CAFE Computer Algebra for Functional Equations Progress Report 2002-2006 November 14th, 2006
CAFE: Computer Algebra for Functional Equations Team: Manuel Bronstein, scientific leader • Alban Quadrat, Evelyne Hubert, research scientists. • Maria Przybylska, Jacques-Arthur Weil, visiting scientists. • Thomas Cluzeau, postdoc. • Min Wu, Daniel Robertz, Nicolas Le Roux, PhD students. Goal: Foster the use of symbolic methods in science and engineering by producing the programs and tools necessary to apply them to academic and industrial problems. Method: functional equation are in the realm of analysis symbolic computation are based on algebraic structures Analysis ↔ Algebra Algorithms Software and libraries of examples. View on: Mathematical physics and control theory.
2002 objectives • Development of efficient algorithms • Linear systems of ordinary differential equations • Non-linear differential systems • Develop mathematical web services • Study more general classes of systems • Solving systems of linear partial differential equations • Finite-dimensional • Infinite-dimensional • Nonlinear systems invariant under a group of symmetry
Outline 1 Applications of Differential Galois Theory 2 General linear functional systems 3 Nonlinear Differential Systems and Invariants 4 Objectives for the future
Solving ordinary differential equations • To a linear ordinary differential equation ˙ X = A ( t ) X is associated a linear algebraic group • Properties of the solution set are then equivalent to group-theoretic properties • Algorithms to solve the equation in closed form are based on this correspondence M. Bronstein, P. Sol´ e, Linear recurrences with polynomial coefficients , Journal of Complexity, (2004). E. Compoint, J.-A. Weil. Absolute reducibility of diffal operators and Galois groups , J. of Algebra, (2004) Σ it and the web service powered by Bernina
Integrable Hamiltonian systems p = ∂ H q = − ∂ H ˙ ˙ p = ( p 1 , . . . , p n ) , q = ( q 1 , . . . , q n ) ∂ q ∂ p Integrability: the system possess n first integrals M-R: the Hamiltonian system is integrable ⇒ the Galois group of the linearisation is Abelian For systems with parameters this enables to find the values for which system might be integrable. 7 systems, essentially in celestial mechanics, analyzed by M. Przybylska and J.-A. Weil in collaboration with D. Boucher, A. Maciejewski, & T. Stachowiak n General results for H = 1 � p 2 i + V ( q ) , V ( q ) homogeneous 2 i =1 A. Maciejewski, M. Przybylska, Physics Letters. A, (2004) × 2.
Outline 1 Applications of Differential Galois Theory 2 General linear functional systems 3 Nonlinear Differential Systems and Invariants 4 Objectives for the future
Factorization & Decomposition problems Goal : Simplify the integration of the system R z = 0. Example : A tank containing a fluid � y 1 ( t − 2 h ) + y 2 ( t ) − 2 ˙ y 3 ( t − h ) = 0 , y 1 ( t ) + y 2 ( t − 2 h ) − 2 ˙ y 3 ( t − h ) = 0 , � y 1 � δ 2 − 2 d 1 dt δ = 0 , y 2 ⇔ δ 2 − 2 d 1 dt δ y 3 d dt y ( t ) = ˙ y ( t ) , δ y ( t ) = y ( t − h ) , Theoretical basis: D-module Algorithmic tool: Gr¨ obner bases
Factorization & Decomposition problems Goal : Simplify the integration of the system R z = 0. Factorisation : R = R 1 R 2 . Example : � � 1 � δ 2 − 2 d � � δ 2 � 1 dt δ 1 − 1 0 = . δ 2 + 1 − 2 δ d δ 2 − 2 d 1 dt δ 1 1 0 dt
Factorization & Decomposition problems Goal : Simplify the integration of the system R z = 0. Factorisation : R = R 1 R 2 . � � � � 0 R 11 R 12 R 11 Decomposition: V R U − 1 = or 0 R 22 0 R 22 U,V invertible Example: � δ 2 − 1 � δ 2 − 2 d � � 1 0 0 dt δ U − 1 = V . δ 2 + 1 δ 2 − 2 d − 4 d 0 1 dt δ dt δ
Results 1 Difference/differential linear finite-dimensional systems: • Factorization and decomposition by means of the eigenring. • Algorithms for the computations of hypergeometric solutions. Bronstein, Li , Wu, Picard-Vessiot extensions for linear functional systems, ISSAC 2005 2 General linear systems : General conditions for the existence of factorizations and decompositions by homological algebra. Cluzeau, Quadrat, Using morphism computations for factoring and decomposing, MTNS 2006 OreModule , Morphisms Implications: control theory and mathematical physics.
Parametrizing linear functional systems Parametrization : R z = 0 z = S y . ⇔ Examples: ∇ .� � B = � � ∇ ∧ � B = 0 , A , ⇔ E + ∂� ∇ V − ∂� B A ∇ ∧ � � ∂ t = � E = − � � 0 , ∂ t . x 1 ( t ) + a x 1 ( t ) − k a x 2 ( t − h ) = 0 , ˙ x 2 ( t ) − x 3 ( t ) = 0 , ˙ x 3 ( t ) + ω 2 x 2 ( t ) + 2 ζ ω x 3 ( t ) − ω 2 u ( t ) = 0 , ˙ x 1 ( t ) = ω 2 a k ξ ( t − h ) , x 2 ( t ) = ω 2 ˙ ξ ( t ) + ω 2 a ξ ( t ) , ⇔ x 3 ( t ) = ω 2 ˙ ξ ( t ) + ω 2 a ˙ ξ ( t ) , ξ ( t ) + ( ω 2 + 2 a ζ ω ) ˙ ξ ( t ) + a ω 2 ξ ( t ) . u ( t ) = ξ (3) ( t ) + (2 ζ ω + a ) ¨
Results • Determination of obstructions by module theory and effective homological algebra. • Computation of (minimal) parametrizations. • Applications to multidimensional systems theory (first integrals, controllability, flat outputs, optimal control). Chyzak-Quadrat-Robertz (AAECC05/MTNS04/IFAC03), Quadrat-Robertz (MTNS06/CDC05/IFAC05), Fabia´ nska-Quadrat (MTNS06). OreModules , Stafford , Quillen-Suslin .
Outline 1 Applications of Differential Galois Theory 2 General linear functional systems 3 Nonlinear Differential Systems and Invariants 4 Objectives for the future
Nonlinear differential systems Goal: Differential system triangulation decomposition → canonical forms − − − − − − − − − − − − − − − − − E. Hubert, 2 chapters in LNCS 2630, (2003). diffalg : in Maple • Computing power series solutions by a Newton method E. Hubert, N. Le Roux,. ISSAC (2003). • Systems of ordinary differential equations are equivalent to a single equation ⇒ new canonical form, probabilistic algorithms T. Cluzeau, E. Hubert. AAECC (2003). T. Cluzeau, E. Hubert, HAL INRIA 89287, (2006)
Symmetric systems s ( φ xx + φ yy ) + s x φ x + s y φ y + φ = 0 S s ( ψ xx + ψ yy ) + s x ψ x + s y ψ y + ψ = 0 ψ x φ x + ψ y φ y = 0 What are the conditions on s for the system to have a solution? [G. Metivier] Observe: S invariant under a 7 dimensional algebraic group action Idea: factor out the symmetry = take into account the geometry Project: Algebraic and algorithmic foundation for the reduction Philo: The (differential) invariants of the symmetry group incorporate the geometry in an algebraic way
Differential Algebras of Invariants • Algorithm to compute the (differential) invariants In : A group and its action Optional: a cross-section to the orbits 1. A generating set of rational invariants Out : 2. A rewriting algorithm E. Hubert & I. Kogan, Journal of Symbolic Computation (2006). aida - Algebraic Invariants and their Differential Algebras • Structure of algebraic/smooth invariant: Invariants ↔ Functions on the cross-section E. Hubert & I. Kogan, Smooth and Algebraic Invariants. Local and Global construction , submitted.
Differential Algebras of Invariants • Algorithm to compute the (differential) invariants E. Hubert & I. Kogan, Journal of Symbolic Computation (2006). aida - Algebraic Invariants and their Differential Algebras • Structure of algebraic/smooth invariant: Invariants ↔ Functions on the cross-section E. Hubert & I. Kogan, Smooth and Algebraic Invariants. Local and Global construction , submitted. • Non-commuting derivations act on the algebra of differential invariants E. Hubert, Journal of Pure and Applied Algebra (2005). diffalg inria.fr/cafe/Evelyne.Hubert/diffalg
Outline 1 Applications of Differential Galois Theory 2 General linear functional systems 3 Nonlinear Differential Systems and Invariants 4 Objectives for the future
Status A. Quadrat plans to join the APICS project-team E. Hubert is in sabbatical leave at IMA, Minneapolis.
Objectives - A. Quadrat 1 OreModules based on Singular: Plural (V. Levandoskyy, D. Robertz, E. Zerz, Aachen University, Germany). 2 Study of stabilization problems of infinite-dimensional systems (C. Bonnet (INRIA, SOSSO), S. Niculescu (CNRS, LSS)). 3 Study of variational formulations based on Lie (pseudo)groups coming in mechanics, continuum mechanics. . . (F. Boyer, IRCCyN) . His research topics will certainly be subject to developments and new collaborations are under way (e.g., with J. Leblond on some PDEs coming from the study of the tokamaks (ITER)).
Objectives - E. Hubert 1 Invariants of algebraic groups • Construction of cross-section of degree 1. (E. Schost, U. of Western Ontario) • Efficient and specific algorithms: inductive construction by subgroups, lifting construction, joint invariants. 2 Structure of the algebra of differential invariants • Finite dimensional groups • Pseudo-group = groups determined by a PDE systems. (P. Olver, U. of Minnesota) aida : Algebraic Invarariants and their Differential Algebra
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