COMPOSITION MAGNUS INTEGRATORS. Sergio Blanes. Lie Group Methods and Control Theory Edinburgh 28 June 1 July Joint work with P.C.

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1 COMPOSITION MAGNUS INTEGRATORS Sergio Blanes Departament de Matemàtiques, Universitat Jaume I Castellón) sblanes@mat.uji.es Lie Group Methods and Control Theory Edinburgh 28 June 1 July 24 Joint work with P.C. Moan

2 Report Documentation Page Form Approved OMB No Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 124, Arlington VA Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 3 JAN TITLE AND SUBTITLE Composition Magnus Integrators 2. REPORT TYPE N/A 3. DATES COVERED - 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHORS) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAMES) AND ADDRESSES) University of Valencia 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAMES) AND ADDRESSES) 1. SPONSOR/MONITOR S ACRONYMS) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release, distribution unlimited 11. SPONSOR/MONITOR S REPORT NUMBERS) 13. SUPPLEMENTARY NOTES See also ADM1749, Lie Group Methods And Control Theory Workshop Held on 28 June 24-1 July 24., The original document contains color images. 14. ABSTRACT 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT UU a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified 18. NUMBER OF PAGES 3 19a. NAME OF RESPONSIBLE PERSON Standard Form 298 Rev. 8-98) Prescribed by ANSI Std Z39-18

3 The Goal To consider the Magnus series expansion for numerically solving the non-linear ODE dx dt = fx,t), x) = x R d The Magnus expansion and other geometric numerical integrators have been successfully used to solve the following eqs. separately dx dt = At)x dx dt = fx) Different versions of the ME have proved highly efficient in a number of problems: Quantum Mechanics Highly oscillatory problems Eigenvalue problems etc. Is it possible to use Magnus for the nonautonomous non-linear problem? Zanna) Is it possible to combine both of them for numerically solving the problem? 1

4 Standard Procedure: to consider t = x t on the vector-field and to solve the autonomous equation dy dt = Fy) d { } { } x fx, xt ) = dt x t 1 using an standard algorithm Problems: i) In general, many evaluations of fx,t) at different times, t 1,..., t s. The algorithm can be expensive. ii) If the t-dependent functions in fx,t) and the x-dependent functions in fx,t) evolve at different time-scales it seems convenient to treat them differently. iii) Some times, the structure of the vectorfield fx,t) is simpler than the structure of Fy), allowing to use more efficient algorithms Solution? To use explicit or implicit RK methods with Gaussian or Lobatto quadrature points. This solves i) but not ii) and iii) 2

5 Expected Advantages on using Magnus i) Efficient treatment of the explicit timedependent functions on the vector-field fx,t) in the following cases: * costfx,t 1 )+fx,t 2 )) costfx 1,t)+fx 2,t)) i.e. ẋ = f t) + f 1 t)x + f 2 t)x 2 + f 3 t)x 3 Given t 1,..., t s with t i = t + c i h, and c i the quadrature points at order n with fx,t 1 ),..., fx,t s ) allows methods of order n ii) The t-dependent functions in fx,t) and the x-dependent functions in fx,t) are treated separately. iii) It is of particular interest for those problems where efficient integrators for the autonomous vector-field fx,τ) are known with τ a constant). iv) The geometric properties are preserved. 3

6 Magnus expansion for linear systems dx dt = At)x xh) = eωh) x, where Ω = k=1 Ω k with Ω 1 = Ω 2 = 1 2 Ω 3 = 1 6 h Ω 4 = 1 12 h h At)dt h t1 dt 1 dt 2 [A t1, A t2 ] t1 t2 { dt 1 dt 2 dt 3 t1 t2 t3 { dt 1 dt 2 dt 3 dt 4 [A t1, A t2, A t3 ] + [A t3, A t2, A t1 ] [A t1, A t4, A t3, A t2 ] +[A t2, A t3, A t4, A t1 ] + [A t1, A t2, A t3, A t4 ] [A t4, A t3, A t2, A t1 ] } }. where A ti At i ), [A 1, A 2, A 3 ] [A 1, [A 2, A 3 ]], etc. This expansion is of great interest. However, for building efficient integrators it is convenient to write the truncated Magnus series in a more appropriate way 4

7 To take profit of the time-symmetry, let us consider the Taylor expansion of At) around t 1/2 = t + h/2, At) = i= a i t t1/2)i, ai = 1 i! then, taking b i a i 1 h i, i = 1, 2, 3 Ω = b [b 1, b 2 ] + Oh 5 ) d i At) dt i Ω = b b [b 1, b 2 ] [b 2, b 3 ] [b 1, [b 1, b 3 ]] 1 24 [b 2, [b 1, b 2 ]] [b 1, [b 1, [b 1, b 2 ]]] + Oh 7 ) t=t1/2 {b 1, b 2, b 3 } generators of a graded free Lie algebra with grades 1, 2, 3 Munthe-Kaas & Owren). [b i, b j, b k ] [b i, [b j, b k ]], of order Oh i+j+k ). It is easier to work with. The time-symmetry is clear 5

8 In practice, it seems convenient to consider A 1 = At 1 ),..., A s = At s ) at some quadrature points with A ) = s β j A j = j=1 h Then constants β i) A i) = s j=1 β i) j A j = 1 h h i j At)dt + Oh n+1 ), such that t h 2) i At)dt + Oh n+1 ), i =, 1, 2 Iserles & Nørsett; B, Casas & Ros ) Up to order 6: e Ω[n] = e Ω + Oh n+1 ) Ω [2] = A ) Ω [4] = A ) + [A 1),A ) ] Q = [A 1), 3 2 A) 6A 2) ] Ω [6] = A ) + Q + [A ),[A ), 1 2 A2) 1 6 Q]] [A1),Q] 6

9 This is valid for any n-th order quadrature. We can also use different quadrature formulas for each component, A i,j t). Examples: Gaussian Quadratures Fourth order Consider A i = Ac i h), i = 1, 2 with c 1,2 = Then h A ) = h 2 A 1 + A 2 ) At)dt 3h A 1) = 12 A 2 A 1 ) 1 h t h ) At)dt h 2 Sixth order Consider A i = Ac i h), i = 1, 2, 3 with c 1,3 = , c 2 = 1/2. Then h A ) = h 18 5A 1 + 8A 2 + 5A 3 ) At)dt 15h A 1) = 36 A 3 A 1 ) 1 h t h ) At)dt h 2 A 2) = h 24 A 1 + A 3 ) 1 h t h 2 At)dt h 2) 7

10 This is equivalent to: 1. Consider the equation: x = fx, t)= At)x 2. Frozen the x coordinates of the vectorfield, fx, t) 3. Take the Magnus time-average f Ω x)= Ωh)x autonomous vector-field) 4. Consider the equation: y = f Ω y)= Ωh)y 5. Evaluate the 1-flow: y1) = e Ω xh) Question: Is it possible to follow these steps in a simple way for the non-linear problem? Answer: YES, but only for the steps 1 to 4. In general, due to the complexity of f Ω, step 5 requires the evaluation of a very complicated map Solution: To approximate the complicate map by a composition of simpler maps Composition Magnus Integrators 8

11 Non-linear autonomous system ẋ = fx) 1) with solution xt) = Φ t f x ). Let us consider the Lie operador L f = n i=1 Eq. 1) can be written as f i x i. ẋ = L fx) x d dt Φt f x ) = L f Φ t f x ) and in terms of the evolution operator d dt Φt f = Φt f L fx ) We have rewritten eq. 1) as a infinite dimensional) linear eq. with formal solution: Φ t f = exptl f ) Sol. dif. eq. Lie Transformation ẋ = fx) xt) = exptl fy) )y) y=x 9

12 Non-autonomous System dx dt = fx,t) with solution xt) = Φ t f x ). The evolution operator satisfy the eq. d dt Φt f = Φt f L fx,t) It is possible to make use of the Magnus expansion and to write the solution as Φ t f = expl f Ω x,t) ), with f Ω = i f Ω i where f Ω 1 x, t) = t fx, s)ds t s1 f2 Ω x, t) = 1 2 ds 1 ds 2fx, s 1 ), fx, s 2 )) being h = f, g) the Lie bracket h i = f, g) i = L f g i L g f i = n j=1 f j g i x j g j f i x j 1 )

13 The formal solution at t = h is given by xh) = expl f Ω y,h) )y) y=x, the 1-flow solution of the autonomous DE f i) = s j=1 i =, 1, 2 ẋ = f Ω x, h), x) = x β i) j f j = 1 h h i t h 2) i fx, t)dt + Oh n+1 ), Up to order 6: exp L f Ω[n]) = exp Lf Ω) +Oh n+1 ) f Ω[2] = f ) f Ω[4] = f ) f 1),f ) ) Q = f 1), 3 2 f ) 6f 2) ) f Ω[6] = f ) + Q + f ),f ), 1 2 f 2) 1 6 Q)) f 1),Q) 11

14 The vector-field containing Lie brackets are, in general, very complicate and their maps are computationally expensive. Solution: To approximate this map by a composition of simpler maps This is closely related to the commutatorfree methods Celledoni & Owren) Which simpler maps can be used? A map is the 1-flow solution of a differential equation. Which differential equations can be exactly or efficiently computed? This dependes on the particular problem. 12

15 Given the quadrature points, t 1,..., t s, we consider the following cases: i) Suppose the autonomous equation can be efficiently solved up to t = h ẋ = α 1 fx, t 1 ) + + α s fx, t s ) i.e. suppose it is easy to approximate the map xh) = exp hα 1 L fx,t1 ) + +hα sl fx,ts )) x ) ii) Given f = f A + f B, suppose we can solve ẋ = α 1 f A x, t 1 ) + + α s f A x, t s ) ẋ = β 1 f B x, t 1 ) + + β s f B x, t s ) i.e. suppose we can approximate the maps xh) = exp hα 1 L fa x,t 1 ) + + hα sl fa x,t s )) x ) xh) = exp hβ 1 L fb x,t 1 ) + + hβ sl fb x,t s )) x ) 13

16 Problem to solve expω) = exp b 1 1 ) 12 [b 1, b 2 ] + Oh 5 ) = m exp ) α i,1 b 1 + α i,2 b 2 + Oh 5 ) i=1 expω) = exp b b [b 1, b 2 ] [b 2, b 3 ] [b 1, [b 1, b 3 ]] 1 24 [b 2, [b 1, b 2 ]] + 1 ) 72 [b 1, [b 1, [b 1, b 2 ]]] + Oh 7 ) = m exp ) α i,1 b 1 + α i,2 b 2 + α i,3 b 3 + Oh 7 ) i=1 The coefficients α i,j have to solve a system of non-linear equations. Time-symmetry. It can be preserved with α m+1 i,1 = α i,1 α m+1 i,2 = α i,2 α m+1 i,3 = α i,3 14

17 Order Conditions Given the time-symmetric composition e α 1,1b 1 +α 1,2 b 2 +α 1,3 b 3 e α 2,1b 1 +α 2,2 b 2 +α 2,3 b 3 e α 2,1b 1 α 2,2 b 2 +α 2,3 b 3 e α 1,1b 1 α 1,2 b 2 +α 1,3 b 3 and a general time-symmetric) element Cβ k) ) = β k) 1 b 1 + β k) + β k) 5 2 b 3 + β k) 3 [1, 1, 3] + βk) 6 [1, 2] + βk) 4 [2, 3] [2, 1, 2] + βk) 7 [1, 1, 1, 2] the order conditions can be easily obtained from the recursive relation e xb 1+yb 2 +zb 3 e Cβk)) e xb 1 yb 2 +zb 3 = e Cβk+1) x,y,z)) Next, we must solve the equations = β m) 1, 1, β m) 2, β m) 3, β m) 4, β m) 5, β m) 6, β m) 1 12, 1 12, 1 24, 1 36, 1 24, ) ) Finally, we write the solution in terms of A ), A 1), A 2) 15

18 Example: Fourth-order composition methods B & Moan) Linear problem e Ω exp b ) 12 [b 2, b 1 ] exp A ) + [A 1), A ) ] ) 1 exp 2 A) + 2A 1) exp 2 A) 2A 1) exp A 1)) exp A )) exp A 1)) ) 1 ) Non-linear problem expl f Ω) exp L f ) f 1),f ) ) ) ) exp L 12 f ) 2f 1) exp L 12 f )+2f 1) exp L f 1)) exp Lf )) exp Lf 1) ) ) 16

19 ẋ = fx, t) ) x 2 1) = exp L 12 f ) 2f 1) ) exp L 12 f )+2f 1) x ) ẋ 1 = 1 2 f ) x 1 ) 2f 1) x 1 ), x 1 ) = x, ẋ 2 = 1 2 f ) x 2 ) + 2f 1) x 2 ), x 2 ) = x 1 1) x 2 1) = xh) + Oh 5 ) x 3 1) = exp L f 1)) exp Lf )) exp Lf 1)) x ) ẋ 1 = f 1) x 1 ), x 1 ) = x, ẋ 2 = f ) x 2 ), x 2 ) = x 1 1), ẋ 3 = f 1) x 3 ), x 3 ) = x 2 1) x 3 1) = xh) + Oh 5 ) 17

20 Illustrative Example: ẋ = f t) + f 1 t)x + f 2 t)x 2 + f 3 t)x 3 Evaluate the constants f ) h i f it)dt, f 1) i 1 h t h ) f i t)dt, h 2 i =,..., 3 using a fourth-order quadrature) a i = 1 2 f ) i 2f 1) i, b i = 1 2 f ) i + 2f 1) i Finally, we must solve the autonomous eqs. ẋ 1 = a + a 1 x 1 + a 2 x a 3x 3 1, x 1) = x ẋ 2 = b + b 1 x 2 + b 2 x b 3x 3 2, x 2) = x 1 1) Solution from the method x 2 1) = xh) + Oh 5 ) 18

21 Optimization It is possible to improve the accuracy following different procedures Suppose the constant part of the vector-field, b 1, is the dominant term, then j b 1 s {}}{ Ω = b [b 1, b 2 ] + + c j [ b 1,..., b 1, b 2 ] b [b 1, b 4 ] [b 2, b 3 ] + where the coefficients c k are given by k c k x k = 1 x 2 x x e x 1 = gx). The dominating error term is then E j 1 = c j[b 1,..., b 1, b 2 ] + c j+1 [b 1,..., b 1, b 1, b 2 ] + We can use additional exponentials to cancel the lowest order terms of E j 1 or to remove its first singularities 19

22 Separable Problem: ẋ = Ct)x + Dt)x with [Ct), Dt)] =. Then, we can consider the graded free Lie algebra generated by {c 1, d 1, c 2, d 2, c 3, d 3 } with b i = c i +d i, i = 1, 2, 3 and [c i, c j ] = [d i, d j ] =. Ω = c 1 + d c d [c 1, d 2 ] 1 12 [d 1, c 2 ] [c 2, d 3 ] [d 2, c 3 ] + 1 [c1, c 1, d 3 ] + [c 1, d 1, c 3 ] + [d 1, c 1, d 3 ] + [d 1, d 1, c 3 ] ) 36 [c2, c 1, d 2 ] + [c 2, d 1, c 2 ] + [d 2, c 1, d 2 ] + [d 2, d 1, c 2 ] ) [c1, c 1, c 1, d 2 ] + [c 1, c 1, d 1, c 2 ] + [c 1, d 1, c 1, d 2 ] +[c 1, d 1, d 1, c 2 ] + [d 1, c 1, c 1, d 2 ] + [d 1, c 1, d 1, c 2 ] +[d 1, d 1, c 1, d 2 ] + [d 1, d 1, d 1, c 2 ] ) + Oh 7 ). exp Ω ) can be approximated by m exp ) ) α i,1 c 1 + α i,2 c 2 + α i,3 c 3 exp βi,1 d 1 + β i,2 d 2 + β i,3 d 3 i=1 Fourth-order: easy Sixth-order: very complicate 2

23 For the particular case ẋ = Cx + Dt)x the problem simplifies Ω = c 1 + d d [c 1, d 2 ] + 1 [c1, c 1, d 3 ] + [d 1, c 1, d 3 ] ) 36 1 [d2, c 1, d 2 ] ) + 1 [c 1, c 1, c 1, d 2 ] + [c 1, d 1, c 1, d 2 ] ) +[d 1, c 1, c 1, d 2 ] + [d 1, d 1, c 1, d 2 ] + Oh 7 ). and sixth-order methods can be obtained. Other cases can also be considered. On the other hand, for the autonomous case ẋ = Cx + Dx many efficient splitting methods are known e C+D = m i=1 Then, for our composition e α ic e β id m exp ) ) α i,1 c 1 + α i,2 c 2 + α i,3 c 3 exp βi,1 d 1 + β i,2 d 2 + β i,3 d 3 i=1 we can take α i,1 = α i, β i,1 = β i which solve many the most complicate) order conditions 21

24 Hamiltonian Systems This is just a particular, but very important, case. Let us consider Hq, p, t) : R 2l R R, where q, p R l, the Hamilton eqs. are dq dt = H p ; or, equivalently { } d q = dt p I I dp dt = H q ) H q H p If we denote x = q, p), it corresponds to the particular case f = J H x Lie bracket Poisson bracket The Hamiltonian H = T p, t) + V q, t) appears in many problem: is time-dependent + separable The Poisson brackets destroy the separability Use of composition Magnus integrators 22

25 NUMERICAL EXAMPLES: HAMILTONIAN SYSTEMS Perturbed oscillator by a plane wave H = 1 2 p2 + q 2 ) + ε cosq)g 1 t) + sinq)g 2 t)) with g 1 t) = k i=1 cosw k t), g 2 t) = k i=1 sinw i t) q = p = 11,275 ω i = iω ω = 1/1 k = 1 ε =,25 1,25 We are interested in the Poincaré section and consider different methods. We start with a large time-step, h and repeat the computations reducing h until we get the correct picture. For this time-step we measure the computational cost 23

26 a) 13.5 b) ε= ε= p 11.7 p q q ε =,25 ε = 1,25 CPU N CPU N 2EXq EXq S S RKN RK Given the time step h = 2π/N we show the minimum value of N such that δ < 1 3. For these values we calculated the CPU time in seconds. 24

27 The Duffing Problem or, equivalently q = At) q q V q, t) Ṁ = At)M d dt { q p } = M M 1 ) { q V q, t) p } The one-dimensional equation q = ɛ q + q q 3 + δ coswt) can be obtained from the Hamiltonian H = e ɛtp2 2 + eɛt q 4 4 q2 2 δcosωt)q ) which is separable in two time-dependent parts, being both of them solvable. The evaluation of the time-dependent functions is the most computational costly part 25

28 q = 1,75 p = ɛ = 1 1 δ = 1 ω = RY LOGERROR) 3 4 RK4 FLA 5 2EX LOGT) Errors in positions for different splitting methods and the two-exponential CMI. Time step chosen such that all methods require the same computational cost. 26

29 The Schrödinger Equation i t ux, t) = 1 2µ 2 + V x, t) x2 Spatial semidiscretisation ux, t) ut) = ux, t) ux 1, t). ux N, t) Then, we have to solve. iu t = Ht)u ) ux, t) with H C N N hermitic usually real and symmetric). If we consider then d dt { q p } = u = q + ip Ht) Ht) ) { q p } V x, t) = D1 e αx ) 2 + xft) i) ft) = G coswt) Laser field ii) ft) = Collision with an atom Gw cosh 2 F wt) 27

30 Conclusions The time and the coordinates in the vectorfield fx, t) play different roles on the evolution of the system for many problems it is convenient to treat them differently We have generalized the Magnus integrators for linear systems to be used in nonlinear problems Composition Magnus integrators can be used in tandem with other geometric integrators of different order). An important case being the splitting methods for separable systems. The final methods are still Geometric Integrators Additional stages can be considered for optimization purposes 28

31 Work in Progress To analyze the efficiency of fourth-order methods optimized following different criterions To build different sixth-order composition Magnus integrators and to analyze their performances To consider particular cases, like the separable problem, and to build efficient composition methods for them To look for interesting problems where these methods can be of interest polynomial vector-fields, linear non-homogeneous systems, Riccati equation, separable Hamiltonian systems, some oscillatory problems, etc.) 29

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