CRM CAMP in Nonlinear Analysis

Computer-Assisted Mathematical Proofs in Nonlinear Analysis

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

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.

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.

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

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.

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.
 

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.

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. 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.

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.

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

Ninth seminar of the series

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

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

Tenth seminar of the series

Seminar presented by Maxime Breden (École Polytechnique, France)

Past seminars

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.

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.

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)

Recording NOW available HERE on the CRM's YouTube channel

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.

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List of registered participants

Currently, there are 166 registered participants.

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.

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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.

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.

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.

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.

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.