Rigorous Continuation in Infinite Dimensions

As most physical and biological models depend on parameters, it is important to understand how the solutions change as the parameters vary. While there is a variety of continuation methods to study finite dimensional problems, applying the same techniques to infinite dimensional dynamical systems such as PDEs and DDEs require first considering a finite dimensional projection, which naturally lead to the fundamental question of validity of the outputs. To address this issue, one aspect of my research consists of developing rigorous continuation methods in infinite dimensions.

[1] J.B. van den Berg, J.-P. Lessard and K. Mischaikow. Global smooth solution curves using rigorous branch following. Mathematics of Computation, 79 (271), 1565–1584, 2010. (pdf)


[2] J.-P. Lessard. Continuation of solutions and studying delay differential equations via rigorous numerics. To appear in Proceedings of Symposia in Applied Mathematics. (pdf)


[3] M. Breden, J.-P. Lessard and M. Vanicat. Global bifurcation diagram of steady states of systems of PDEs via rigorous numerics: a 3-component reaction-diffusion system. Acta Applicandae Mathematicae, 128(1): 113-152, 2013. (pdf)


[4] M. Gameiro, J.-P. Lessard and A. Pugliese. Computation of smooth manifolds via rigorous multi-parameter continuation in infinite dimensions, Foundations of Computational Mathematics, 16(2): 531-575, 2016. (pdf)

Branches of equilibria in cross-diffusion PDEs. In [3], we use rigorous numerics to compute several global smooth branches of steady states for a system of three reaction-diffusion PDEs introduced by Iida et al. [J. Math. Biol., 53, 617–641 (2006)], which was introduced to study the effect of cross-diffusion in competitive interactions. An explicit and mathematically rigorous construction of a global bifurcation diagram is done, except in small neighborhoods of the bifurcations. The proposed method introduces analytic estimates, a gluing-free approach for the construction of global smooth branches and provides a detailed analysis of the choice of the parameters to be made in order to maximize the chances of performing successfully the computational proofs.

Global smooth branches of solutions. In [1,2], we introduce methods to rigorously compute smooth branches of zeros of nonlinear operator equation posed on general Banach spaces, which can be finite or infinite dimensional. The methods are introduced first for parameter continuation and then generalized to pseudo-arclength continuation. Examples in the context of ordinary, partial and delay differential equations are given.

Rigorous multi-parameter continuation in infinite dimensions. In [4], we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by zeros of infinite dimensional nonlinear functions. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite dimensional domain of the function. The construction of the smooth charts is done using the radii polynomials to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a two-dimensional manifold of equilibria of the Cahn-Hilliard equation.

Contact


Department of Math. and Stat.

McGill University

Burnside Hall, Room 1119

805 Sherbrooke West

Montreal, QC, H3A 0B9, CANADA

jp.lessard@mcgill.ca

Phone: (514) 398-3804


Positions Available


@ Ph.D. level:

  1. • I recommend that you read some of my papers before contacting me.

  2. • I will not reply to generic emails.

@ Postdoc level:

  1. • Openings are available through the
    CRM-ISM Postdoctoral program.

Jean-Philippe Lessard

Associate Professor

McGill University

Department of Mathematics and Statistics

Research Projects