B-Series and their Cousins - A. Contemporary Geometric View

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1 B-Series and their Cousins - A Contemporary Geometric View Hans Munthe-Kaas 1 Department of Mathematics University of Bergen Norway url: hans.munthe-kaas.no mail: hans.munthe-kaas@uib.no SciCADE Bath, September 2017 Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

2 Introduction B-series: Butcher 72, Hairer Wanner 74,... B(α, y) = y + τ T h τ σ(τ) α(τ)f f (τ)(y). T = {,,,,,,,,...} h : time step τ : Number of nodes in τ, e.g. = 3 σ(τ) : symmetry factor, e.g. σ( ) = 2 α(τ) R f : vector field, e.g. right-hand-side of y (t) = f (y(t)) F f (τ) : elementary differentials, e.g. F f ( ) = f i f j f k i,j Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

3 Geometric view: Parallel transport on R n B-series Γ y,t1 v(y 0 ) = v(y 1 ) Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

4 Connection (covariant derivation): f g f g Connection = infinitesimal parallel transport (f g)(y 0 ) := t Γ y,t g(y 0 ), t=0 where ẏ(t) = f (y(t)). From t Γ y,tg = Γ y,t f g Taylor series for parallel transport Γ y,t g = g +f g f (f g)+ 1 1 f (f (f g))+ f (f (f (f g))) Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

5 Curvature and Torsion Curvature R f,g (h) = f (g h) g (f h) f, g h Torsion T (f, g) = f g g f f, g R and T relate via the associator R f,g (h) T (f, g) h = a(f, g, h) a(g, f, h) a(f, g, h) := f (g h) (f g) h Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

6 Euclidean connections and pre-lie algebras Euclidean space Natural connection: (f g) i = l f l gi x l = f gi has R = 0 and T = 0 a(f, g, h) = a(g, f, h). Definition A pre-lie algebra is a vector space with a product satisfying a(f, g, h) = a(g, f, h) (where a(f, g, h) = f (g h) (f g) h) Note: f g g f = f, g (Jacobi bracket) Examples: Vector fields on R n with canonical connection. The free pre-lie algebra (trees with grafting). Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

7 Why trees? The Free pre-lie algebra: prelie({ }) = (T, ) T = span R ({,,,,,,,,...}) with grafting product: = + +. Theorem (T, ) is pre-lie (T, ) is universal Elementary differentials: F f ( ) = f F f (x y) = F f (x) F f (y) Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

8 Pre-Lie Butcher series (Ebrahimi-Fard & al.) α: T R, B(α) = τ T α(τ) σ(τ) F f (τ) F f (τ) are given in terms of the connection: F f ( ) = f F f ( ) = F f ( ) = f f F f ( ) = F f (( ) ) = (f f ) f F f ( ) = F f ( ( ) ( ) ) = f (f f ) (f f ) f Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

9 Enveloping algebra: S (prelie({ })) (Higher order diff. operators) S (prelie({ })) = span R (F), F = {I,,,,,,,,,...)} Some natural operations on enveloping algebra: Concatenation (associative): = Grafting (non-assoc.): = Composition (assoc.): f (g h) = (f g) h, f g = fg + f g Taylor series for parallel transport (revisited) Γ y,t g = g + f g f (f g) + 1 f (f (f g)) + 6 = g + f g (f f ) g (f f f ) g + = exp (f ) g. Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

The fundamental problem of numerical integration Two exponentials: (sending vector fields to flows) Exact solution: exp ( ) = I + + 1 2 + 1 6 + Geodesic flow (Euler): exp ( ) = I

10 The fundamental problem of numerical integration Two exponentials: (sending vector fields to flows) Exact solution: exp ( ) = I Geodesic flow (Euler): exp ( ) = I The Fundamental Problem of Numerical Integration Approximate exp ( ) with exp ( ) Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

11 Runge Kutta methods: Symbolic in S(preLie({ })) Mapped down to R n for i = 1,..., s s k i = exp ( a i,j k j ) j=1 Φ RK = exp ( b j k j ) j Φ RK exp ( ) for i = 1,..., s s k i = f (y n + a i,j k j ) j=1 y n+1 = y n + b j k j j y n+1 y(t n+1 ) (Order theory [Butcher 63] can be derived very elegantly from left formulation!) Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

12 Butchers order theory, pre-lie formulation Runge Kutta: (K is s-tuple) K = exp (AK ) Φ RK = exp (b T K ) Order conditions Exact solution: exp ( )(τ) = 1 τ! RK - character: Φ RK (τ) = b T K (τ) K ( ) = I R R = (R s, ) (Hadamard product) A Lin(R, R) b T Lin(R, R) K : T R K ((τ 1 τ p ) )) = AK (τ 1 ) AK (τ 2 ) AK (τ p ) Order conditions: Φ RK (τ) = 1 τ! for all τ < p + 1 Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

Geometric characterisation of B-series methods Definition B-series method: Flowmap which can be expanded in a B-series Affine symmetries ϕ(x) = Ax + b preserve Euclidean (ϕ f ) (ϕ g) =

13 Geometric characterisation of B-series methods Definition B-series method: Flowmap which can be expanded in a B-series Affine symmetries ϕ(x) = Ax + b preserve Euclidean (ϕ f ) (ϕ g) = ϕ (f g) Hence: B-series (methods) are affine equivariant! Q-McL question: Affine equivariant B-series? Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

14 First cousin: Aromatic B-series Iserles-Quispel-Tse 07, Chartier-Murua 07 Theorem (MK Verdier FoCM 2016) A flow map Φ on R n which is local and affine equivariant can be expanded in an Aromatic B-series Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

Hand-waving proof: Φ: X(R n ) X(R n ) (backward error of method) Locality Taylor terms of Φ(f ) takes form (Peetre theorem): aijkl mno (x)f i fmf j k fno l shift invar. = aijkl mno f i fmf j k fno.

15 Hand-waving proof: Φ: X(R n ) X(R n ) (backward error of method) Locality Taylor terms of Φ(f ) takes form (Peetre theorem): aijkl mno (x)f i fmf j k fno l shift invar. = aijkl mno f i fmf j k fno. l Theorem (Invariant tensor thm:) A GL(V ) equivariant tensor T : V V V V V V is a linear sum of permutations. a mno ijkl f i f j mf k f l no invar. tensor = f i f j l f k fij l = Terms indexed by graphs such that each node has 1 edge out, except a root node with 0 out. Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

16 Answer to Quispel-McLachlan question: Theorem (McL-Modin-MK-Verdier Numer. Math. 2016) A family of flow maps {Φ n } n=1 on Rn for all n has a unique B-series expansion if and only if the family is strongly affine equivariant. Strongly means: The family {Φ n } respects all affine projections and inclusions between spaces of different dimensions. Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

17 WAKE UP! Summary pre-lie An algebraic theory of B-series can be developed from just the geometric properties R = T = 0 of a Euclidean connection. prelie({ }) and its coordinate ring H BCK (Butcher Connes Kreimer Hopf algebra) contain just the right algebraic structures to represent vector fields and flows, parallel transport, analytical and numerical flow maps and other important geometric properties of flows on Euclidean spaces in a universal manner. B-series are THE universal Taylor type expansions which respect symmetries of Euclidean geometry! Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

18 Second cousin: Lie Butcher series; Klein geometries Lie group G: Think SO(n) : Orthogonal matrices. Lie algebra g = T e G: Think so(n): skew matrices, A T = A exp g : g G: Think matrix exponential Homogeneous space M: Think of sphere, acted upon by SO(3) Lie group equation: ẏ(t) = f (y(t)) y(t), f : M g Lie-Euler method: y n+1 = exp g (hf (y n )) y n. Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

19 Post-Lie algebras Parallel transport by left (or right) multiplication in a Lie group yields a connection with zero curvature and constant torsion R = 0, T = 0. The torsion is given by a Lie bracket T (f, g) = [g, f ]. Algebraic description: post-lie algebra. Definition (Post-Lie algebra) A vectorspace with a product and a Lie bracket [_, _] such that 1 f [g, h] = [f g, h] + [g, f h] 2 [f, g] h = a(f, g, h) a(g, f, h) Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

20 Some facts: Pre-Lie is post-lie with [f, g] = 0 Post-Lie always come with: two Lie structures: g : [f, g] g : f, g = f g g f + [f, g]. Free post-lie: postlie({ }) = span R (Lie(OT )), where OT = {,,,,,,...}, Lie(OT ) = {,, [, ],...} and is left grafting. Enveloping algebra U(postLie({ })) comes with two associative products: (concatenation) and (Grossman Larson), and two exponentials exp and exp. The Fundamental Problem of Numerical Lie Group Integration Approximate exact flow exp ( ) with Lie-Euler exp ( ) Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

Summary post-lie An algebraic theory of LB-series can be developed from just the geometric properties R = 0, T = 0 of the canonical connection on a Lie group.

21 Summary post-lie An algebraic theory of LB-series can be developed from just the geometric properties R = 0, T = 0 of the canonical connection on a Lie group. postlie({ }) and its coordinate ring H MKW contain just the right algebraic structures to represent vector fields and flows, parallel transport, analytical and numerical flow maps on Lie groups and homogeneous spaces in a universal manner. Not all properties of pre-lie have yet been generalised to post-lie. Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

22 Geometry Algebra Series Geometry Connection Algebra Hopf alg. Series Autonom Series Control/ R.P. Manifold none Lie H Lie Chen Fliess/ Lyons Euclid R = 0 pre-lie H BCK B-series T = 0 Branched rough paths Klein / R = 0 post-lie H MKW LB-series Lie group T = 0 Planar Branched R.P. Riemannian R = 0 post-lie-?? symmetric T = 0 triple Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

23 Application: Rough paths on Homogeneous Spaces (Curry, Ebrahimi-Fard, Manchon, MK) Controlled differential equation on homogeneous manifold: dy t = d i=1 f i (Y t ) Y t dx i t where Y t M, f i : M g, X : R R d. Solution in the Lie Butcher group: The equation above can be lifted to Y t G(H MKW ) as: dy t = d i=1 Y t i dx i t An explicit solution can be written in terms of iterated integrals. Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

24 Concluding remarks Algebraic and geometric understanding of B-series type expansions on model geometries, Euclid, Klein and Riemann symmetric, are emerging. Algebras describing invariant connections are central in these theories. These are NEW algebraic structures discovered in the last decade, with fundamental mathematical importance. Interaction between pure mathematics and computational mathematics is an important source of inspiration. We are hoping for new geometric and computational insight into rough DEs, regularity structures and problems in geometric mechanics. Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

25 Want to know more? Minisymposium MS05, Friday 10:30-12:30 (Room CB 1.10): Adrien Laurent : Exotic Aromatic B-series Yvain Bruned : Algebraic Renormalisation of Regularity Structures Kurusch Ebrahimi-Fard : Rough Diff. Eqns. on Homogeneous Spaces Olivier Verdier : Geometry of prelie algebras Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

26 Some References I K. Ebrahimi-Fard, A. Lundervold, I. Mencattini, and H. Z. Munthe-Kaas. Post-Lie algebras and isospectral flows. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 11, K. Ebrahimi-Fard, A. Lundervold, and H. Munthe-Kaas. On the Lie enveloping algebra of a post-lie algebra. Journal of Lie theory, 25(4): , G. Fløystad and H. Munthe-Kaas. Pre-and post-lie algebras: The algebro-geometric view. arxiv preprint arxiv: , A. Lundervold and H. Munthe-Kaas. Backward error analysis and the substitution law for Lie group integrators. Foundations of Computational Mathematics, 13(2): , R. I. McLachlan, K. Modin, H. Munthe-Kaas, and O. Verdier. B-series methods are exactly the affine equivariant methods. Numerische Mathematik, 133(3): , H. Munthe-Kaas and O. Verdier. Aromatic Butcher series. Foundations of Computational Mathematics, 16: , Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

27 Some References II H. Munthe-Kaas and W. Wright. On the Hopf algebraic structure of Lie group integrators. Foundations of Computational Mathematics, 8(2): , H. Z. Munthe-Kaas and K. K. Føllesdal. Lie-Butcher series, geometry, algebra and computation. arxiv preprint arxiv: , H. Z. Munthe-Kaas and A. Lundervold. On post-lie algebras, Lie Butcher series and moving frames. Foundations of Computational Mathematics, 13(4): , Hans Munthe-Kaas (Univ. of Bergen) B-Series - Geometric View SciCADE / 27

arxiv: v1 [math.na] 18 Dec 2017

arxiv: v1 [math.na] 18 Dec 2017 What is a post-lie algebra and why is it useful in geometric integration arxiv:1712.09415v1 [math.na] 18 Dec 2017 Charles Curry 1, Kurusch Ebrahimi-Fard 1, and Hans Munthe-Kaas 2 1 Norwegian University