Parallel two-step Runge-Kutta methods - hu-berlin.degaggle/EVENTS/2006/BRENT60... · Parallel...

Parallel two-step Runge-Kutta methods

Helmut Podhaisky, [email protected]

Martin–Luther–University Halle–Wittenberg, Germany

20 July 2006

Outline

Runge–Kutta and multi-step methods

Explicit parallel peer methods

Implicit parallel peer methods

Summary

Explicit Runge–Kutta methods for y ′ = f (y)

classical Runge–Kutta method (1904)

Ym1 = ym−1

Ym2 = ym−1 +1

2hf (Ym1)

Ym3 = ym−1 +1

2hf (Ym2)

Ym4 = ym−1 + hf (Ym3)

ym = ym−1 +h

6f (Ym1) +

h

3f (Ym2) +

h

3f (Ym3) +

h

6f (Ym4)



Ym1 = ym−1

Ym2 = ym−1 +1

2hf (Ym1)

Ym3 = ym−1 +1

2hf (Ym2)

Ym4 = ym−1 + hf (Ym3)

ym = ym−1 +h

6f (Ym1) +

h

3f (Ym2) +

h

3f (Ym3) +

h

6f (Ym4)

Butcher scheme

0

12

12

12 0 1

2

1 0 0 1

16

13

13

16

◮ the computation is sequential



Ym1 = ym−1

Ym2 = ym−1 +1

2hf (Ym1)

Ym3 = ym−1 +1

2hf (Ym2)

Ym4 = ym−1 + hf (Ym3)

ym = ym−1 +h

6f (Ym1) +

h

3f (Ym2) +

h

3f (Ym3) +

h

6f (Ym4)

Butcher scheme

0

12

12

12 0 1

2

1 0 0 1

16

13

13

16

◮ the computation is sequential

◮ Ymk is not an accurate approximation to y(tm−1 + ckh)leading to complex order conditions

◮ linear stability can be studied easily

◮ stepsize changes are trivial

Explicit Runge–Kutta methods, Dormand/Prince (1980)

order 5, much better accuracy than RK4

0

15

15

310

340

940

45

4445 −56

15329

89

193726561 −25360

2187644486561 −212

729

1 90173168 −355

33467325247

49176 − 5103

18656

1 35384 0 500

1113125192 −2187

67841184

35384 0 500

1113125192 −2187

67841184

◮ widely used, Matlab: ode45

◮ not parallelizable

Bashforth and Adams (1883)

explicit multistep methods, e.g. order 4:

y [m] = y [m−1] +h

24(55fm−1 − 59fm−2 + 37fm−3 − 9fm−4)

◮ the computation in sequential

◮ the order conditions are rather simple, just interpolation

◮ linear stability is more difficult to analyze

◮ implementing a robust variable stepsize–variable order code isquite a challenge, Matlab ode113

Implicit methods

Runge–Kutta methods

◮ low stage order =⇒ orderreduction

◮ high stage order =⇒coupled iterations (and agood chance for someparallelization, bottleneck:LU-decomposition of theJacobian)

Multistep methods

◮ second Dahlquist barrier: anA-stable method cannothave order p > 2

◮ not parallelizable

Questions:

◮ What limitations can be overcome with more generalmethods? In particular: what about parallelism?

◮ Is it possible to be competitive in applications?

Explicit parallel peer methods

Idea: re-use stages from the last step

Ym =

Ym1

Ym2

...

Yms

, Ymk = y(tm + ckh) + O(hp+1)

Scheme: A and B full s × s matrices

Ym = hAf (Ym−1) + BYm−1

◮ the computation is completely parallel

◮ the order conditions are simple as long as p ≤ s − 1(interpolation, B = B(A))

◮ we want zero-stability with no parasitic roots, σ(B) = {1, 0}

◮ difficult: optimizing stability and variable stepsize behaviour

order conditions for Ym = hAf (Ym−1) + BYm−1

Taylor series expansion

With z = h ddt

, we have Ym = exp(cz)y(tm) and

exp(cz) = Az exp((c − 1)z) + B exp((c − 1)z) + O(zp+1)

which can be satisfied for B easily if p = s − 1.



With z = h ddt



which can be satisfied for B easily if p = s − 1.ordered by powers of z

exp(cz) ∼

1 c1c212 . . .

cs−11

s−1!

1 c2c222 . . .

cs−12

s−1!...

...... . . .

...

1 csc2s

2 . . . cs−1s

s−1!



With z = h ddt




exp((c − 1)z) ∼

1 c1 − 1(c−1)21

2 . . . (c1−1)s−1

s−1!

1 c2 − 1(c−1)22

2 . . . (c2−1)s−1

s−1!...

...... . . .

...

1 cs − 1 (c−1)2s2 . . . (cs−1)s−1

s−1!



With z = h ddt




z exp((c − 1)z) ∼

0 1 c1 − 1 . . . (c1−1)s−2

s−2!

0 1 c2 − 1 . . . (c2−1)s−2

s−2!...

...... . . .

...

0 1 cs − 1 . . . (cs−1)s−2

s−2!

Explicit parallel peer methods, linear stability

Let y ′ = λy and z = hλ.We obtain

Ym = M(z)Ym−1 = (B + zA)Ym−1

◮ M(z) is called stability or amplification matrix.

◮ zero stability: B = M(0) must be power-bounded

◮ stability domain: {z : M(z) is power-bounded}

Explicit parallel peer methods, linear stability

Let y ′ = λy and z = hλ.We obtain

Ym = M(z)Ym−1 = (B + zA)Ym−1

◮ M(z) is called stability or amplification matrix.

◮ zero stability: B = M(0) must be power-bounded

◮ stability domain: {z : M(z) is power-bounded}

◮ example s = 6

−2 −1.5 −1 −0.5 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1 compute boundary:M(z)v = exp(φi)vleads togeneralized EV problem:zA = (exp(φi)I − B)v

Explicit parallel peer methods, optimization

inner loop: linear conditions

◮ order p ≤ s − 1

◮ zero-parasitic eigenvalues: Schur-form QT B̂Q ist strictlyupper triangular,Q = (I + S)(I − S)−1 Cayley-transform, S – free skew matrix

◮ linear least squares, e.g. ‖A‖F and ‖B‖F

outer loop: nonlinear conditions

◮ ρ(M(z)), shape of the stability domain

◮ higher order errors p ≥ s, super-convergence conditions

After all, the method has to be reformulated for variable stepsizes.

Explicit parallel peer methods, optimization

inner loop: linear conditions

◮ order p ≤ s − 1

◮ zero-parasitic eigenvalues: Schur-form QT B̂Q ist strictlyupper triangular,Q = (I + S)(I − S)−1 Cayley-transform, S – free skew matrix

◮ linear least squares, e.g. ‖A‖F and ‖B‖F

outer loop: nonlinear conditions

◮ ρ(M(z)), shape of the stability domain

◮ higher order errors p ≥ s, super-convergence conditions

After all, the method has to be reformulated for variable stepsizes.

⇒ work in progress

implicit parallel peer methods


parallel+implicit

Ymi depends on hf (Ymi )

Ym = Ghf (Ym) + hAf (Ym−1) + BYm−1, G = diag(g1, . . . , gs)

stiff accuracy

M(z) = (I − zG )−1(zA + B) vanishes for z → −∞ iff A = 0

example: method PP3, 3 stages, order 2

L-stable, stiffly accuratec = (−0.1, 0.7, 1), G = diag(1.7, 0.2670, 0.4454)


parallel+implicit

Ymi depends on hf (Ymi )

Ym = Ghf (Ym) + hAf (Ym−1) + BYm−1, G = diag(g1, . . . , gs)

stiff accuracy

M(z) = (I − zG )−1(zA + B) vanishes for z → −∞ iff A = 0

example: method PP3, 3 stages, order 2

L-stable, stiffly accuratec = (−0.1, 0.7, 1), G = diag(1.7, 0.2670, 0.4454)

parasitic roots: B =

−0.21 6.79 −5.57

0.27 −2.46 3.18

0.31 −2.99 3.68

,

σ(B) = {1,−0.0005 ± 0.2202i} (for constant stepsize)

Implementation

◮ one Newton iteration per stage, linearly implicit method

◮ Krylov subspace K := span{b, Tb, . . . ,Tκ−1}, with T = fyFind x̄ for (I − γhT )x = b with x̄ ∈ K and (Ax̄ − b) ⊥ K.

◮ Fortran 90 using OpenMP for loop parallelization

◮ experiments on a SunFire (shared memory)

◮ spatial discretizations with finite differences and finitevolumes, respectively

Example 1: Diffu, ut = uxx + uyy + f (t, x , y)

with 0 ≤ t, x , y ≤ 1 and exact solutionsin(πx) sin(πy)(1 + 4xy sin t)

101

102

−8

−7.5

−7

−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

computing time (seconds)

log(

err)

VODPK



101

102

−8

−7.5

−7

−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5


log(

err)

VODPKROWMAP



101

102

−8

−7.5

−7

−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5


log(

err)

VODPKROWMAPPP3 (1 CPU)



101

102

−8

−7.5

−7

−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5


log(

err)

VODPKROWMAPPP3 (1 CPU)PP3 (3 CPUs)

Example 2: Radiation-Diffusion

Mousseau, Knoll & Rider (2000), Verwer, Hundsdorfer (2003)

Et = ∇ · (Dr∇E ) + σ(T 4 − E ), Dr = (3σ + (1/E )|∂E/∂x |)−1,

Tt = ∇ · (Dt∇T ) − σ(T 4 − E ), Dt= kT 5/2, k = 0 or k = 0.1

strongly nonlinear, conservative second-order discretization

101

102

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1


log(

err)

VODPK



Et = ∇ · (Dr∇E ) + σ(T 4 − E ), Dr = (3σ + (1/E )|∂E/∂x |)−1,



101

102

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1


log(

err)

VODPKROWMAP



Et = ∇ · (Dr∇E ) + σ(T 4 − E ), Dr = (3σ + (1/E )|∂E/∂x |)−1,



101

102

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1


log(

err)

VODPKROWMAPPP3 (1 CPU)



Et = ∇ · (Dr∇E ) + σ(T 4 − E ), Dr = (3σ + (1/E )|∂E/∂x |)−1,



101

102

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1


log(

err)

VODPKROWMAPPP3 (1 CPU)PP3 (2.5 cheated)

Summary

◮ classical methods cannot be parallelized, we need multi-stage,multi-step methods

◮ methods with parallel stages scale can be implemented easilyon shared memory computers (with, say, 2 to 8 CPUs/cores)

◮ order conditions can be satisfied by interpolation, however,super-convergence is more difficult to achieve

◮ the main difficulty lies in finding robust methods with smallcoefficients and moderate error constants

◮ sequential general linear methods are worth to be studied, too

John Butcher (NZ), Adrian Hill (UK),Caren Tischendorf (Matheon/Colone),group in Halle, . . .

Parallel two-step Runge-Kutta methods - hu-berlin.degaggle/EVENTS/2006/BRENT60... · Parallel...

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