Geometric Integration Integration Lotka-Volterra Poisson - PowerPoint PPT Presentation

Geometric Integration and Parareal Martin J. Gander Geometric Geometric Integration Integration Lotka-Volterra Poisson Integrator and the Parareal Algorithm Energy Conservation Positivity Parareal Geometric Parareal? Harmonic Oscillator Martin J. Gander Kepler Problem H´ enon-Heiles martin.gander@unige.ch Derivative Parareal Conclusions Joint work with Ernst Hairer University of Geneva July 2015

Geometric The Lotka-Volterra Equations Integration and Parareal Lotka Volterra System of differential equations with predator Martin J. Gander y and prey x 1 Geometric Integration − xy ∂ H Lotka-Volterra x ˙ = x xy = ∂ y ; x (0) = ˆ x , − Poisson Integrator xy ∂ H Energy Conservation ˙ = + = ∂ x ; y (0) = ˆ y − y xy y . Positivity Parareal Geometric Parareal? with the function H ( x , y ) = x + y − ln x − ln y . The exact Harmonic Oscillator solution is thus a cycle, and is known in closed form. 2 Kepler Problem H´ enon-Heiles Derivative Parareal Discretization by Forward Euler: Conclusions x n +1 − x n = x n x n y n ; x 0 = ˆ x , − ∆ t y n +1 − y n = − y n + x n y n ; y 0 = ˆ y . ∆ t 1 Alfred J. Lotka, Elements of Physical Biology (1925), and Vito Volterra, Variazioni e fluttuazioni del numero d’individui in specie animali conviventi (1927) 2 A. Steiner and M. Arrigoni, “Die L¨ osung gewisser R¨ auber-Beute-Systeme”, Studia Biophysica vol. 123 (1988) No. 2.

Geometric Forward Euler Solution (exact solution dashed) Integration and Parareal 3 Martin J. Gander Geometric Integration 2.5 Lotka-Volterra Poisson Integrator Energy Conservation Positivity Parareal 2 Geometric Parareal? Harmonic Oscillator Kepler Problem H´ enon-Heiles Derivative Parareal y 1.5 Conclusions 1 0.5 0 0 0.5 1 1.5 2 2.5 3 x

Geometric A Geometric Method for Lotka Volterra Integration and Parareal Using a small modification 3 Martin J. Gander Geometric x n +1 − x n Integration = x n x n y n ; x 0 = ˆ x , − ∆ t Lotka-Volterra y n +1 − y n Poisson Integrator = − y n + x n +1 y n ; y 0 = ˆ y , Energy Conservation ∆ t Positivity Parareal leads to a physically correct so called Poisson Integrator . Geometric Parareal? Harmonic Oscillator Geometric Numerical Integration , Hairer, Lubich, Kepler Problem H´ enon-Heiles Derivative Parareal Wanner, Springer Verlag, 2002: Conclusions “The subject of this book is numerical methods that preserve geometric properties of the flow of a differential equation: symplectic integrators for Hamiltonian systems, symmetric integrators for reversible systems, methods preserving first integrals and numerical methods on manifolds, including Lie group methods and integrators for constrained mechanical systems, and methods for problems with highly oscillatory solutions.” 3 A Non Spiraling Integrator for the Lotka Volterra Equation, G., Il Volterriano No. 4, pp. 21–28, Liceo Cantonale e Biblioteca Cantonale di Mendrisio, 1994.

Geometric Poisson Integrator for Lotka Volterra Integration and Parareal 2.5 Martin J. Gander Geometric Integration Lotka-Volterra 2 Poisson Integrator Energy Conservation Positivity Parareal Geometric Parareal? Harmonic Oscillator 1.5 Kepler Problem H´ enon-Heiles Derivative Parareal y Conclusions 1 0.5 0 0 0.5 1 1.5 2 2.5 x

Geometric Preservation of the Hamiltonian Integration and Parareal y α corrector step Martin J. Gander ( x n +1 , y n +1 ) . Geometric (X,Y) Integration Euler predictor step Lotka-Volterra Poisson Integrator ( x n , y n ) Energy Conservation Positivity Parareal Geometric Parareal? Harmonic Oscillator Kepler Problem Level set of H ( x , y ) H´ enon-Heiles x Derivative Parareal Conclusions x n +1 + α∂ H X = ∂ x ( X , Y ) y n +1 + α∂ H Y = ∂ y ( X , Y ) For the new approximation ( X , Y ), determine α such that H ( X ( x n +1 , y n +1 , ∆ t , α ) , Y ( x n +1 , y n +1 , ∆ t , α )) = H ( x n , y n ) . (could also evaluate gradient at x n +1 , y n +1 )

Geometric Hamiltonian Preserving Integrator Integration and Parareal Martin J. Gander 2.5 Geometric Integration Lotka-Volterra Poisson Integrator 2 Energy Conservation Positivity Parareal Geometric Parareal? 1.5 Harmonic Oscillator Kepler Problem H´ enon-Heiles Derivative Parareal 1 Conclusions 0.5 0 0 0 0 0.5 1 1.5 2 2.5 However: Tupper (2005): A test problem for molecular dynamics integration: “The computed covariance function is clearly not converging to C as n → ∞ ”

Geometric Positivity as a Geometric Property Integration and Parareal 2.5 100 Martin J. Gander 90 Geometric Integration 2 80 Lotka-Volterra Poisson Integrator 70 Energy Conservation Positivity Parareal 1.5 60 Geometric Parareal? Harmonic Oscillator 50 v Kepler Problem H´ enon-Heiles 1 40 Derivative Parareal Conclusions 30 0.5 20 10 0 0 0 0.5 1 1.5 2 2.5 u On the Positivity of Poisson Integrators for the Lotka-Volterra Equations , M. Beck and M.J. Gander, BIT Numerical Mathematics, Vol. 55, No. 2, pp. 319–340, 2015.

Geometric The Parareal Algorithm Integration and Parareal Martin J. Gander J-L. Lions, Y. Maday, G. Turinici (2001): A “Parareal” in Time Discretization of PDEs Geometric Integration Lotka-Volterra The parareal algorithm for the model problem Poisson Integrator Energy Conservation u ′ = f ( u ) Positivity Parareal Geometric Parareal? is defined using two propagation operators: Harmonic Oscillator Kepler Problem 1. G ( t 2 , t 1 , u 1 ) is a rough approximation to u ( t 2 ) with H´ enon-Heiles Derivative Parareal initial condition u ( t 1 ) = u 1 , Conclusions 2. F ( t 2 , t 1 , u 1 ) is a more accurate approximation of the solution u ( t 2 ) with initial condition u ( t 1 ) = u 1 . Starting with a coarse approximation U 0 n at the time points t 1 , t 2 , . . . , t N , parareal performs for k = 0 , 1 , . . . the correction iteration U k +1 n +1 = G ( t n +1 , t n , U k +1 )+ F ( t n +1 , t n , U k n ) − G ( t n +1 , t n , U k n ) . n

Geometric Geometric Parareal Algorithms ? Integration and Parareal ◮ Bal and Wu (DD17, 2008): Symplectic Parareal. Martin J. Gander Non-iterative: “the two-step IPC scheme can be arbitrarily Geometric accurate” Integration Lotka-Volterra ◮ Audouze, Massot, Volz (2009): Symplectic Poisson Integrator Energy Conservation multi-time step parareal algorithms applied to molecular Positivity dynamics. “We also prove the symplecticity of this method, Parareal Geometric Parareal? which is an expected behavior of the molecular dynamics Harmonic Oscillator Kepler Problem integrators” H´ enon-Heiles Derivative Parareal ◮ Jim´ enez-P´ erez, Laskar (2011): A time-parallel Conclusions algorithm for almost integrable Hamiltonian systems. “In this paper we propose a refinement of the SST97 algorithm to accelerate the solution and to preserve the accuracy of the sequential integrator” ◮ Dai, Le Bris, Legoll, Maday (2013): Symmetric parareal algorithms for Hamiltonian systems. “Using a symmetrization procedure and/or a projection step, we introduce here several variants of the original plain parareal in time algorithm”

Geometric Harmonic Oscillator Integration and Parareal Martin J. Gander H ( p , q ) = 1 p 2 + q 2 � � q (0) = 1 , p (0) = 0 , Geometric 2 Integration Lotka-Volterra 10 0 Poisson Integrator 10 −3 Energy Conservation Positivity 10 −6 Parareal 10 −9 Geometric Parareal? Harmonic Oscillator Kepler Problem 10 0 10 1 10 2 10 3 10 4 10 5 H´ enon-Heiles 10 0 Derivative Parareal Conclusions 10 −3 10 −6 10 −9 10 0 10 1 10 2 10 3 10 4 10 5 St¨ ormer-Verlet for ∆ T = 0 . 1 and ∆ T = 0 . 01: Theorem (G, Hairer 2014) For the harmonic oscillator with G of order ε , convergence can be achieved on a time window of length O ( ε − 1 ) .

Geometric Kepler Problem (Completely Integrable) Integration and Parareal Martin J. Gander H ( p , q ) = 1 1 p 2 1 + p 2 � 2 ) − 2 � Geometric q 2 1 + q 2 Integration 2 Lotka-Volterra 10 0 Poisson Integrator Energy Conservation 10 −3 Positivity Parareal 10 −6 Geometric Parareal? 10 −9 Harmonic Oscillator Kepler Problem H´ enon-Heiles 10 0 10 1 10 2 10 3 Derivative Parareal 10 0 Conclusions 10 −3 10 −6 10 −9 10 0 10 1 10 2 10 3 Simulations for ∆ T = 0 . 1 and ∆ T = 0 . 01: Theorem (G, Hairer 2014) For integrable systems with G of order ε , convergence can be achieved on a time window of length O ( ε − 1 / 2 ) .

Geometric H´ enon-Heiles Equation (Chaotic) Integration and 1 Parareal p 2 1 + p 2 � � H ( p , q ) = + U ( q 1 , q 2 ) 2 Martin J. Gander 2 1 1 q 2 − 1 Geometric q 2 1 + q 2 + q 2 3 q 3 � � U ( q 1 , q 2 ) = Integration 2 2 2 Lotka-Volterra Poisson Integrator 10 0 Energy Conservation 10 −3 Positivity 10 −6 Parareal Geometric Parareal? 10 −9 Harmonic Oscillator 10 −12 Kepler Problem H´ enon-Heiles 10 0 10 1 10 2 10 3 Derivative Parareal 10 0 Conclusions 10 −3 10 −6 10 −9 10 −12 10 0 10 1 10 2 10 3 Simulations for ∆ T = 0 . 01 and ∆ T = 0 . 001: Theorem (G, Hairer 2014) For general systems with G of order ε , convergence can be achieved only on a time window of length O (1) .