A block-step version of KS regularization
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The document presents a block-step version of Kustaanheimo–Stiefel (KS) regularization implemented in the N-body dynamics code nbody6/gpu, highlighting its effectiveness in eliminating serial bottlenecks. It discusses an accurate hermite integrator for harmonic oscillators and the process of coordinate transformation to achieve a harmonic oscillator form. Additionally, the paper includes acknowledgments and details about certain mathematical transformations and algorithms utilized in the regularization method.
![A block-step version
of KS regularization
Keigo Nitadori
KS regularization
Hermite integrator
KS block-step
Summary
3 steps of regularization
1. Time transformation. dt = krkd, (hereafter, [˙] = d
dt [ ],
d [ ] = krk[˙])
and [ ]0 = d
I r() is already a harmonic oscillator (+bias ).
I But equation of motion r00 = f (r; r0) is numerically
dangerous.
2. Write the equation of motion in conserved quantities.
ex. r00 = f (E;e; r) in Sparing–Burdeet–Heggie
regularization.
I Perturbation to E and e are integrated independently.
3. Coordinate transformation from r to u. Finally, it becomes
a harmonic oscillator, u00 = 12
hu.
I h is an energy integral and available as h(u;u0), but should
be integrated independently.](https://image.slidesharecdn.com/talk-141015121126-conversion-gate02/85/A-block-step-version-of-KS-regularization-5-320.jpg)
A block-step version of KS regularization
- 1. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary A block-step version of KS regularization at Steller N-body Dynamics, Sexten Keigo Nitadori Co-Design Team, Exascale Computing Project RIKEN Advanced Institute for Computational Sciencd September 10, 2014
- 2. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Acknowledgement My sincerest thanks go to Sverre Aarseth and IoA for inviting me 7 times, 07, 08, 09, 10, 11, 12, and 13, in which the most important and creative collaboration on the development of NBODY6/GPU was done.
- 3. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Abstract I talk about a new future implemented in NBODY6/GPU: A block-step version of KS regularized binary. I To eliminate serial bottleneck I Very accurate with fully conservational Hermite integrator Amazingly, it is already working! Public code is available as nbody6b and nbody7b
- 4. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Topics 1. Brief introduction of Kustaanheimo–Stiefel (KS) regularization (with Hamilton’s quaternion numbers) 2. A very accurate variant of Hermite integrator for harmonic oscillators (and regularized binaries) 3. Block stepping in real time t
- 5. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary 3 steps of regularization 1. Time transformation. dt = krkd, (hereafter, [˙] = d dt [ ], d [ ] = krk[˙]) and [ ]0 = d I r() is already a harmonic oscillator (+bias ). I But equation of motion r00 = f (r; r0) is numerically dangerous. 2. Write the equation of motion in conserved quantities. ex. r00 = f (E;e; r) in Sparing–Burdeet–Heggie regularization. I Perturbation to E and e are integrated independently. 3. Coordinate transformation from r to u. Finally, it becomes a harmonic oscillator, u00 = 12 hu. I h is an energy integral and available as h(u;u0), but should be integrated independently.
- 6. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Levi-Civita transformation in a complex plane 1.5 1 0.5 0 -0.5 -1 -1.5 harmonic Kepler F1 F2 -1.5 -1 -0.5 0 0.5 1 1.5 Ellipse of harmonic oscillator: u = A cos + iB sin (1) Ellipse of Kepler motion: r =u2 = A2 B2 2 + A2 + B2 2 cos 2 + i2AB sin 2 =a(e + cos ) + ia p 1 e2 sin with = 2; a = A2 + B2 2 ; e = A2 B2 A2 + B2 (2)
- 7. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Hamilton’s quaternion I The complex numbers algebra can be applied to the two-dimensional geometry I We can add, subtract, multiply, and divide 2D vectors I What about on 3D vectors? I According to a letter Hamilton wrote later to his son Archibald: Every morning in the early part of October 1843, on my coming down to breakfast, your brother William Edward and yourself used to ask me: Well, Papa, can you multiply triples? Whereto I was always obliged to reply, with a sad shake of the head, No, I can only add and subtract them. I Finally, he discovered that quadruple numbers can be applied to the arithmetics of 3D vectors
- 8. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary cont. Introduce a new imaginary unit j that anti-commutes with i, e.g. ij = ji, and let k = ij. Then, i2 = j2 = k2 = ijk = 1 or ij = ji = k; jk = kj = i; ki = ik = j defines the (non-commutative) quaternion algebra. A quaternion is H 3 q = s + ix + jy + kz (s;x;y; z 2 R) s is referred to as scalar and ix + jy + kz vector.
- 9. Broom Bridge
- 10. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Usage of quaternion A unit quaternion rotates a vector. rrot = r ¯; (3) with a unit quaternion and a vector r = ix + jy + kz. Example: ek=2(ix + jy + kz)ek=2 =ek (ix + jy) + kz =ek (x + ky)i + kz (4) iek=2 = ek=2i; jek=2 = ek=2j; kek=2 = ek=2k = ek=2 = cos 2 + k sin 2 rotated a vector about the z-axis by an angle .
- 11. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Let’s rotate the orbit Let Kepler orbit on the xy-plane x + iy = u2; (5) with LC coordinate u = u0 + iu1. By multiplying j from right, we move the orbit on the yz-plane, (x + iy)j = jx + ky = u2j = uj¯u: (6) With arbitrary rotational quaternions and
- 12. , (jx + ky)¯ = (uju¯ )¯ = (u
- 13. ¯)(
- 14. j
- 15. ¯)(
- 16. u¯¯ ) (7) Now we can write r = q ¯q; (8) with a three-dimensional orbit r, a quaternion q = u
- 17. ¯, and a unit vector =
- 18. j
- 19. ¯ (is either i, j, or k).
- 20. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Comparison of LC and KS LC KS 12 12 algebra complex, C quaternion, 12 12H phys. coord. r = x + iy r = ix + jy + kz reg. coord. u = u0 + iu1 u = u0 + iu1 + ju2 + ku3 transformation r = u2 r = u u ¯eq. of motion u00 = hu + juj2fu u00 = hu kuk2fu I f is a perturbation vector I Original 2 2 real matrix formulation is possible for LC I Original 4 4 real matrix or 2 2 complex matrix (Pauli matrices) formulation is possible for KS
- 21. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary A very accurate Hermite integrator (Accidentally) it turn out that the following symmetric corrector form ucorr =uold + 2 (u0 corr + u0 old) 2 12 (u00 new u00 old); u0 corr =u0 old + 2 (u00 new + u00 old) 2 12 (u000 new u000 old); (9) after P(EC)n convergence 5 of 6 orbital elements (except for the phase) conserve in machine accuracy. If you use another form for the position ucorr =uold + 2 (u0 corr + u0 old) 2 10 (u00 new u00 old) + 3 120 (u000 new + u000 old); (10) this property is lost.
- 22. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Why it works Consider a harmonic oscillator ¨x = !x, and let vi = ˙xi and fi = ¨xi. An analytical solution for one step t writes, !x1 v1 ! = cos!t sin!t sin!t cos!t ! !x0 v0 ! : (11) It is equivalent to v1 v0 x1 x0 ! = tan(!t=2) !t=2 ! t 2 f1 + f0 v1 + v0 ! : (12) A second order integrator approximates this as v1 v0 x1 x0 ! = t 2 f1 + f0 v1 + v0 ! : (13) The dierence is only a small factor on the stepsize t.
- 23. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Fourth- and sixth-order version The 4th-order Hermite integrator is equivalent to v1 v0 x1 x0 ! = *.. , 1 1 13 !t 2 2 +// - t 2 f1 + f0 v1 + v0 ! ; (14) and a 6th-order one v1 v0 x1 x0 ! = *.. , 1 1 15 !t 2 2 1 25 !t 2 2 +// - t 2 f1 + f0 v1 + v0 ! : (15) The factor to t is approaching to tan(!t=2) !t=2 ! , of exact solution. Phase error shared among 4 waves of KS ) 5 of 6 orbital elements conserve
- 24. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Phase correction If one likes to conserve also the phase, emphasize the step-size t. x1 =x0 + c2t 2 (v1 + v0); x1 =x0 + c4t 2 (v1 + v0) t2 12 (f1 f0); x1 =x0 + c6t 2 (v1 + v0) t2 10 (f1 f0) + c6t3 120 (˙f1 + ˙f0); (16) (so on the velocity), with c2 = tan ; c4 = tan 1 1 3 2 ! ; c6 = tan 1 25 2 1 1 15 2 ; (17) and = !t=2. Now the results agree with the analitic trigonometric functions in machine accuracy.
- 25. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Block stepping I Usually, fixed , 30 step/orbit is enough for weakly perturbed binary. I Then, real step t = Z + dt d d = Z + krkd, becomes a varying real (non-quantized) number. I This prevents parallelization, we hope t restricted to 2n (n 2 Z) (McMillan, 1986; Makino 1992).
- 26. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Truncation procedure From given natural step nat, we compute a truncated step bs ( nat) for which the corresponding step tbs = 2n obeys the block-step criterion. nat ! integrate tnat ! truncate tbs ! solve bs (18) We integrate tnat = X6 n=1 thni (nat)n n! ; thni = dn dn t ! : (19) After truncation tnat ! tbs, we solve bs from tbs + X6 n=1 thni (bs)n n! = 0; (20) by using of Newton–Raphson iterations (unique solution exists).
- 27. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Derivatives of physical time t I After a 4th-order Hermite integration, up to 3rd derivative of force (5th derivative of coordinate) is available I Up to 6th derivative of time is available t0 =u u; t00 =2(u u0); t000 =2(u u00 + u0 u0); th4i =2(u u000) + 6(u0 u00); th5i =2(u uh4i) + 8(u0 u000) + 6(u00 u00); th6i =2(u uh5i) + 10(u0 uh4i) + 20(u00 u000): (21) I tnext is explicitly available at the end of this step I Predictor only, no corrector is applied for t
- 28. A block-step version of KS regularization Keigo Nitadori KS regularization Hermite integrator KS block-step Summary Summary I The block-step version of KS is working in Sverre’s code I Some Hermite integrators turned out to integrate harmonic oscillators very accurately with P(EC)n iterations I Regularization is not acrobatics, nor a black magic (at least for two-body).