General Techniques for Constructing Variational Integrators Melvin - PowerPoint PPT Presentation

General Techniques for Constructing Variational Integrators Melvin Leok Mathematics, University of California, San Diego Joint work with James Hall, Cuicui Liao, Tatiana Shingel, Joris Vankerschaver, Jingjing Zhang. Rough Paths and Combinatorics in Control Theory, UCSD, July, 2011. arXiv:1001.1408 arXiv:1101.1995 arXiv:1102.2685 Supported in part by NSF DMS-0726263, DMS-1001521, DMS-1010687 (CAREER).

2 Geometry and Numerical Methods � Dynamical equations preserve structure • Many continuous systems of interest have properties that are con- served by the flow: ◦ Energy ◦ Symmetries, Reversibility, Monotonicity ◦ Momentum - Angular, Linear, Kelvin Circulation Theorem. ◦ Symplectic Form ◦ Integrability • At other times, the equations themselves are defined on a mani- fold, such as a Lie group, or more general configuration manifold of a mechanical system, and the discrete trajectory we compute should remain on this manifold, since the equations may not be well-defined off the surface.

3 Motivation: Geometric Integration � Main Goal of Geometric Integration: Structure preservation in order to reproduce long time behavior. � Role of Discrete Structure-Preservation: Discrete conservation laws impart long time numerical stability to computations, since the structure-preserving algorithm exactly conserves a discrete quantity that is always close to the continuous quantity we are interested in.

4 Geometric Integration: Energy Stability � Energy stability for symplectic integrators Control�on�global�error Continuous�energy Isosurface Discrete�energy Isosurface

5 Geometric Integration: Energy Stability � Energy behavior for conservative and dissipative systems 0.35 0.3 Midpoint Newmark 0.3 0.25 0.25 0.2 Variational Benchmark Explicit Newmark Variational 0.2 Energy Energy 0.15 non-variational Runge-Kutta 0.15 0.1 0.1 Runge-Kutta 0.05 0.05 Benchmark 0 0 0 100 200 300 400 500 600 700 800 900 1000 0 200 400 600 800 1000 1200 1400 1600 Time Time (a) Conservative mechanics (b) Dissipative mechanics

6 Geometric Integration: Energy Stability � Solar System Simulation • Forward Euler q k +1 = q k + h ˙ q ( q k , p k ) , p k +1 = p k + h ˙ p ( q k , p k ) . • Inverse Euler q k +1 = q k + h ˙ q ( q k +1 , p k +1 ) , p k +1 = p k + h ˙ p ( q k +1 , p k +1 ) . • Symplectic Euler q k +1 = q k + h ˙ q ( q k , p k +1 ) , p k +1 = p k + h ˙ p ( q k , p k +1 ) .

7 Geometric Integration: Energy Stability � Forward Euler 30 20 10 0 2144 −10 −20 −30 30 20 30 10 20 0 10 0 −10 −10 −20 −20 −30 −30 1 0.5 0 1980 2000 2020 2040 2060 2080 2100 2120 2140 2160 Energy error

8 Geometric Integration: Energy Stability � Inverse Euler 30 20 10 0 2077 −10 −20 −30 30 20 30 10 20 0 10 0 −10 −10 −20 −20 −30 −30 50 0 −50 1990 2000 2010 2020 2030 2040 2050 2060 2070 2080 Energy error

9 Geometric Integration: Energy Stability � Symplectic Euler 30 20 10 0 2268 −10 −20 −30 30 20 30 10 20 0 10 0 −10 −10 −20 −20 −30 −3 −30 x 10 2 0 −2 1950 2000 2050 2100 2150 2200 2250 2300 Energy error

10 Introduction to Computational Geometric Mechanics � Geometric Mechanics • Differential geometric and symmetry techniques applied to the study of Lagrangian and Hamiltonian mechanics. � Computational Geometric Mechanics • Constructing computational algorithms using ideas from geometric mechanics. • Variational integrators based on discretizing Hamilton’s principle, automatically symplectic and momentum preserving.

11 Symplecticity in the Planar Pendulum 2 2 2 2 0 0 2 2 4 4 6 6 8 8 0 0 2 2 4 4 6 6 8 8 explicit Euler Runge, order 2 −2 −2 −2 −2 2 2 2 2 0 0 2 2 4 4 6 6 8 8 0 0 2 2 4 4 6 6 8 8 symplectic Euler −2 −2 Verlet −2 −2 2 2 2 2 0 0 2 2 4 4 6 6 8 8 0 0 2 2 4 4 6 6 8 8 implicit Euler midpoint rule −2 −2 −2 −2 Images courtesy of Hairer, Lubich, Wanner, Geometric Numerical Integration , 2nd Edition, Springer, 2006.

12 Lagrangian Variational Integrators � Discrete Variational Principle q i varied�curve varied�point q t ( ) Q Q q b (��) q N � q t ( ) � q i q 0 q a (��) • Discrete Lagrangian � h L d ( q 0 , q 1 ) ≈ L exact � � ( q 0 , q 1 ) ≡ L q 0 , 1 ( t ) , ˙ q 0 , 1 ( t ) dt, d 0 where q 0 , 1 ( t ) satisfies the Euler–Lagrange equations for L and the boundary conditions q 0 , 1 (0) = q 0 , q 0 , 1 ( h ) = q 1 . • This is related to Jacobi’s solution of the Hamilton–Jacobi equation .

13 Lagrangian Variational Integrators � Discrete Variational Principle • Discrete Hamilton’s principle � δ S d = δ L d ( q k , q k +1 ) = 0 , where q 0 , q N are fixed. � Discrete Euler–Lagrange Equations • Discrete Euler-Lagrange equation D 2 L d ( q k − 1 , q k ) + D 1 L d ( q k , q k +1 ) = 0 . • The associated discrete flow ( q k − 1 , q k ) �→ ( q k , q k +1 ) is automati- cally symplectic, since it is equivalent to, p k = − D 1 L d ( q k , q k +1 ) , p k +1 = D 2 L d ( q k , q k +1 ) , which is the Type I generating function characterization of a symplectic map.

14 Lagrangian Variational Integrators � Main Advantages of Variational Integrators • Discrete Noether’s Theorem If the discrete Lagrangian L d is (infinitesimally) G -invariant under the diagonal group action on Q × Q , L d ( gq 0 , gq 1 ) = L d ( q 0 , q 1 ) then the discrete momentum map J d : Q × Q → g ∗ , � � � J d ( q k , q k +1 ) , ξ � ≡ D 1 L d ( q k , q k +1 ) , ξ Q ( q k ) is preserved by the discrete flow.

15 Lagrangian Variational Integrators � Main Advantages of Variational Integrators • Variational Error Analysis Since the exact discrete Lagrangian generates the exact solution of the Euler–Lagrange equation, the exact discrete flow map is formally expressible in the setting of variational integrators. • This is analogous to the situation for B-series methods, where the exact flow can be expressed formally as a B-series. • If a computable discrete Lagrangian L d is of order r , i.e., L d ( q 0 , q 1 ) = L exact ( q 0 , q 1 ) + O ( h r +1 ) d then the discrete Euler–Lagrange equations yield an order r accu- rate symplectic integrator.

16 Constructing Discrete Lagrangians � Systematic Approaches • The theory of variational error analysis suggests that one should aim to construct computable approximations of the exact discrete Lagrangian. • There are two equivalent characterizations of the exact discrete Lagrangian: ◦ Euler–Lagrange boundary-value problem characterization, ◦ Variational characterization, which lead to two general classes of computable discrete Lagrangians: ◦ Shooting-based discrete Lagrangians. ◦ Galerkin discrete Lagrangians,

17 Shooting-Based Variational Integrators � Boundary-Value Problem Characterization of L exact d • The classical characterization of the exact discrete Lagrangian is Jacobi’s solution of the Hamilton–Jacobi equation, and is given by, � h L exact � � ( q 0 , q 1 ) ≡ L q 0 , 1 ( t ) , ˙ q 0 , 1 ( t ) dt, d 0 where q 0 , 1 ( t ) satisfies the Euler–Lagrange boundary-value problem. � Shooting-Based Discrete Lagrangians • Replaces the solution of the Euler–Lagrange boundary-value prob- lem with the shooting-based solution from a one-step method . • Replace the integral with a numerical quadrature formula .

18 Shooting-Based Variational Integrators � Shooting-Based Discrete Lagrangian • Consider a one-step method Ψ h : TQ → TQ , and a numerical quadrature formula � h n � f ( x ) dx ≈ h b i f ( x ( c i h )) , 0 i =0 with quadrature weights b i and quadrature nodes 0 = c 0 < c 1 < . . . < c n − 1 < c n = 1. • We construct the shooting-based discrete Lagrangian , � n i =0 b i L ( q i , v i ) , L d ( q 0 , q 1 ; h ) = h where q 0 = q 0 , q n = q 1 . ( q i +1 , v i +1 ) = Ψ ( c i +1 − c i ) h ( q i , v i ) ,

19 Shooting-Based Variational Integrators � Implementation Issues • While one can view the implicit definition of the discrete Lagrangian separately from the implicit discrete Euler–Lagrange equations, p 0 = − D 1 L d ( q 0 , q 1 ; h ) , p 1 = D 2 L d ( q 0 , q 1 ; h ) , in practice, one typically considers the two sets of equations to- gether to implicitly define a one-step method: � n i =0 b i L ( q i , v i ) , L d ( q 0 , q 1 ; h ) = h ( q i +1 , v i +1 ) = Ψ ( c i +1 − c i ) h ( q i , v i ) , i = 0 , . . . n − 1 , q 0 = q 0 , q n = q 1 , p 0 = − D 1 L d ( q 0 , q 1 ; h ) , p 1 = D 2 L d ( q 0 , q 1 ; h ) .

20 Shooting-Based Variational Integrators � Shooting-Based Implementation • Given ( q 0 , p 0 ), we let q 0 = q 0 , and guess an initial velocity v 0 . • We obtain ( q i , v i ) n i =1 by setting ( q i +1 , v i +1 ) = Ψ ( c i +1 − c i ) h ( q i , v i ). • We let q 1 = q n , and compute p 1 = D 2 L d ( q 0 , q 1 ; h ). • Unless the initial velocity v 0 is chosen correctly, the equation p 0 = − D 1 L d ( q 0 , q 1 ; h ) will not be satisfied, and one needs to compute the sensitivity of − D 1 L d ( q 0 , q 1 ; h ) on v 0 , and iterate on v 0 so that p 0 = − D 1 L d ( q 0 , q 1 ; h ) is satisfied. • This gives a one-step method ( q 0 , p 0 ) �→ ( q 1 , p 1 ). • In practice, a good initial guess for v 0 can be obtained by inverting the continuous Legendre transformation p = ∂L/∂v .