a block-step version of ks regularization

A block-step versionof KS regularization

Keigo Nitadori

KS regularization

Hermite integrator

KS block-step

Summary

A block-step version of KS regularizationat Steller N-body Dynamics, Sexten

Keigo Nitadori

Co-Design Team, Exascale Computing ProjectRIKEN Advanced Institute for Computational Sciencd

September 10, 2014


Keigo Nitadori

KS regularization

Hermite integrator

KS block-step

Summary

Acknowledgement

My sincerest thanks go to Sverre Aarseth and IoA for invitingme 7 times, 07, 08, 09, 10, 11, 12, and 13, in which the mostimportant and creative collaboration on the development ofNBODY6/GPU was done.


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 bottleneckI Very accurate with fully conservational Hermite integrator

Amazingly, it is already working!

Public code is available as nbody6b and nbody7b


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 harmonicoscillators (and regularized binaries)

3. Block stepping in real time ∆t


Keigo Nitadori

KS regularization

Hermite integrator

KS block-step

Summary

3 steps of regularization

1. Time transformation. dt = ‖r‖dτ, (hereafter, [ ] = ddt [ ],

and [ ]′ = ddτ [ ] = ‖r‖ [ ])

I r(τ) is already a harmonic oscillator (+bias ).I But equation of motion r′′ = f (r,r′) is numerically

dangerous.

2. Write the equation of motion in conserved quantities.ex. r′′ = f (E,e,r) in Sparing–Burdeet–Heggieregularization.I Perturbation to E and e are integrated independently.

3. Coordinate transformation from r to u. Finally, it becomesa harmonic oscillator, u′′ = 1

2 hu.I h is an energy integral and available as h(u,u′), but should

be integrated independently.


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

-1.5 -1 -0.5 0 0.5 1 1.5

harmonicKepler

F1F2

Ellipse of harmonic oscillator:

u = A cos θ + iB sin θ (1)

Ellipse of Kepler motion:

r =u2

=A2 − B2

2+

A2 + B2

2cos 2θ + i2AB sin 2θ

=a(e + cos φ) + ia√

1 − e2 sin φ

with

φ = 2θ, a =A2 + B2

2, e =

A2 − B2

A2 + B2(2)


Keigo Nitadori

KS regularization

Hermite integrator

KS block-step

Summary

Hamilton’s quaternion

I The complex numbers algebra can be applied to thetwo-dimensional geometry

I We can add, subtract, multiply, and divide 2D vectorsI 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 comingdown to breakfast, your brother William Edward and yourselfused to ask me: "Well, Papa, can you multiply triples?" WheretoI 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 tothe arithmetics of 3D vectors


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

orij = −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 ∈ R)

s is referred to as scalar and ix + jy + kz vector.

Broom Bridge


Keigo Nitadori

KS regularization

Hermite integrator

KS block-step

Summary

Usage of quaternion

A unit quaternion rotates a vector.

rrot = αrα, (3)

with a unit quaternion α and a vector r = ix + jy + kz.Example:

ekφ/2(ix + jy + kz)e−kφ/2 =ekφ (ix + jy) + kz

=ekφ (x + ky)i + kz (4)(ie−kφ/2 = ekφ/2i, je−kφ/2 = ekφ/2j, ke−kφ/2 = e−kφ/2k

)α = ekφ/2

(= cos φ2 + k sin φ

2

)rotated a vector about the z-axis

by an angle φ.


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 = uju. (6)

With arbitrary rotational quaternions α and β,

α(jx + ky)α = α(uju)α = (αu β)(βj β)(βuα) (7)

Now we can writer = qκq, (8)

with a three-dimensional orbit r, a quaternion q = αu β, anda unit vector κ = βj β (is either i, j, or k).


Keigo Nitadori

KS regularization

Hermite integrator

KS block-step

Summary

Comparison of LC and KS

LC KSalgebra complex, C quaternion, Hphys. coord. r = x + iy r = ix + jy + kzreg. coord. u = u0 + iu1 u = u0 + iu1 + ju2 + ku3transformation r = u2 r = uκueq. of motion u′′ = 1

2 hu + 12 |u|

2fu u′′ = 12 hu − 1

2 ‖u‖2fuκ

I f is a perturbation vectorI Original 2 × 2 real matrix formulation is possible for LCI Original 4 × 4 real matrix or 2 × 2 complex matrix (Pauli

matrices) formulation is possible for KS


Keigo Nitadori

KS regularization

Hermite integrator

KS block-step

Summary

A very accurate Hermite integrator

(Accidentally) it turn out that the following symmetriccorrector form

ucorr =uold +∆τ

2(u′corr + u′old) −

∆τ2

12(u′′new − u′′old),

u′corr =u′old +∆τ

2(u′′new + u′′old) −

∆τ2

12(u′′′new − u′′′old), (9)

after P(EC)n convergence 5 of 6 orbital elements (except for thephase) conserve in machine accuracy.If you use another form for the position

ucorr =uold +∆τ

2(u′corr + u′old) −

∆τ2

10(u′′new − u′′old) +

∆τ3

120(u′′′new + u′′′old),

(10)

this property is lost.


Keigo Nitadori

KS regularization

Hermite integrator

KS block-step

Summary

Why it works

Consider a harmonic oscillator x = −ωx, and let vi = xi andfi = xi.An analytical solution for one step ∆t writes,(

ωx1v1

)=

(cosω∆t sinω∆t− sinω∆t cosω∆t

) (ωx0v0

). (11)

It is equivalent to(v1 − v0x1 − x0

)=

(tan(ω∆t/2)ω∆t/2

)∆t2

(f1 + f0v1 + v0

). (12)

A second order integrator approximates this as(v1 − v0x1 − x0

)=∆t2

(f1 + f0v1 + v0

). (13)

The difference is only a small factor on the stepsize ∆t.


Keigo Nitadori

KS regularization

Hermite integrator

KS block-step

Summary

Fourth- and sixth-order version

The 4th-order Hermite integrator is equivalent to(v1 − v0x1 − x0

)=*..,

1

1 − 13

(ω∆t

2

)2

+//-

∆t2

(f1 + f0v1 + v0

), (14)

and a 6th-order one(v1 − v0x1 − x0

)=*..,

1 − 115

(ω∆t

2

)2

1 − 25

(ω∆t

2

)2

+//-

∆t2

(f1 + f0v1 + v0

). (15)

The factor to ∆t is approaching to(

tan(ω∆t/2)ω∆t/2

), of exact

solution.Phase error shared among 4 waves of KS⇒ 5 of 6 orbital elements conserve


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 +c2∆t

2(v1 + v0),

x1 =x0 +c4∆t

2(v1 + v0) −

∆t2

12(f1 − f0),

x1 =x0 +c6∆t

2(v1 + v0) −

∆t2

10(f1 − f0) +

c6∆t3

120(f1 + f0), (16)

(so on the velocity), with

c2 =tan φφ

, c4 =tan φφ

(1 −

13φ2

), c6 =

tan φφ

1 − 25φ

2

1 − 115φ

2, (17)

and φ = ω∆t/2. Now the results agree with the analitictrigonometric functions in machine accuracy.


Keigo Nitadori

KS regularization

Hermite integrator

KS block-step

Summary

Block stepping

I Usually, fixed ∆τ, ∼ 30 step/orbit is enough for weaklyperturbed binary.

I Then, real step ∆t =

∫ τ+∆τ

τ

dtdτ

dτ =

∫ τ+∆τ

τ‖r‖dτ,

becomes a varying real (non-quantized) number.I This prevents parallelization, we hope ∆t restricted to

2n (n ∈ Z) (McMillan, 1986; Makino 1992).


Keigo Nitadori

KS regularization

Hermite integrator

KS block-step

Summary

Truncation procedure

From given natural step ∆τnat, we compute a truncated step∆τbs(< ∆τnat) for which the corresponding step ∆tbs = 2n obeysthe block-step criterion.

∆τnat −−−−−−→integrate

∆tnat −−−−−−→truncate

∆tbs −−−−→solve

∆τbs (18)

We integrate

∆tnat =

6∑n=1

t〈n〉(∆τnat)n

n!,

(t〈n〉 =

dn

dτn t). (19)

After truncation ∆tnat → ∆tbs, we solve ∆τbs from

−∆tbs +

6∑n=1

t〈n〉(∆τbs)n

n!= 0, (20)

by using of Newton–Raphson iterations (unique solutionexists).


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 derivativeof force (5th derivative of coordinate) is available

I Up to 6th derivative of time is available

t′ =u · u,t′′ =2(u · u′),

t′′′ =2(u · u′′ + u′ · u′),

t〈4〉 =2(u · u′′′) + 6(u′ · u′′),

t〈5〉 =2(u · u〈4〉) + 8(u′ · u′′′) + 6(u′′ · u′′),

t〈6〉 =2(u · u〈5〉) + 10(u′ · u〈4〉) + 20(u′′ · u′′′). (21)

I tnext is explicitly available at the end of this stepI Predictor only, no corrector is applied for t


Keigo Nitadori

KS regularization

Hermite integrator

KS block-step

Summary

Summary

I The block-step version of KS is working in Sverre’s codeI Some Hermite integrators turned out to integrate harmonic

oscillators very accurately with P(EC)n iterationsI Regularization is not acrobatics, nor a black magic

(at least for two-body).

a block-step version of ks regularization

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