Faà di Bruno Hopf Algebras, Dyson–Schwinger Equations, and Lie–Butcher Series

Numerical analysis of time-integration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher-Connes-Kreimer Hopf algebra first appeared in Butcher's work on composition of integration methods in 1972. In more recent years, the analysis of structure preserving algorithms, geometric integration techniques and integration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey structures that have found applications within these areas. This includes pre-Lie structures for the geometry of flat and torsion free connections appearing in the analysis of numerical flows on vector spaces. The much more recent post-Lie and D-algebras appear in the analysis of flows on manifolds with flat connections with constant torsion. Dynkin and Eulerian idempotents appear in the analysis of non-autonomous flows and in backward error analysis. Non-commutative Bell polynomials and a non-commutative Fa\`a di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds.

[1]  Bruno Vallette,et al.  Homology of generalized partition posets , 2007 .

[2]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[3]  Kurusch Ebrahimi-Fard,et al.  Twisted dendriform algebras and the pre-Lie Magnus expansion , 2009, 0910.2166.

[4]  B. Leimkuhler,et al.  Simulating Hamiltonian Dynamics , 2005 .

[5]  G. Quispel,et al.  Geometric integrators for ODEs , 2006 .

[6]  RUNGE{KUTTA METHODS ON MANIFOLDS , 1999 .

[7]  John C. Butcher,et al.  An algebraic theory of integration methods , 1972 .

[8]  H. Munthe-Kaas.  Runge-Kutta methods on Lie groups , 1998 .

[9]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[10]  Christian Brouder,et al.  Runge–Kutta methods and renormalization , 2000 .

[11]  I. Gel'fand,et al.  Determinants of matrices over noncommutative rings , 1991 .

[12]  Andrei A. Agrachev,et al.  Chronological algebras and nonstationary vector fields , 1981 .

[13]  R. Palais.  A Global Formulation of the Lie Theory of Transformation Groups , 1957 .

[14]  H. Munthe-Kaas,et al.  Computations in a free Lie algebra , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[15]  A. Iserles,et al.  On the solution of linear differential equations in Lie groups , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[16]  Arieh Iserles,et al.  B-SERIES METHODS CANNOT BE VOLUME-PRESERVING , 2007 .

[17]  Dominique Manchon,et al.  Hopf Algebras in Renormalisation , 2008 .

[18]  A. Iserles,et al.  On the Implementation of the Method of Magnus Series for Linear Differential Equations , 1999 .

[19]  Fr'ed'eric Chapoton,et al.  A rooted-trees q-series lifting a one-parameter family of Lie idempotents , 2008, 0807.1830.

[20]  Saad Zagloul Rida,et al.  Noncommutative Bell polynomials , 1996, Int. J. Algebra Comput..

[21]  S. Blanes,et al.  The Magnus expansion and some of its applications , 2008, 0810.5488.

[22]  A. Murua.  Formal series and numerical integrators, part I: Systems of ODEs and symplectic integrators , 1999 .

[23]  Frederic Chapoton,et al.  Pre-Lie algebras and the rooted trees operad , 2000 .

[24]  Kurusch Ebrahimi-Fard,et al.  Two interacting Hopf algebras of trees: A Hopf-algebraic approach to composition and substitution of B-series , 2008, Adv. Appl. Math..

[25]  Alain Connes,et al.  Hopf Algebras, Renormalization and Noncommutative Geometry , 1998 .

[26]  E. Hairer.  Backward analysis of numerical integrators and symplectic methods , 1994 .

[27]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[28]  P. Crouch,et al.  Numerical integration of ordinary differential equations on manifolds , 1993 .

[29]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[30]  Kenth Engø-Monsen,et al.  Numerical Integration of Lie-Poisson Systems While Preserving Coadjoint Orbits and Energy , 2001, SIAM J. Numer. Anal..

[31]  Jerrold E. Marsden,et al.  Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems , 1999 .

[32]  R. Abraham,et al.  Manifolds, Tensor Analysis, and Applications , 1983 .

[33]  Ernst Hairer,et al.  On the Butcher group and general multi-value methods , 1974, Computing.

[34]  A. Iserles,et al.  Lie-group methods , 2000, Acta Numerica.

[35]  J. M. Sanz-Serna,et al.  Canonical B-series , 1994 .

[36]  J. Butcher.  Coefficients for the study of Runge-Kutta integration processes , 1963, Journal of the Australian Mathematical Society.

[37]  Kurusch Ebrahimi-Fard,et al.  A Magnus- and Fer-Type Formula in Dendriform Algebras , 2007, Found. Comput. Math..

[38]  G. Quispel,et al.  Foundations of Computational Mathematics: Six lectures on the geometric integration of ODEs , 2001 .

[39]  Frederic Chapoton.  Rooted trees and an exponential-like series , 2002 .

[40]  Pierre Cartier,et al.  A Primer of Hopf Algebras , 2007 .

[41]  Brynjulf Owren,et al.  Order conditions for commutator-free Lie group methods , 2006 .

[42]  Arne Marthinsen,et al.  Runge-Kutta Methods Adapted to Manifolds and Based on Rigid Frames , 1999 .

[43]  Loic Foissy,et al.  Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations , 2007, 0707.1204.

[44]  Elena Celledoni,et al.  Energy-Preserving Integrators and the Structure of B-series , 2010, Found. Comput. Math..

[45]  Elena Celledoni,et al.  Commutator-free Lie group methods , 2003, Future Gener. Comput. Syst..

[46]  Richard G. Larson,et al.  Hopf-algebraic structure of families of trees , 1989 .

[47]  J. H. Wilkinson.  Error analysis of floating-point computation , 1960 .

[48]  C. Löfwall,et al.  Trees, free right-symmetric algebras, free Novikov algebras and identities , 2002 .

[49]  S. Chern,et al.  Differential Geometry: Cartan's Generalization of Klein's Erlangen Program , 2000 .

[50]  Alexander Lundervold.  Lie–Butcher series and geometric numerical integration on manifolds , 2011 .

[51]  C. Reutenauer.  Free Lie Algebras , 1993 .

[52]  A. Marthinsen,et al.  Modeling and Solution of Some Mechanical Problems on Lie Groups , 1998 .

[53]  Dirk Kreimer,et al.  On the Hopf algebra structure of perturbative quantum field theories , 1997 .

[54]  Dominique Manchon,et al.  A short survey on pre-Lie algebras , 2011 .

[55]  Hans Z. Munthe-Kaas,et al.  Backward Error Analysis and the Substitution Law for Lie Group Integrators , 2011, Foundations of Computational Mathematics.

[56]  Ernst Hairer,et al.  Important Aspects of Geometric Numerical Integration , 2005, J. Sci. Comput..

[57]  W. Magnus.  On the exponential solution of differential equations for a linear operator , 1954 .

[58]  Hans Z. Munthe-Kaas,et al.  Foundations of Computational Mathematics on the Hopf Algebraic Structure of Lie Group Integrators , 2022 .

[59]  J. Wisdom,et al.  Symplectic maps for the N-body problem. , 1991 .

[60]  C. Budd,et al.  Geometric integration and its applications , 2003 .

[61]  H. Munthe-Kaas.  Lie-Butcher theory for Runge-Kutta methods , 1995 .

[62]  H. Munthe-Kaas.  High order Runge-Kutta methods on manifolds , 1999 .

[63]  Antonella Zanna,et al.  Numerical solution of isospectral flows , 1997, Math. Comput..

[64]  Michael E. Hoffman.  Combinatorics of rooted trees and Hopf algebras , 2003 .

[65]  Peter J. Olver,et al.  Geometric Integration Algorithms on Homogeneous Manifolds , 2002, Found. Comput. Math..

[66]  Arthur Cayley,et al.  The Collected Mathematical Papers: On the Theory of the Analytical Forms called Trees , 2009 .

[67]  Murray Gerstenhaber,et al.  The Cohomology Structure of an Associative Ring , 1963 .

[68]  Kenth Engø.  On the Construction of Geometric Integrators in the RKMK Class , 2000 .