Computer Assisted Proof in Analysis
Part of my research deals with nonlinear analysis of operator equations coming from applied mathematics and dynamical systems theory. The computer is a powerful tool for studying these equations, and I am especially interested in methods for obtaining mathematically rigorous results in collaboration with the computer. Many of these methods have an `a-posteriori' flavor, i.e. first we computer a good numerical approximation and then we try to get explicit bounds between the approximate and the true solution. In practice this requires a delicate balance between deliberate numerics and pen and paper mathematical analysis.
[1] Persistence of Periodic Orbits under State-Dependent Delayed Perturbations: Computer-Assisted Proofs With Jiaqi Yang, Joan Gimeno, and J.P. Lessard. (Submitted).
[2] Validated Numerical Approximation of Stable Manifolds for Parabolic Partial Differential Equations With Jonathan Jaquette, and J.B. van den Berg. \\ Journal of Dynamics and Differential Equations (Accepted Feb 2022) \\ https://doi.org/10.1007/s10884-022-10146-1 (appeared online March 2022).
[3] Parameterization of unstable manifolds for DDEs: formal series solutions and validated error bounds with Olivier Henot and J.P. Lessard \\ Journal of Dynamics and Differential Equations (Accepted April 2021) \\ https://doi.org/10.1007/s10884-021-10002-8 (appeared online May 2021).
[4] Computer assisted proof of drift orbits along normally hyperbolic manifolds With Maciej Capinski, Jorge Gonzalez, and Jean-Pierre Marco. Communications in Nonlinear Science and Numerical Simulations. Volume 106 (2022), Paper No. 105970, 21pp. \\ https://doi.org/10.1016/j.cnsns.2021.105970
[5] A rigorous implicit C1 Chebyshev integrator for delay equations With J.P. Lessard. Journal of Dynamics and Differential Equations (Accepted July 2020) \\ https://doi.org/10.1007/s10884-020-09880-1 (appeared online August 2020)
[6] A functional analytic approach to validated numerics for eigenvalues of delay equations. With J.P. Lessard. Journal of Computational Dynamics. Vol 7, Num 1, pp. 123-158. June 2020.
[7] Computer assisted proofs of contracting invariant tori for ODEs. With Maciej Capinski and Emmanuel Fleurantin. Discrete and Continuous Dynamical Systems. December 2020, 40(12):6681 - 6707.
[8] Torus Knot Choreographies in the N-body problem. With Renato Calleja, Carlos Garcia-Azpeitia, and J.P. Lessard. Nonlinearity. Vol 34, Num 1, January 2021.
[9] Spatial periodic orbits in the equilateral circular restricted four body problem: computer assisted proofs of existence. With Jaime Burgos-Garcia and J.P. Lessard. Celestial Mechanics and Dynamical Astronomy. 131:2 (2019).
[10] Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem. With Shane Kepley. Journal of Differential Equations. Volume 266, Issue 4, February 2019. Pages 1709-1755.
[11] Validated numerics for continuation and bifurcation of connecting orbits for maps. With Ronald Adams. Qualitative Theory of Dynamical Systems 18(2), July 2017.
[12] Analytic continuation of local (un)stable manifolds with rigorous computer assisted error bounds. With Shane Kepley and Bill Kalies. SIAM Journal on Applied Dynamical Systems, Vol 17, No. 1, pp. 157--202 (2018)
[13] Validated numerics for equilibria of analytic vector fields: invariant manifolds and connecting orbits. Chapter 2 of the AMS Proceedings of Symposia in Applied Mathematics, Volume 74.
[14] Parameterization of invariant manifolds for periodic orbits (II): a-posteriori analysis and computer assisted error bounds. With Roberto Castelli and J.P. Lessard. The Journal of Dynamics and Differential Equations. 30, 1525-1581 (2018).
[15] Fourier-Taylor Parameterization of Unstable Manifolds for Parabolic Partial Differential Equations: Formalism, Implementation, and Rigorous Validation with Christian Reinhardt. Indagationes Mathematicae. Volume 30, Issue 1, January 2019. Pages 39-80.
[16] Computer assisted Fourier analysis in sequence spaces of varying regularity with J.P. Lessard. SIAM Journal on Mathematical Analysis, Vol 49, Issue 1, pp. 530-561. (2017).
[17] Connecting orbits for compact infinite dimensional maps: computer assisted proofs of existence with Rafael de la Llave. SIAM Journal on Applied Dynamical Systems, Vol. 15, No. 2, pp. 1268-1323 (2016).
[18] Fourier-Taylor Approximation of Unstable Manifolds for Compact Maps: Numerical Implementation and Computer Assisted Error Bounds. Foundations of Computational Mathematics, Vol 17, Issue 6, pp. 1467-1523 (2017).
[19] Analytic Enclosure of the Fundamental Matrix Solution. With Roberto Castelli and J.P. Lessard. Applications of Mathematics, vol 60 (2015), issue 6, pp. 617--636.
[20] Stationary coexistence of hexagons and rolls via rigorous computations. With J.B. van den Berg, A. Deschenes, and J.P. Lessard SIAM Journal on Applied Dynamical Systems, Vol 14, No. 2, pp. 942-979. (2015)
Click here for the associated codes.
[21] Computer Assisted Proof of Transverse Saddle-to-Saddle Connecting Orbits for First Order Vector Fields. With J.-P. Lessard and C. Reinhardt. Journal of Dynamics and Differential Equations. Volume 26, Issue 2, pp 267--313. (2014)
[22] Rigorous Numerics for Analytic Solutions of Differential Equations: the Radii Polynomial Approach. With Allan Hungria and J.P. Lessard. Mathematics of Computation, Volume 85, Number 299, May 2016, Pages 1427-1459.
[23] Automatic differentiation for Fourier series and the radii polynomial approach. With Julian Ransford and J.P. Lessard. Physica D: Nonlinear Phenomena, Vol 334, Number 1, pp. 174-186, November (2016)
[24] Rigorous A-Posteriori Computation of (Un)Stable Manifolds and Connecting Orbits for Analytic Maps. With K. Mischaikow. SIAM Journal on Applied Dynamical Systems, Volume 12, Number 2, pp. 957-1006. (2013)
[25] Computer Assisted Error Bounds for Linear Approximation of (Un)Stable Manifolds and Rigorous Validation of Higher Dimensional Transverse Connecting Orbits. Communications in Nonlinear Science and Numerical Simulation, Vol 22, pp. 1102-1133. (2015)
[26] Rigorous Numerics for Symmetric Connecting Orbits: Even Homoclinics of the Gray-Scott Equation. With J. B. van den Berg, J.-P. Lessard, and K. Mischaikow. SIAM Journal on Mathematical Analysis, Volume 43, Issue 4, pp. 1557-1594. (2011)
[27] Computational Proofs in Dynamics. With K. Mischaikow. Appears in the Springer Encyclopedia of Applied and Computational Mathematics. Bjorn Engquist Editor.