical system with additive noise associated to the BZ map, as the noise amplitude increases the Lyapunov exponent of the model transitions from positive to negative. The proof is obtained through rigorous approximation of the stationary measure using Ulam method.

I will also show a sufficient condition for the existence of NIO in a wide family of one-dimensional examples.

[1] S. Galatolo, M. Monge, I. Nisoli "Existence of Noise Induced Order: a computer aided proof", Nonlinearity 33(9)
[2] I. Nisoli "Sufficient Conditions for Noise Induced Order in 1-dimensional systems", arXiv:2003.08422

A proof of Noise Induced Order in the BZ map, and some remarks on the phenomenon

September 1, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Computer assisted proofs of Arnold Diffusion

Seminar presented by Maciej Capiński (AGH University of Science and Technology, Poland)

We will present three methods that can be used for computer assisted proofs of Arnold diffusion in Hamiltonian systems. The first is the classical Melnikov method; the second is based a shadowing lemma in the setting of the scattering map theory; the last is based on topological shadowing using correctly aligned windows and cones. We will also discuss an application in the setting of the Planar Elliptic Restricted Three Body Problem.

Computer assisted proofs of Arnold Diffusion

August 25, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems

Seminar presented by Maxime Breden (École Polytechnique, France), Maximilian Engel (Freie Universität Berlin, Germany)

In this talk, we discuss a long-standing conjecture concerning shear-induced chaos in stochastically perturbed systems exhibiting a Hopf bifurcation. Using the recently developed theory of conditioned Lyapunov exponents on bounded domains, we reformulate the problem into the rigorous computation of eigenvectors of some elliptic PDEs, namely the Kolmogorov/Fokker-Planck equations describing distributions of the underlying stochastic process, and are thus able to prove that  the first Lyapunov exponent is positive for certain parameter regimes.

Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems

August 18, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Rigorously validated estimation of statistical properties of expanding maps

Seminar presented by Caroline Wormell (University of Sydney, Australia)

Full-branch uniformly expanding maps and their long-time statistical quantities serve as common models for chaotic dynamics, as well as having applications to number theory. I will present an efficient method to compute important statistical quantities such as physical invariant measures, which can obtain rigorously validated bounds. To accomplish this, a Chebyshev Galerkin discretisation of transfer operators of these maps is constructed; the spectral data at the eigenvalue 1 is then approximated from this discretisation. Using this method we obtain validated estimates of Lyapunov exponents and diffusion coefficients that are accurate to over 100 decimal places. These methods may also fruitfully be extended to non-uniformly expanding maps of Pomeau-Manneville type, which have largely been altogether resistant to numerical study.

Rigorously validated estimation of statistical properties of expanding maps

August 11, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Torus knot choreographies in the N-body problem

Seminar presented by Renato Calleja (Universidad Nacional Autonoma de Mexico, Mexico)

N-body choreographies are periodic solutions to the N-body equations in which equal masses chase each other around a fixed closed curve. In this talk I will present a systematic approach for proving the existence of spatial choreographies in the gravitational body problem with the help of the digital computer. These arise from the polygonal system of bodies in a rotating frame of reference. In rotating coordinates, after exploiting the symmetries, the equation of a choreographic configuration is reduced to a delay differential equation (DDE) describing the position and velocity of a single body. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, correspond to choreographies.

Torus knot choreographies in the N-body problem

August 4, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Rigorous numerical investigation of chaos and stability of periodic orbits in the Kuramoto-Sivashinsky PDE

Seminar presented by Daniel Wilczak (Jagiellonian University, Poland)

We give a computer-assisted proof of the existence of symbolic dynamics for a certain Poincaré map in the one-dimensional Kuramoto-Sivashinsky PDE. In particular, we show the existence of infinitely many (countably) periodic orbits (POs) of arbitrary large principal periods. We provide also a study of the stability type of some POs and show the existence of a countable infinity of geometrically different homoclinic orbits to a periodic solution. The proof utilizes pure topological results (variant of the method of covering relations on compact absolute neighbourhood retracts) with rigorous integration of the PDE and the associated variational equation. This talk is based on the recent results [1,2].

[1] D. Wilczak and P. Zgliczyński. A geometric method for infinite-dimensional chaos: symbolic dynamics for the Kuramoto-Sivashinsky PDE on the line, Journal of Differential Equations, Vol. 269 No. 10 (2020), 8509-8548.
[2] D. Wilczak and P. Zgliczyński. A rigorous C1-algorithm for integration of dissipative PDEs based on automatic differentiation and the Taylor method, in preparation.

Rigorous numerical investigation of chaos and stability of periodic orbits in the Kuramoto-Sivashinsky PDE

July 28, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

A modification of Schiffer's conjecture, and a proof via finite elements

Seminar presented by Nilima Nigam (Simon Fraser University, Canada)

Approximations via conforming and non-conforming finite elements can be used to construct validated and computable bounds on eigenvalues for the Dirichlet Laplacian in certain domains. If these are to be used as part of a proof, care must be taken to ensure each step of the computation is validated and verifiable. In this talk we present a long-standing conjecture in spectral geometry, and its resolution using validated finite element computations.  Schiffer’s conjecture states that if a bounded domain Ω in R^n has any nontrivial Neumann eigenfunction which is a constant on the boundary, then Ω must be a ball. This conjecture is open. A modification of Schiffer’s conjecture is: for regular polygons of at least 5 sides, we can demonstrate the existence of a Neumann eigenfunction which does not change sign on the boundary. In this talk, we provide a recent proof using finite element calculations for the regular pentagon. The strategy involves iteratively bounding eigenvalues for a sequence of polygonal subdomains of the triangle with mixed Dirichlet and Neumann boundary conditions. We use a learning algorithm to find and optimize this sequence of subdomains, and use non-conforming linear FEM to compute validated lower bounds for the lowest eigenvalue in each of these domains. The linear algebra is performed within interval arithmetic. This talk is based on the following paper, which is a joint work with Bartlomiej Siudeja and Ben Green at University of Oregon.

A modification of Schiffer's conjecture, and a proof via finite elements

July 21, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Solution verification for the stationary Navier-Stokes equation over bounded non-convex 3D domains

Seminar presented by Xuefeng Liu (Niigata University, Japan)

We consider the solution verification for the stationary Navier-Stokes equation over a bounded non-convex 3D domain Ω. In 1999, M.T. Nakao, et al., reported a solution existence verification example for the 2D square domain.  However, it has been a difficult problem to deal with general 2D domains and 3D domains, due to the bottleneck problem in the  a priori error estimation for the linearized NS equation. Recently, by extending the hypercircle method (Prage-Synge's theorem) to deal with the divergence-free condition in the Stokes equation, the explicit error estimation is constructed successfully based on a conforming finite element approach [arXiv:2006.02952]. Further,  we succeeded in the solution existence verification for the stationary NS equation in several nonconvex 3D domains.  In this talk, I will show the latest progress on this topic, including the rigorous estimation of the eigenvalue of Stokes operator in 3D domains.

Solution verification for the stationary Navier-Stokes equation over bounded non-convex 3D domains

July 14, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Uniqueness of Whitham's highest cusped wave

Seminar presented by Javier Gómez-Serrano (Brown University, USA & University of Barcelona, Spain)

Whitham’s equation of shallow water waves is a non-homogeneous non-local dispersive equation. As in the case of the Stokes wave for the Euler equation, non-smooth traveling waves with greatest height between crest and trough have been shown to exist. In this talk I will discuss uniqueness of solutions to the Whitham equation and show that there exists a unique, even and periodic traveling wave of greatest height, that moreover has a convex profile between consecutive stagnation points, at which there is a cusp. Joint work with Alberto Enciso and Bruno Vergara.

Uniqueness of Whitham's highest cusped wave

July 7, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Computer-assisted existence and multiplicity proofs for semilinear elliptic problems on bounded and unbounded domains

Seminar presented by Michael Plum (Karlsruhe Institute of Technology, Germany)

Many boundary value problems for semilinear elliptic partial differential equations allow very stable numerical computations of approximate solutions, but are still lacking analytical existence proofs. In this lecture, we propose a method which exploits the knowledge of a "good" numerical approximate solution, in order to provide a rigorous proof of existence of an exact solution close to the approximate one. This goal is achieved by a fixed-point argument which takes all numerical errors into account, and thus gives a mathematical proof which is not "worse" than any purely analytical one. A crucial part of the proof consists of the computation of eigenvalue bounds for the linearization of the given problem at the approximate solution. The method is used to prove existence and multiplicity statements for some specific examples, including cases where purely analytical methods had not been successful.

Computer-assisted existence and multiplicity proofs for semilinear elliptic problems on bounded and unbounded domains

June 30, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

An overabundance of breathers in a nonlinear Schrödinger equation without gauge invariance

Seminar presented by Jonathan Jaquette (Boston University, USA)

In this talk we study the nonlinear Schrödinger equation  posed on the 1-torus. Based on their numerics, Cho, Okamoto, & Shōji conjectured in their 2016 paper that: (C1) any singularity in the complex plane of time arising from inhomogeneous initial data is a branch singularity, and (C2) real initial data will exist globally in real time. If true, Conjecture 1 would suggest a strong incompatibility with the Painlevé property, a property closely associated with integrable systems. While Masuda proved (C1) in 1983 for close-to-constant initial data, a generalization to other initial data is not known. Using computer assisted proofs we establish a branch singularity in the complex plane of time for specific, large initial data which is not close-to-constant.

Concerning (C2), we demonstrate an open set of initial data which is homoclinic to the 0-homogeneous-equilibrium, proving (C2) for close-to-constant initial data. This proof is then extended to a broader class of nonlinear Schrödinger equation without gauge invariance, and then used to prove the non-existence of any real-analytic conserved quantities. Indeed, while numerical evidence suggests that most initial data is homoclinic to the 0-equilibrium, there is more than meets the eye. Using computer assisted proofs, we establish an infinite family of unstable nonhomogeneous equilibria, as well as heteroclinic orbits traveling between these nonhomogeneous equilibria and the 0-equilibrium.

An overabundance of breathers in a nonlinear Schrödinger equation without gauge invariance

June 23, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

Stable periodic patterns in 3D for the Ohta-Kawasaki problem

Seminar presented by Jan Bouwe van den Berg (VU Amsterdam, Netherlands)

In this talk we discuss a mathematically rigorous computational method to compare local minimizers of the Ohta-Kawasaki free energy, describing diblock copolymer melts. This energy incorporates a nonlocal term to take into account the bond between the monomers.

Working within an arbitrary space group symmetry, we explore the phase space, computing candidates both with and without experimentally observed symmetries. We validate the phase diagram, identifying regions of parameter space where different spatially periodic structures have the lowest energy. These patterns may be lamellar layers, hexagonally packed cylinders, body-centered or close-packed spheres, as well as double gyroids and 'O70' arrangements. Each computation is validated by a mathematical theorem, where we bound the truncation errors and apply a fixed point argument to establish a computer-assisted proof. The method can be applied more generally to symmetric space-time periodic solution of many partial differential equations.

Stable periodic patterns in 3D for the Ohta-Kawasaki problem

Start time Title Speaker
2020-11-24 10:00 Symmetry breaking and Hopf bifurcations for the planar Navier-Stokes equation Gianni Arioli (Politecnico di Milano, Italy)
2020-11-17 10:00 Stable periodic orbits for the Mackey-Glass equation Ferenc Bartha (University of Szeged, Hungary)
2020-11-10 10:00 Recent progress in proving stability of traveling waves in the 1D Navier-Stokes equations using rigorous computations Blake Barker (Brigham Young University, USA)
2020-11-03 16:00 Stability and approximation of statistical limit laws Gary Froyland (UNSW Sydney, Australia)
2020-10-27 10:00 Equilibrium validation in models for pattern formation based on Sobolev embeddings Evelyn Sander (George Mason University, USA)
2020-10-20 10:00 Formally verified computer-assisted mathematical proofs Assia Mahboubi (Inria, France & VU Amsterdam, Netherlands)
2020-10-13 10:00 Rigorous computation of periodic solutions and Floquet multipliers in delay differential equations with time-forced discontinuities Kevin Church (McGill University, Canada)
2020-10-06 10:00 Computer-assisted proofs for finding the monodromy of hypergeometric differential equations Akitoshi Takayasu (University of Tsukuba, Japan)
2020-09-29 10:00 A computer-assisted proof of Kazhdan’s property (T) for automorphism groups of free groups Piotr Nowak (Polish Academy of Sciences, Poland)
2020-09-22 10:00 Computing and validating collisions, ejections, and homoclinics for the three body problem Shane Kepley (Rutgers University, USA)
2020-09-15 10:00 Validating Hopf bifurcations in the Kuramoto-Sivashinsky PDE Elena Queirolo (Rutgers University, USA)
2020-09-08 10:00 A proof of Noise Induced Order in the BZ map, and some remarks on the phenomenon Isaia Nisoli (Universidade Federal do Rio de Janeiro, Brazil)
2020-09-01 10:00 Computer assisted proofs of Arnold Diffusion Maciej Capiński (AGH University of Science and Technology, Poland)
2020-08-25 10:00 Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems Maxime Breden (École Polytechnique, France), Maximilian Engel (Freie Universität Berlin, Germany)
2020-08-18 10:00 Rigorously validated estimation of statistical properties of expanding maps Caroline Wormell (University of Sydney, Australia)
2020-08-11 10:00 Torus knot choreographies in the N-body problem Renato Calleja (Universidad Nacional Autonoma de Mexico, Mexico)
2020-08-04 10:00 Rigorous numerical investigation of chaos and stability of periodic orbits in the Kuramoto-Sivashinsky PDE Daniel Wilczak (Jagiellonian University, Poland)
2020-07-28 10:00 A modification of Schiffer's conjecture, and a proof via finite elements Nilima Nigam (Simon Fraser University, Canada)
2020-07-21 10:00 Solution verification for the stationary Navier-Stokes equation over bounded non-convex 3D domains Xuefeng Liu (Niigata University, Japan)
2020-07-14 10:00 Uniqueness of Whitham's highest cusped wave Javier Gómez-Serrano (Brown University, USA & University of Barcelona, Spain)
2020-07-07 10:00 Computer-assisted existence and multiplicity proofs for semilinear elliptic problems on bounded and unbounded domains Michael Plum (Karlsruhe Institute of Technology, Germany)
2020-06-30 10:00 An overabundance of breathers in a nonlinear Schrödinger equation without gauge invariance Jonathan Jaquette (Boston University, USA)
2020-06-23 10:00 Stable periodic patterns in 3D for the Ohta-Kawasaki problem Jan Bouwe van den Berg (VU Amsterdam, Netherlands)
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Videos

November 17, 2020

Stable periodic orbits for the Mackey-Glass equation

Ferenc Bartha

November 10, 2020

Recent progress in proving stability of traveling waves in the 1D Navier-Stokes equations using rigorous computations

Blake Barker

November 3, 2020

Stability and approximation of statistical limit laws

Gary Froyland

October 27, 2020

Equilibrium validation in models for pattern formation based on Sobolev embeddings

Evelyn Sander

October 13, 2020

Rigorous computation of periodic solutions and Floquet multipliers in DDEs with time-forced discontinuities

Kevin Church

October 6, 2020

Computer-assisted proofs for finding the monodromy of hypergeometric differential equations

Akitoshi Takayasu

September 29, 2020

A computer-assisted proof of Kazhdan’s property (T) for automorphism groups of free groups

Piotr Nowak

September 22, 2020

Computing and validating collisions, ejections, and homoclinics for the three body problem

Shane Kepley

September 15, 2020

Validating Hopf bifurcations in the Kuramoto-Sivashinsky PDE

Elena Queirolo

September 8, 2020

A proof of Noise Induced Order in the BZ map, and some remarks on the phenomenon

Isaia Nisoli

September 1, 2020

Computer assisted proofs of Arnold Diffusion

Maciej Capiński

August 25, 2020

Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems

Maxime Breden, Maximilian Engel

August 18, 2020

Rigorously validated estimation of statistical properties of expanding maps

Caroline Wormell

August 11, 2020

Torus knot choreographies in the N-body problem

Renato Calleja

August 4, 2020

Rigorous numerical investigation of chaos and stability of periodic orbits in the Kuramoto-Sivashinsky PDE

Daniel Wilczak

July 28, 2020

A modification of Schiffer's conjecture, and a proof via finite elements

Nilima Nigam

July 21, 2020

Solution verification for the stationary Navier-Stokes equation over bounded non-convex 3D domains

Xuefeng Liu

July 14, 2020

Uniqueness of Whitham's highest cusped wave

Javier Gómez-Serrano

July 7, 2020

Computer-assisted existence and multiplicity proofs for semilinear elliptic problems on bounded and unbounded domains

Michael Plum

June 30, 2020

An overabundance of breathers in a nonlinear Schrödinger equation without gauge invariance

Jonathan Jaquette

June 23, 2020

Stable periodic patterns in 3D for the Ohta-Kawasaki problem

Jan Bouwe van den Berg

Open Problems Series

This is a series of talks focusing on either open problems concerning techniques of computer-assisted proof, or more broadly open problem in mathematics where the speaker believes there could be a computer-assisted solution. Talks range from 5 minutes to an hour, and can be proposed at any level. When an open problem is solved, or when substantial progress is made, we provide citation and links to the relevant work.

Stay up-to-date with the group's activities and receive invites to the seminars.

Register

Other Scientific Activities

Here, the different scientific activities related to the research group will be grouped together. First, the upcoming activities, then the past activities.

Upcoming Scientific Activities

December 3, 2020 from 10:00 to 10:00 (Montreal/Miami time) Zoom meeting

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computation of tight enclosures for Laplacian eigenvalues

Seminar presented by Joel Dahne (Uppsala University, Sweden)

Recently, there has been interest in high-precision approximations of the fundamental eigenvalue of the Laplace-Beltrami operator on spherical triangles for combinatorial purposes. We present computations of improved and rigorous enclosures to these eigenvalues. This is achieved by applying the method of particular solutions in high precision, the enclosure being obtained by a combination of interval arithmetic and Taylor models. The index of the eigenvalue can be certified by exploiting the monotonicity of the eigenvalue with respect to the domain. The classically troublesome case of singular corners we handle by combining expansions from all singular corners and an expansion from an interior point.

December 3, 2020 from 10:00 to 10:00 (Montreal/Miami time) Zoom meeting

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted existence proofs for Navier-Stokes equations on an unbounded strip with obstacle

Seminar presented by Jonathan Wunderlich (Karlsruhe Institute of Technology, Germany)

The incompressible stationary 2D Navier-Stokes equations are considered on an unbounded strip domain with a compact obstacle. In order to get existence and error bounds (in a Sobolev space) for a solution, an approximate solution (using divergence-free finite elements), a bound for its defect, and a norm bound for the inverse of the linearization at the approximate solution are computed. For the latter, bounds for the essential spectrum and for eigenvalues play a crucial role, especially for the eigenvalues “close to” zero. Note that, on an unbounded domain, the only possible method for computing the desired norm bound appears to be via eigenvalue bounds. In this way, the first rigorous existence proof for the Navier-Stokes problem under consideration is obtained.

December 3, 2020 from 10:00 to 11:00 (Montreal/Miami time) Zoom meeting

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted proofs of two-dimensional attracting invariant tori for ODEs

Seminar presented by Emmanuel Fleurantin (Florida Atlantic University, USA)

We study the existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We consider separately two important cases of rotational and resonant tori for which we describe how we apply our methods. This is a joint work with Maciej Capinski and J.D. Mireles-James.

December 3, 2020 from 11:00 to 11:00 (Montreal/Miami time) Zoom meeting

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted proofs for a nonlinear Laplace-Beltrami equation on the sphere

Seminar presented by Gabriel William Duchesne (McGill University, Canada)

We prove the existence and local uniqueness of radially symmetric solutions of nonlinear Laplace-Beltrami equation on the sphere by using the Radii Polynomial Theorem on Banach spaces with a combination of Taylor and Chebyshev coefficients of the solutions.

December 3, 2020 from 11:00 to 11:00 (Montreal/Miami time) Zoom meeting

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Parameterized invariant manifold and applications in celestial mechanics

Seminar presented by Maxime Murray (Florida Atlantic University, USA)

The parameterization method is a well-known framework with proven value to parameterize hyperbolic manifolds attached to periodic solutions of ordinary differential equations. Using a Taylor expansion, one can rewrite the computation of the manifold into a recursive system of linear differential equations describing the coefficients. I will discuss this approach and how to obtain an interval enclosure of the truncated solution to the system. I will then show how validated manifolds are used to compute cycle-to-cycle connections in the case of the circular restricted three-body problem and Hill's four-body problem.

December 3, 2020 from 11:00 to 12:00 (Montreal/Miami time) Zoom meeting

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Periodic orbit for Brusselator system with diffusion

Seminar presented by Jakub Banaśkiewicz (Jagiellonian University, Poland)

We will present numerical evidence for the existence of periodic solutions to a one-dimensional Brusselator system with diffusion and Dirichlet boundary conditions. Then we discuss a plan for proving their existence by a rigorous integration of differential inclusion corresponding to the first modes of the Galerkin projection and dissipative estimations on further modes.

December 3, 2020 from 12:00 to 12:00 (Montreal/Miami time) Zoom meeting

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Rigorous computation of the unstable manifold for equilibria of delay differential equations

Seminar presented by Olivier Hénot (McGill University, Canada)

We will review the parameterization method to obtain the unstable manifold of equilibria of Delay Differential Equations. Then, we will discuss how to compute rigorous error bounds for this parameterization.

About

The CRM CAMP seminar series explores the interplay between scientific computing and rigorous mathematical analysis, spotlighting research in areas like dynamical systems theory and nonlinear analysis. The field has developed rapidly over the last several decades and, with researchers spread around the globe, there is a growing need for a regular forum to share results, pose interesting questions, and discuss new directions. This project is envisioned as a kind of online community center for weekly gatherings, as well as a repository for educational materials. In addition to the weekly lecture series the program also serves as a mechanism for organizing workshops, tutorials, and other scientific activities. We hope that, by increasing the visibility of this research, the project will stimulate collaborations across existing groups and between our community and mathematicians working in other areas.

Contact

Scientific coordinators

Jean-Philippe Lessard (McGill University, Canada)

Jason D. Mireles James (Florida Atlantic University, USA)

Jan Bouwe van den Berg (VU Amsterdam, Netherlands)

Related links

CAPA – Computer-aided proofs in analysis

The CAPD group

Institute for Reliable Computing

Jean-Philippe Lessard

Associate Professor

McGill University, Canada

Biography

Jean-Philippe Lessard is an associate professor at McGill University since 2017. He obtained his Ph.D. from Georgia Tech in 2007 under the supervision of Konstantin Mischaikow. He spent some time as a postdoctoral researcher at Rutgers University, at VU University Amsterdam, was awarded a fellowship from the IAS in Princeton and was a group leader at the Basque Center for Applied Mathematics. He then became a professor at Laval University, where he stayed for six years. In 2016, he was awarded the CAIMS/PIMS Early Career Award in Applied Mathematics and is currently CRM’s deputy director of scientific programs. In his research, he combines numerical analysis, topology and functional analysis to study finite and infinite dimensional dynamical systems.

Jason D. Mireles James

Associate Professor

Florida Atlantic University, USA

Biography

Jason D. Mireles James received his Ph.D. from the University of Texas at Austin in 2009, where he worked with Rafael de la Llave. He moved to Rutgers University where he was first a postdoc from 2010 to 2011, and then a Hill Assistant Professor in the Mathematics Department from 2011-2014. During this time, he worked closely with the group of Konstantin Mischaikow. In 2014 he joined the Department of Mathematics at Florida Atlantic University, where he currently holds the rank of associate professor. His research focuses on problems in nonlinear analysis, drawing on tools from computational mathematics, approximation theory, and functional analysis.

Jan Bouwe van den Berg

Full Professor

VU Amsterdam, Netherlands

Biography

Jan Bouwe van den Berg is a full professor at Vrije Universiteit Amsterdam since 2007. He obtained his Ph.D. from Leiden University in 2000 under the supervision of Bert Peletier. He spent two years as a postdoc in Nottingham and has held visiting positions at Simons Fraser University and at McGill. He was awarded an NWO-Vici grant in 2012 and he was a CRM-Simons visiting professor in 2019. Jan Bouwe’s research revolves around dynamical systems and nonlinear partial differential equations, where he use techniques ranging from topological and variational analysis to (rigorous) computational methods to study the dynamics of patterns.

Videos

June 23, 2020

Stable periodic patterns in 3D for the Ohta-Kawasaki problem

Jan Bouwe van den Berg

Books

Survey articles

SeMA, 76, pages 459–484, 2019

Computer-assisted proofs in PDE: a survey

Javier Gómez-Serrano

Notices of the American Mathematical Society, Volume 62 (9), pages 1057-1061, 2015

Rigorous Numerics in Dynamics

Jean-Philippe Lessard, Jan Bouwe van den Berg

Acta Numerica, Volume 19, pages 287-449, 2010

Verification methods: rigorous results using floating-point arithmetic

Siegfried M. Rump

SIAM Review, Volume 38 (4), pages 565-604, 1996

Computer-assisted proofs in analysis and programming in logic: a case study

Hans Koch, Alain Schenkel, Peter Wittwer

Schools

August 1, 2018 https://mym.iimas.unam.mx/renato/curso.html

Computer-Assisted Proofs in Nonlinear Dynamics

Jason D. Mireles James, Jean-Philippe Lessard

The main question addressed in this course is: suppose we have computed a good numerical approximation of a solution of nonlinear equation -- can we establish the existence of a true solution nearby? Combining tools from functional analysis, complex analysis, numerical analysis, and interval computing, we see that for many of the problems mentioned above the answer is yes. A broad and example driven overview of the field of validated numerics is given.

Youtube Videos

November 17, 2020

Stable periodic orbits for the Mackey-Glass equation

Ferenc Bartha

November 10, 2020

Recent progress in proving stability of traveling waves in the 1D Navier-Stokes equations using rigorous computations

Blake Barker

November 3, 2020

Stability and approximation of statistical limit laws

Gary Froyland

October 27, 2020

Equilibrium validation in models for pattern formation based on Sobolev embeddings

Evelyn Sander

October 13, 2020

Rigorous computation of periodic solutions and Floquet multipliers in DDEs with time-forced discontinuities

Kevin Church

October 6, 2020

Computer-assisted proofs for finding the monodromy of hypergeometric differential equations

Akitoshi Takayasu

September 29, 2020

A computer-assisted proof of Kazhdan’s property (T) for automorphism groups of free groups

Piotr Nowak

September 22, 2020

Computing and validating collisions, ejections, and homoclinics for the three body problem

Shane Kepley

September 15, 2020

Validating Hopf bifurcations in the Kuramoto-Sivashinsky PDE

Elena Queirolo

September 8, 2020

A proof of Noise Induced Order in the BZ map, and some remarks on the phenomenon

Isaia Nisoli

September 1, 2020

Computer assisted proofs of Arnold Diffusion

Maciej Capiński

August 25, 2020

Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems

Maxime Breden, Maximilian Engel

August 18, 2020

Rigorously validated estimation of statistical properties of expanding maps

Caroline Wormell

August 11, 2020

Torus knot choreographies in the N-body problem

Renato Calleja

August 4, 2020

Rigorous numerical investigation of chaos and stability of periodic orbits in the Kuramoto-Sivashinsky PDE

Daniel Wilczak

July 28, 2020

A modification of Schiffer's conjecture, and a proof via finite elements

Nilima Nigam

July 21, 2020

Solution verification for the stationary Navier-Stokes equation over bounded non-convex 3D domains

Xuefeng Liu

July 14, 2020

Uniqueness of Whitham's highest cusped wave

Javier Gómez-Serrano

July 7, 2020

Computer-assisted existence and multiplicity proofs for semilinear elliptic problems on bounded and unbounded domains

Michael Plum

June 30, 2020

An overabundance of breathers in a nonlinear Schrödinger equation without gauge invariance

Jonathan Jaquette

June 23, 2020

Stable periodic patterns in 3D for the Ohta-Kawasaki problem

Jan Bouwe van den Berg

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