Validated Explicit and Implicit Runge-Kutta Methods

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1 Validated Explicit and Implicit Runge-Kutta Methods Alexandre Chapoutot joint work with Julien Alexandre dit Sandretto and Olivier Mullier U2IS, ENSTA ParisTech 8th Small Workshop on Interval Methods, Praha June, 205

2 Initial Value Problem of Ordinary Differential Equations Consider an IVP for ODE, over the time interval [0, T ] ẏ = f (y) with y(0) = y 0 IVP has a unique solution y(t; y 0 ) if f : R n R n is Lipschitz in y but for our purpose we suppose f smooth enough i.e., of class C k Goal of numerical integration Compute a sequence of time instants: t 0 = 0 < t < < t n = T Compute a sequence of values: y 0, y,..., y n such that i [0, n], y i y(t i ; y 0 ). s.t. y n+ y(t n + h; y n ) with an error O(h p+ ) where h is the integration step-size p is the order of the method true with localization assumption i.e., yn = y(t n; y 0). 2 / 26

3 Validated solution of IVP for ODE Goal of validated numerical integration Compute a sequence of time instants: t 0 = 0 < t < < t n = T Compute a sequence of values: [y 0 ], [y ],..., [y n ] such that i [0, n], [y i ] y(t i ; y 0 ). A two-step approach Exact solution of ẏ = f (y(t)) with y(0) Y 0 Safe approximation at discrete time instants Safe approximation between time instants 3 / 26

4 State of the art Taylor methods They have been developed since 60 s (Moore, Lohner, Makino and Berz, Rhim, Jackson and Nedialkov, etc.) prove the existence and uniqueness: high order interval Picard-Lindelöf works very well on various kinds of problems: non stiff and moderately stiff linear and non-linear systems, with thin uncertainties on initial conditions with (a writing process) thin uncertainties on parameters very efficient with automatic differentiation techniques wrapping effect fighting: interval centered form and QR decomposition many software: AWA, COSY infinity, VNODE-LP, CAPD, etc. Some extensions Taylor polynomial with Hermite-Obreskov (Jackson and Nedialkov) Taylor polynomial in Chebyshev basis (T. Dzetkulic) 4 / 26

5 One question Why bother to define new methods? 5 / 26

6 Answer : it may fail A chemical reaction simulated with VNODE-LP ẏ = z { y(0) = 0 ż = z 2 3 with z(0) = y 2 and t [0, 50] Result: it is stuck around t = with various order between 5 and 40. With validated Lobatto-3C (order 4) method with tolerance 0 0, we get in about 7.6s (Intel i7 3.4Ghz) width(y (50.0)) = width(y 2 (50.0)) = Note: CAPD can solve this problem 6 / 26

7 Answer 2: there is no silver bullet Numerical solutions of IVP for ODEs are produced by Adams-Bashworth/Moulton methods BDF methods Runge-Kutta methods etc. each of these methods is adapted to a particular class of ODEs Runge-Kutta methods have strong stability properties for various kinds of problems (A-stable, L-stable, algebraic stability, etc.) may preserve quadratic algebraic invariant (symplectic methods) can produce continuous output (polynomial approximation of y(t)) Can we benefit these properties in validated computations? 7 / 26

8 Examples of Runge-Kutta methods k 2 = f(t 0 + h, y 0 + hk ) ( ) y = y 0 + h 2 k + k 2 k y These methods have a nice geometric interpre two pictures of Fig..2 for a famous problem Single-step fixed step-size explicit Runge-Kutta of polygonal lines, method which assume the slopes p evaluated at previous points. e.g. explicit Trapzoidal method (or Heun s method) Idea is of defined Heun (900) by: and Kutta (90):compu ing at y 0 and assuming the various slopes k j o which are proportional to some given constan k = f (t n, y n ), k 2 = f (t n + h n, y n + hkgon ) evaluate a new0 slope k ( y n+ = y n + h 2 k + ) i.thelastofthes mines the numerical 2 k solution y (see the third 2 the class of explicit Runge Kutta methods, i. for i j. 2 2 Intuition ẏ = t 2 + y 2 y 0 = 0.46 h =.0 dotted line is the exact solution. y k y 0 expl. trap. rule 2 t y k 2 y expl. midp. k k 2 y 0 2 Fig..2. Runge Kutta methods for ẏ = t 2 + y 2, y example coming from Geometric Numerical Integration, Hairer, Lubich and Wanner. 8 / 26

9 Examples of Runge-Kutta methods Single-step fixed step-size implicit Runge-Kutta method e.g. Runge-Kutta Gauss method (order 4) is defined by: k = f k 2 = f ( ( t n + t n + ) 3 2 h n, 6 ) h n, 6 ) ( ( ( y n+ = y n + h 2 k + 2 k 2 y n + h y n + h ( ( ) )) 4 k k 2 6 (( ) )) k k 2 (a) (b) (c) Remark: A non-linear system of equations must be solved at each step. 8 / 26

10 Runge-Kutta methods s-stage Runge-Kutta methods are described by a Butcher tableau c a a 2 a s.... c s a s a s2 a ss b b 2 b s b b 2 b s (optional) Which induces the following recurrence: ( ) s k i = f t n + c i h n, y n + h a ij k j j= Explicit method (ERK) if a ij = 0 is i j y n+ = y n + h i j s b i k i (2) Diagonal Implicit method (DIRK) if a ij = 0 is i j and at least one a ii 0 Implicit method (IRK) otherwise i= 9 / 26

11 Validated Runge-Kutta methods Challenges. Computing with sets of values taking into account dependency problem and wrapping effect; 2. Bounding the approximation error of Runge-Kutta formula. Our approach Problem is solved using affine arithmetic avoiding centered form and QR decomposition Problem 2 is solved by bounding the Local truncation error of Runge-Kutta method based on B-series We focus on Problem 2 in this talk 0 / 26

12 Order condition for Runge-Kutta methods Method order of Runge-Kutta methods and Local Truncation Error (LTE) we want to bound this! y(t n ; y n ) y n = C O ( h p+) with C R. Order condition This condition states that a method of Runge-Kutta family is of order p iff the Taylor expansion of the exact solution and the Taylor expansion of the numerical methods have the same p + first coefficients. Consequence The LTE is the difference of Lagrange remainders of two Taylor expansions... but how to compute it? / 26

13 A quick view of Runge-Kutta order condition theory 2 Starting from y (q) = (f (y)) (q ) and with the Chain rule, we have High order derivatives of exact solution y ẏ = f (y) ÿ = f (y)ẏ y (3) = f (y)(ẏ, ẏ) + f (y)ÿ y (4) = f (y)(ẏ, ẏ, ẏ) + 3f (y)(ÿ, ẏ) + f (y)y (3) y (5) = f (4) (y)(ẏ, ẏ, ẏ, ẏ) + 6f (y)(ÿ, ẏ, ẏ) + 4f (y)(y (3), ẏ) + 3f (y)(ÿ, ÿ) + f (y)y (4).. f (y) is a linear map f (y) is a bi-linear map f (y) is a tri-linear map. 2 strongly inspired from Geometric Numerical Integration, Hairer, Lubich and Wanner. 2 / 26

14 A quick view of Runge-Kutta order condition theory 2 Inserting the value of ẏ, ÿ,..., we have: High order derivatives of exact solution y ẏ = f ÿ = f (f ) y (3) = f (f, f ) + f (f (f )) y (4) = f (f, f, f ) + 3f (f f, f ) + f (f (f, f )) + f (f (f (f ))). Elementary differentials, are denoted by F (τ) Remark a tree structure is made apparent in these computations 2 strongly inspired from Geometric Numerical Integration, Hairer, Lubich and Wanner. 2 / 26

15 A quick view of Runge-Kutta order condition theory 2 Rooted trees f is a leaf f is a tree with one branch,..., f (k) is a tree with k branches Example f (f f, f ) is associated to f f f Remark: this tree is not unique e.g., symmetry f 2 strongly inspired from Geometric Numerical Integration, Hairer, Lubich and Wanner. 2 / 26

16 A quick view of Runge-Kutta order condition theory 2 Theorem (Butcher, 963) The qth derivative of the exact solution is given by y (q) = r(τ)=q α(τ)f (τ)(y 0) with We can do the same for the numerical solution r(τ) the order of τ i.e., number of nodes α(τ) a positive integer Theorem 2 (Butcher, 963) The qth derivative of the numerical solution is given by y (q) = r(τ)=q γ(τ)φ(τ)α(τ)f (τ)(y 0) with γ(τ) a positive integer φ(τ) depending on a Butcher tableau Theorem 3, order condition (Butcher, 963) A Runge-Kutta method has order p iff φ(τ) = γ(τ) τ, r(τ) p 2 strongly inspired from Geometric Numerical Integration, Hairer, Lubich and Wanner. 2 / 26

17 LTE formula for explicit and implicit Runge-Kutta From Theorem and Theorem 2, if a Runge-Kutta has order p then y(t n ; y n ) y n = hp+ (p + )! r(τ)=p+ α(τ) [ γ(τ)φ(τ)] F (τ)(y(ξ)) ξ [t n, t n ] α(τ) and γ(τ) are positive integer (with some combinatorial meaning) φ(τ) function of the coefficients of the RK method, Example ( ) φ is associated to s b i a ij c j with c j = i,j= Note: y(ξ) may be over-approximated using Interval Picard-Lindelöf operator. s k= a jk 3 / 26

18 Implementation of LTE formula Elementary differentials F (τ)(y) = f (m) (y) (F (τ )(y),..., F (τ m )(y)) for τ = [τ,..., τ m ] translate as a sum of partial derivatives of f associated to sub-trees Notations n the state-space dimension p the order of a Rung-Kutta method Two ways of computing F (τ). Direct form (current): complexity O(n p+ ) 2. Factorized form (under test): complexity O(n(p + ) 5 2 ) based on the work of Ferenc Bartha and Hans Munthe-Kaas Computing of B-series by automatic differentiation, / 26

19 Experimentation Toy example ) ( ) (ẏ y2 = ẏ 2 y with ( ) y (0) = [0, 0.] y 2 (0) = [0.95,.05] Validated RK4 method with tolerance 0 8 we get in about 3s (Intel i7 3.4Ghz) width(y (00.0)) = width(y 2 (00.0)) = / 26

20 Experimentation Usefulness of affine arithmetic ẏ =, y (0) = 0 ẏ 2 = y 3, y 2 (0) = 0 ẏ 3 = 6 y 2 3 y sin(p y ) with p [2.78, 2.79], y 3 (0) = 0. Validated RK4 method with tolerance 0 6 we get in about 2.3s (Intel i7 3.4Ghz) width(y (0.0)) = width(y 2 (0.0)) = width(y 3 (0.0)) = Note: none of the method in the Vericomp benchmark can reach 0s Note 2: CAPD can solve it 6 / 26

21 Experimentation Based on Vericomp benchmark 3 (around 70 problems) simple (A) Uncertain (U) or not linear (L) moderate (B) IVP non-stiff (P.I) complicate (C) nonlinear (L) idem with the following metrics: c5t: user time taken to simulate the problem for second. c5w: the final diameter of the solution (infinity norm is used). c6t: the time to breakdown the method with a maximal limit of 0 seconds. c6w: the diameter of the solution a the breakdown time / 26

22 Summary RK vs Vnode-LP c5w Vnode-LP: order 5, 20, 25 (tolerances 0 4 ) RK4, LC3, LA3: tolerances 0 8 to 0 4 (order 4) 8 / 26

23 Summary RK vs Vnode-LP c6w Vnode-LP: order 5, 20, 25 (tolerances 0 4 ) RK4, LC3, LA3: tolerances 0 8 to 0 4 (order 4) 9 / 26

24 Conclusion We presented a new approach to validate Runge-Kutta methods a new formula to compute LTE based on B-series fully parametrized by a Butcher tableau affine arithmetic avoiding QR decomposition implementation as a plugin of IBEX, code name DynIbex, available at chapoutot/dynibex/ Future work finish testing the implementation of LTE with automatic differentiation implement new a priori enclosure methods based on Runge-Kutta define new methods mixing different Runge-Kutta in one simulation solve new IVP problems such as for DAE (next talk) or DDE 20 / 26

25 BACKUP 2 / 26

26 Note on the number of trees (up to order (left)): Number of Rooted Trees (total 3047) 22 / 26

27 Quick remainder: Taylor series method Taylor series development of y(t) (assume y(t n ) [y n ]) Challenges N h i d i y y(t n+ ) = y(t n ) + i! dt i (t n) + hn n+ d Nx N! dt N (t ) i= N [y n ] + h i f [i ]( y(t n ) ) + h N f [N ]( y(t ) ) i= N [y n ] + h i f [i ] ([y n ]) + h N f [N ] ([ỹ n ]) [y n+ ] i= Computation of [ỹ n ] such that t [t n, t n+ ], y(t) [ỹ n ] Solution: interval Picard-Lindelöf operator With that formula: width([y n+ ]) width([y n ]) Solutions: interval centered form + QR decomposition 23 / 26

28 Variable step-size explicit Runge-Kutta methods Single-step variable step-size explicit Runge-Kutta method e.g. Bogacki-Shampine (ode23) is defined by: k = f (t n, y n ) k 2 = f (t n + 2 h n, y n + 2 hk ) k 3 = f (t n h n, y n hk 2) ( 2 y n+ = y n + h 9 k + 3 k ) 9 k 3 k 4 = f (t n + h n, y n+ ) ( 7 z n+ = y n + h 24 k + 4 k k 3 + ) 8 k Remark: the step-size h is adapted following y n+ z n+ tol / 26

29 Gauss-Legendre methods Single-step fixed step-size implicit Runge-Kutta method e.g. Runge-Kutta Gauss method (order 4) is defined by: k = f k 2 = f ( ( t n + t n + ( ) 3 2 h n, 6 ) h n, 6 ) ( ( y n+ = y n + h 2 k + 2 k 2 y n + h y n + h ( ( ) )) 4 k k 2 6 (( ) )) k k 2 (3a) (3b) (3c) Remark: A non-linear system of equations must be solved at each step. 25 / 26

30 Note on building IRK Gauss method tn+ ẏ = f (y) with y(0) = y 0 y(t) = y 0 + f (y(s))ds t n We solve this equation using quadrature formula. IRK Gauss method is associated to a collocation method (polynomial approximation of the integral) such that for i, j =,..., s: a ij = ci 0 t c k c j c k l j (t)dt and b j = 0 l j (t)dt with l j (t) = k j the Lagrange polynomial. And the c i are chosen as the solution of the Shifted Legendre polynomial of degree s: s ( )( ) s s + k P s (x) = ( ) s ( x) k k s k=0 Example:, 2x, 6x 2 6x +, 20x 3 30x 2 + 2x, etc. 26 / 26

Runge-Kutta Theory and Constraint Programming Julien Alexandre dit Sandretto Alexandre Chapoutot. Department U2IS ENSTA ParisTech SCAN Uppsala

Runge-Kutta Theory and Constraint Programming Julien Alexandre dit Sandretto Alexandre Chapoutot Department U2IS ENSTA ParisTech SCAN 2016 - Uppsala Contents Numerical integration Runge-Kutta with interval