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2. Nowhere coexpanding functions

Author

Cook, Andrew, Hammerlindl, Andy, and Tucker, Warwick

Subjects
Statistics - Machine Learning, Computer Science - Machine Learning, Mathematics - Dynamical Systems, 37E99 (Primary) 37N99 (Secondary)
Abstract

We define a family of $C^1$ functions which we call "nowhere coexpanding functions" that is closed under composition and includes all $C^3$ functions with non-positive Schwarzian derivative. We establish results on the number and nature of the fixed points of these functions, including a generalisation of a classic result of Singer., Comment: 9 pages, 3 figures

Published
2023

3. The number of relative equilibria in the PCR4PB

Author

Figueras, Jordi-Lluís, Tucker, Warwick, and Zgliczynski, Piotr

Subjects
Mathematics - Dynamical Systems, Mathematical Physics, 37
Abstract

The aim of this paper is to present a new, analytical, method for computing the exact number of relative equilibria in the planar, circular, restricted 4-body problem of celestial mechanics. The new approach allows for a very efficient computer-aided proof, and opens a potential pathway to proving harder instances of the n-body problem.

Published
2022

4. Lower bounds on the Hausdorff dimension of some Julia sets

Author

Dudko, Artem, Gorbovickis, Igors, and Tucker, Warwick

Subjects
Mathematics - Dynamical Systems
Abstract

We present an algorithm for a rigorous computation of lower bounds on the Hausdorff dimensions of Julia sets for a wide class of holomorphic maps. We apply this algorithm to obtain lower bounds on the Hausdorff dimension of the Julia sets of some infinitely renormalizable real quadratic polynomials, including the Feigenbaum polynomial $p_{\,\mathrm{Feig}}(z)=z^2+c_{\,\mathrm{Feig}}$. In addition to that, we construct a piecewise constant function on $[-2,2]$ that provides rigorous lower bounds for the Hausdorff dimension of the Julia sets of all quadratic polynomials $p_c(z) = z^2+c$ with $c \in [-2,2]$. Finally, we verify the conjecture of Ludwik Jaksztas and Michel Zinsmeister that the Hausdorff dimension of the Julia set of a quadratic polynomial $p_c(z)=z^2+c$, is a $C^1$-smooth function of the real parameter $c$ on the interval $c\in(c_{\,\mathrm{Feig}},-3/4)$., Comment: 23 pages, 7 figures

Published
2022
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5. Scalable Multivariate Histograms

Author

Sainudiin, Raazesh, Tucker, Warwick, and Wiklund, Tilo

Subjects
Statistics - Computation, Computer Science - Distributed, Parallel, and Cluster Computing, Mathematics - Optimization and Control, 62G07, G.3.6
Abstract

We give a distributed variant of an adaptive histogram estimation procedure previously developed by the first author. The procedure is based on regular pavings and is known to have numerous appealing statistical and arithmetical properties. The distributed version makes it possible to process data sets significantly bigger than previously. We provide prototype implementation under a permissive license.

Published
2020

9. Lower bounds on the Hausdorff dimension of some Julia sets

Author

Dudko, Artem, Gorbovickis, Igors, and Tucker, Warwick

Subjects
Mathematics::Dynamical Systems, Applied Mathematics, FOS: Mathematics, General Physics and Astronomy, Statistical and Nonlinear Physics, Dynamical Systems (math.DS), Mathematics - Dynamical Systems, Mathematical Physics
Abstract

We present an algorithm for a rigorous computation of lower bounds on the Hausdorff dimensions of Julia sets for a wide class of holomorphic maps. We apply this algorithm to obtain lower bounds on the Hausdorff dimension of the Julia sets of some infinitely renormalizable real quadratic polynomials, including the Feigenbaum polynomial $p_{\,\mathrm{Feig}}(z)=z^2+c_{\,\mathrm{Feig}}$. In addition to that, we construct a piecewise constant function on $[-2,2]$ that provides rigorous lower bounds for the Hausdorff dimension of the Julia sets of all quadratic polynomials $p_c(z) = z^2+c$ with $c \in [-2,2]$. Finally, we verify the conjecture of Ludwik Jaksztas and Michel Zinsmeister that the Hausdorff dimension of the Julia set of a quadratic polynomial $p_c(z)=z^2+c$, is a $C^1$-smooth function of the real parameter $c$ on the interval $c\in(c_{\,\mathrm{Feig}},-3/4)$., Comment: 23 pages, 7 figures

Published
2023

11. A note on the convergence of parametrised non-resonant invariant manifolds

Author

Johnson, Tomas and Tucker, Warwick

Subjects
Mathematics - Dynamical Systems, 37D10, 34C20, 37M99, 65G20
Abstract

Truncated Taylor series representations of invariant manifolds are abundant in numerical computations. We present an aposteriori method to compute the convergence radii and error estimates of analytic parametrisations of non-resonant local invariant manifolds of a saddle of an analytic vector field, from such a truncated series. This enables us to obtain local enclosures, as well as existence results, for the invariant manifolds.

Published
2008
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12. Automated computation of robust normal forms of planar analytic vector fields

Author

Johnson, Tomas and Tucker, Warwick

Subjects
Mathematics - Dynamical Systems, 34C20, 37M99, 65G20, 65L70
Abstract

We construct an auto-validated algorithm that calculates a close to identity change of variables which brings a general saddle point into a normal form. The transformation is robust in the underlying vector field, and is analytic on a computable neighborhood of the saddle point. The normal form is suitable for computations aimed at enclosing the flow close to the saddle, and the time it takes a trajectory to pass it. Several examples illustrate the usefulness of this method.

Published
2008
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13. On a computer-aided approach to the computation of Abelian integrals

Author

Johnson, Tomas and Tucker, Warwick

Subjects
Mathematics - Dynamical Systems, 34C07, 37G15, 37M20, 65G20
Abstract

An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is applied to the study of bifurcations of limit cycles arising from a cubic perturbation of an elliptic Hamiltonian of degree four.

Published
2008
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15. Interval Methods

Author

Tucker, Warwick, Gefen, Amit, Series editor, Geris, Liesbet, editor, and Gomez-Cabrero, David, editor

Published
2016
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16. CAMPARY: Cuda Multiple Precision Arithmetic Library and Applications

Author

Joldes, Mioara, Muller, Jean-Michel, Popescu, Valentina, Tucker, Warwick, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Greuel, Gert-Martin, editor, Koch, Thorsten, editor, Paule, Peter, editor, and Sommese, Andrew, editor

Published
2016
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17. Fixed points of arbitrarily deep 1-dimensional neural networks

Author

Cook, Andrew, Hammerlindl, Andy, and Tucker, Warwick

Subjects
FOS: Computer and information sciences, Computer Science - Machine Learning, Statistics - Machine Learning, 37E99 (Primary) 37N99 (Secondary), FOS: Mathematics, Machine Learning (stat.ML), Dynamical Systems (math.DS), Mathematics - Dynamical Systems, Machine Learning (cs.LG)
Abstract

In this paper, we establish a sharp upper bound on the the number of fixed points a certain class of neural networks can have. The networks under study (autoencoders) can be viewed as discrete dynamical systems whose nonlinearities are given by the choice of activation functions. To this end, we introduce a new class $\mathcal{F}$ of $C^1$ activation functions that is closed under composition, and contains e.g. the logistic sigmoid function. We use this class to show that any 1-dimensional neural network of arbitrary depth with activation functions in $\mathcal{F}$ has at most three fixed points. Due to the simple nature of such networks, we are able to completely understand their fixed points, providing a foundation to the much needed connection between application and theory of deep neural networks., Comment: 9 pages, 3 figures

Published
2023
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22. Fundamentals of Chaos

Author

Tucker, Warwick, Kacprzyk, Janusz, editor, Kocarev, Ljupco, editor, Galias, Zbigniew, editor, and Lian, Shiguo, editor

Published
2009
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23. Reconstructing Metabolic Networks Using Interval Analysis

Author

Tucker, Warwick, Moulton, Vincent, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Istrail, Sorin, editor, Pevzner, Pavel, editor, Waterman, Michael, editor, Casadio, Rita, editor, and Myers, Gene, editor

Published
2005
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25. Efficient and Validated Numerical Evaluation of Abelian Integrals

Author

Bréhard, Florent, Brisebarre, Nicolas, Joldeş, Mioara, Tucker, Warwick, Centre National de la Recherche Scientifique (CNRS), Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Arithmetic and Computing (ARIC), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Laboratoire de l'Informatique du Parallélisme (LIP), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS), Équipe Recherche Opérationnelle, Optimisation Combinatoire et Contraintes (LAAS-ROC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), School of Mathematics [Australia], Monash University [Melbourne], ANR-20-CE48-0014,NuSCAP,Sûreté numérique pour les preuves assistées par ordinateur(2020), Brisebarre, Nicolas, and Sûreté numérique pour les preuves assistées par ordinateur - - NuSCAP2020 - ANR-20-CE48-0014 - AAPG2020 - VALID

Subjects
[INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], limit cycles, [INFO.INFO-SC] Computer Science [cs]/Symbolic Computation [cs.SC], [INFO.INFO-AO]Computer Science [cs]/Computer Arithmetic, Abelian integral, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Newton-like operator, [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], rigorous numerics, Hilbert's 16th problem, [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], 2020 AMS classification: Primary: 34C07, Secondary: 37G15, 37M20, 42A10, 42A15, 65G20, [INFO.INFO-NA] Computer Science [cs]/Numerical Analysis [cs.NA], ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, [INFO.INFO-AO] Computer Science [cs]/Computer Arithmetic, trigonometric polynomial interpolation
Abstract

Abelian integrals play a key role in the infinitesimal version of Hilbert's 16th problem. Being able to evaluate such integrals - with guaranteed error bounds - is a fundamental step in computer-aided proofs aimed at this problem. Using interpolation by trigonometric polynomials and quasi-Newton-Kantorovitch validation, we develop a validated numerics method for computing Abelian integrals in a quasi-linear number of arithmetic operations. Our approach is both effective, as exemplified on two practical perturbed integrable systems, and amenable to an implementation in a formal proof assistant, which is key to provide fully reliable computer-aided proofs.

Published
2022

38. A New Lower Bound on the Hilbert Number for Quartic Systems

Author

Bréhard, Florent, Brisebarre, Nicolas, Joldes, Mioara, Tucker, Warwick, Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Arithmetic and Computing (ARIC), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de l'Informatique du Parallélisme (LIP), Centre National de la Recherche Scientifique (CNRS)-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-École normale supérieure - Lyon (ENS Lyon), Preuves et Langages (PLUME), Laboratoire de l'Informatique du Parallélisme (LIP), Université de Lyon-École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Claude Bernard Lyon 1 (UCBL), Computer-aided proofs in analysis [Uppsala] (CAPA), Department of Mathematics [Uppsala], Uppsala University-Uppsala University, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), and Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA]
Published
2019

45. Searching for sinks of Henon map using a multiple-precision GPU arithmetic library

Author

Joldes, Mioara, Popescu, Valentina, Tucker, Warwick, Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Computer-aided proofs in analysis [Uppsala] (CAPA), Department of Mathematics [Uppsala], Uppsala University-Uppsala University, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), and Université de Toulouse (UT)

Subjects
multiple precision library, floating-point arithmetic, [INFO.INFO-AO]Computer Science [cs]/Computer Arithmetic, GPGPU computing, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Computer Science::Mathematical Software, error-free transform, dynamical systems, Henon map
Abstract

7 pages; National audience; Today, GPUs represent an important hardware development platform for many problems in dynamical systems, where massive parallel computations are needed. Beside that, many numerical studies of chaotic dynamical systems require a computing precision higher than common floating point (FP) formats. One such application is locating invariant sets for chaotic dynamical systems. In particular, we focus on rigorously proving the existence of stable periodic orbits for the Henon map for parameter values close to the classical ones. For that, we present a multiple-precision floating-point arithmetic library in CUDA programming language for the NVIDIA GPU platform. Our library extends the precision using so-called FP expansions, where a number is represented as the unevaluated sum of standard machine precision FP numbers. This format offers the advantage of using directly available and highly optimized hardware FP operations. We generalize algorithms used by multiple-precisions libraries such as Bailey's QD, or the analogue GPU version, GQD.

Published
2013

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