Parallel Space-Time Methods M. Neum uller Special Semester - - PowerPoint PPT Presentation

Institute of Computational Mathematics

Parallel Space-Time Methods

• M. Neum¨

uller Special Semester Space-Time Methods for PDEs

• Nov. 7 - 11, 2016
• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 1 / 29

Institute of Computational Mathematics

Outline

Model problem Space-time method Numerical analysis Numerical examples Solvers Standard solvers Space-time multigrid method Conclusions and outlook

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 2 / 29

Institute of Computational Mathematics

Model problem

Heat equation: ∂tu − ∆u = f in Q := Ω × (0, T), u = gD

• n Σ := ∂Ω × (0, T),

u = u0

• n Σ0 := Ω × {0}.

Q x t Σ0 Σ Σ T ΣT

• U. Langer, S.E. Moore and M.N., Space–time isogeometric analysis of parabolic

evolution problems (2016)

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 3 / 29

Institute of Computational Mathematics

Outline

Model problem Space-time method Numerical analysis Numerical examples Solvers Standard solvers Space-time multigrid method Conclusions and outlook

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 4 / 29

Institute of Computational Mathematics

IgA function space

◮ Parameter space-time domain:

Q := (0, 1)d+1

◮ Geometrical mapping:

Φ : Q → Q

Q x T K

• K

ˆ x

• Q

1 1 Φ−1 Φ Ω

• Ω

a b t ˆ t Q ˆ Q

nx

nt

• x1

x2 ˆ x1 ˆ x2 Φ Φ−1 1 1 T Ω

• Ω

t ˆ t

IgA function space: V p

h := span{ϕh,k}k ⊂ Cp−1(Q)

with ϕh,k = Rk,p ◦ Φ−1, V p

0h := {vh ∈ V p h : vh = 0 on Σ ∪ Σ0} .

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 5 / 29

Institute of Computational Mathematics

Variational formulation

Let vh ∈ V p

0h for p ≥ 2 and

wh := vh + θh∂tvh with θ > 0.

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 6 / 29

Institute of Computational Mathematics

Variational formulation

Let vh ∈ V p

0h for p ≥ 2 and

wh := vh + θh∂tvh with θ > 0.

• Q

f whdxdt =

• Q

∂tu whdxdt −

• Q

∆u whdxdt

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 6 / 29

Institute of Computational Mathematics

Variational formulation

Let vh ∈ V p

0h for p ≥ 2 and

wh := vh + θh∂tvh with θ > 0.

• Q

f whdxdt =

• Q

∂tu whdxdt −

• Q

∆u whdxdt =

• Q

∂tu whdxdt +

• Q

∇xu · ∇xwhdxdt −

• ∂Q

nx · ∇xu whds

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 6 / 29

Institute of Computational Mathematics

Variational formulation

Let vh ∈ V p

0h for p ≥ 2 and

wh := vh + θh∂tvh with θ > 0.

• Q

f whdxdt =

• Q

∂tu whdxdt −

• Q

∆u whdxdt =

• Q

∂tu whdxdt +

• Q

∇xu · ∇xwhdxdt −

• ∂Q

nx · ∇xu whds =

• Q

∂tu whdxdt +

• Q

∇xu · ∇xwhdxdt.

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 6 / 29

Institute of Computational Mathematics

Variational formulation

Let vh ∈ V p

0h for p ≥ 2 and

wh := vh + θh∂tvh with θ > 0.

• Q

f whdxdt =

• Q

∂tu whdxdt −

• Q

∆u whdxdt =

• Q

∂tu whdxdt +

• Q

∇xu · ∇xwhdxdt −

• ∂Q

nx · ∇xu whds =

• Q

∂tu whdxdt +

• Q

∇xu · ∇xwhdxdt. Bilinear form: ah(uh, vh) :=

• Q

∂tuh (vh + θh∂tvh) dxdt +

• Q

∇xuh · ∇x (vh + θh∂tvh) dxdt

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 6 / 29

Institute of Computational Mathematics

Variational formulation

Let vh ∈ V p

0h for p ≥ 2 and

wh := vh + θh∂tvh with θ > 0.

• Q

f whdxdt =

• Q

∂tu whdxdt −

• Q

∆u whdxdt =

• Q

∂tu whdxdt +

• Q

∇xu · ∇xwhdxdt −

• ∂Q

nx · ∇xu whds =

• Q

∂tu whdxdt +

• Q

∇xu · ∇xwhdxdt. Bilinear form: ah(uh, vh) :=

• Q

∂tuh (vh + θh∂tvh) dxdt +

• Q

∇xuh · ∇x (vh + θh∂tvh) dxdt Linear form: lh(vh) :=

• Q

f (vh + θh∂tvh) dxdt

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 6 / 29

Institute of Computational Mathematics

Variational formulation

Find uh ∈ Vg := g + V p

0h with g = u0 on Σ0 and g = gD on Σ, such that

ah(uh, vh) = lh(vh) for all vh ∈ V p

0h.

Discrete problem: Find uh = g + u0h with u0h ∈ V p

0h, such that

ah(u0h, vh) = lh(vh) − ah(g, vh) for all vh ∈ V p

0h.

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 7 / 29

Institute of Computational Mathematics

Variational formulation

Find uh ∈ Vg := g + V p

0h with g = u0 on Σ0 and g = gD on Σ, such that

ah(uh, vh) = lh(vh) for all vh ∈ V p

0h.

Discrete problem: Find uh = g + u0h with u0h ∈ V p

0h, such that

ah(u0h, vh) = lh(vh) − ah(g, vh) for all vh ∈ V p

0h.

Bilinear form: ah(uh, vh) :=

• Q

∂tuh (vh + θh∂tvh) dxdt +

• Q

∇xuh · ∇x (vh + θh∂tvh) dxdt Linear form: lh(vh) :=

• Q

f (vh + θh∂tvh) dxdt

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 7 / 29

Institute of Computational Mathematics

Coercivity

For vh ∈ V p

0h with p ≥ 2 we have

ah(vh, vh) =

• Q

∂tvh (vh + θh∂tvh) dxdt +

• Q

∇xvh · ∇x (vh + θh∂tvh) dxdt

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 8 / 29

Institute of Computational Mathematics

Coercivity

For vh ∈ V p

0h with p ≥ 2 we have

ah(vh, vh) =

• Q

∂tvh (vh + θh∂tvh) dxdt +

• Q

∇xvh · ∇x (vh + θh∂tvh) dxdt = θh∂tvh2

L2(Q) + ∇xvh2 L2(Q)

+

• Q

∂tvh vhdxdt + θh

• Q

∇xvh · ∇x∂tvhdxdt

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 8 / 29

Institute of Computational Mathematics

Coercivity

For vh ∈ V p

0h with p ≥ 2 we have

ah(vh, vh) =

• Q

∂tvh (vh + θh∂tvh) dxdt +

• Q

∇xvh · ∇x (vh + θh∂tvh) dxdt = θh∂tvh2

L2(Q) + ∇xvh2 L2(Q)

+

• Q

∂tvh vhdxdt + θh

• Q

∇xvh · ∇x∂tvhdxdt = θh∂tvh2

L2(Q) + ∇xvh2 L2(Q)

+ 1 2

• Q

∂t(vh)2dxdt + θh 2

• Q

∂t |∇xvh|2 dxdt

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 8 / 29

Institute of Computational Mathematics

Coercivity

For vh ∈ V p

0h with p ≥ 2 we have

ah(vh, vh) =

• Q

∂tvh (vh + θh∂tvh) dxdt +

• Q

∇xvh · ∇x (vh + θh∂tvh) dxdt = θh∂tvh2

L2(Q) + ∇xvh2 L2(Q)

+

• Q

∂tvh vhdxdt + θh

• Q

∇xvh · ∇x∂tvhdxdt = θh∂tvh2

L2(Q) + ∇xvh2 L2(Q)

+ 1 2

• Q

∂t(vh)2dxdt + θh 2

• Q

∂t |∇xvh|2 dxdt = θh∂tvh2

L2(Q) + ∇xvh2 L2(Q)

+ 1 2

• ∂Q

nt(vh)2ds + θh 2

• ∂Q

nt |∇xvh|2 dxdt

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 8 / 29

Institute of Computational Mathematics

Coercivity

For vh ∈ V p

0h with p ≥ 2 we have

ah(vh, vh) =

• Q

∂tvh (vh + θh∂tvh) dxdt +

• Q

∇xvh · ∇x (vh + θh∂tvh) dxdt = θh∂tvh2

L2(Q) + ∇xvh2 L2(Q)

+

• Q

∂tvh vhdxdt + θh

• Q

∇xvh · ∇x∂tvhdxdt = θh∂tvh2

L2(Q) + ∇xvh2 L2(Q)

+ 1 2

• Q

∂t(vh)2dxdt + θh 2

• Q

∂t |∇xvh|2 dxdt = θh∂tvh2

L2(Q) + ∇xvh2 L2(Q)

+ 1 2

• ∂Q

nt(vh)2ds + θh 2

• ∂Q

nt |∇xvh|2 dxdt = θh∂tvh2

L2(Q) + ∇xvh2 L2(Q) + 1

2vh2

L2(ΣT ) + θh

2 ∇xvh2

L2(ΣT ).

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 8 / 29

Institute of Computational Mathematics

Numerical analysis

Coercivity: For vh ∈ V p

0h we have

ah(vh, vh) ≥ θh∂tvh2

L2(Q) + ∇xvh2 L2(Q) + 1

2vh2

L2(ΣT ) := vh2 h.

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 9 / 29

Institute of Computational Mathematics

Numerical analysis

Coercivity: For vh ∈ V p

0h we have

ah(vh, vh) ≥ θh∂tvh2

L2(Q) + ∇xvh2 L2(Q) + 1

2vh2

L2(ΣT ) := vh2 h.

Boundedness: For u ∈ H2,1(Q) + V p

0h and vh ∈ V p 0h we have

ah(u, vh) ≤ µbuh,∗vhh, with u2

h,∗ := u2 h + (θh)−1vh2 L2(Q).

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 9 / 29

Institute of Computational Mathematics

Numerical analysis

Coercivity: For vh ∈ V p

0h we have

ah(vh, vh) ≥ θh∂tvh2

L2(Q) + ∇xvh2 L2(Q) + 1

2vh2

L2(ΣT ) := vh2 h.

Boundedness: For u ∈ H2,1(Q) + V p

0h and vh ∈ V p 0h we have

ah(u, vh) ≤ µbuh,∗vhh, with u2

h,∗ := u2 h + (θh)−1vh2 L2(Q).

Consistency: For u ∈ H2,1(Q) we have ah(u, vh) = lh(vh) for all vh ∈ V p

0h.

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 9 / 29

Institute of Computational Mathematics

Error estimates

Erros estimate: u − uhh ≤ Chmin{s,p+1}−1uHs(Q) for s ≥ 2. Proof

◮ triangle inequality ◮ coercivity ◮ consistency → Galerkin orthogonality ◮ boundedness ◮ interpolation error estimates

Incredient: Inverse inequality

• C. Koutschan, M.N., C.S. Radu: Inverse Inequality Estimates with

Symbolic Computation (2016)

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 10 / 29

Institute of Computational Mathematics

Outline

Model problem Space-time method Numerical analysis Numerical examples Solvers Standard solvers Space-time multigrid method Conclusions and outlook

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 11 / 29

Institute of Computational Mathematics

Numerical example for d = 2

Exact solution: u(x1, x2, t) = sin(πx1) sin(πx2) sin(πt) for Q = (0, 1)3

p = 1 p = 2 Dofs u − uhh Rate Dofs u − uhh Rate 8 1.63740 27 2.15440e-01 27 7.39981e-01 1.15 64 2.11247e-01 0.03 125 3.60495e-01 1.04 216 3.98871e-02 2.40 729 1.79065e-01 1.01 1000 9.27926e-03 2.10 4913 8.92787e-02 1.00 5832 2.27556e-03 2.03 35937 4.45779e-02 1.00 39304 5.65772e-04 2.01 p = 3 p = 4 Dofs u − uhh Rate Dofs u − uhh Rate 64 2.16883e-01 125 6.67416e-03 125 2.75120e-02 2.97 216 7.69213e-03 0.03 343 5.09465e-03 2.43 512 3.55820e-03 3.55 1331 5.72742e-04 3.15 1728 3.26623e-05 4.10 6859 6.92964e-05 3.04 8000 2.05215e-06 4.00 42875 8.58214e-06 3.01 46656 1.37611e-07 4.00

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 12 / 29

Institute of Computational Mathematics

Numerical example for d = 2

Exact solution: u(x1, x2, t) = sin(πx1) sin(πx2) sin(πt) for Q = (0, 1)3

p = 1 p = 2 Dofs u − uhL2(Q) Rate Dofs u − uhL2(Q) Rate 8 3.65528e-01 27 2.39153e-02 27 9.56396e-02 1.93 64 2.37388e-02 0.01 125 2.32679e-02 2.03 216 1.99848e-03 3.57 729 5.75358e-03 2.01 1000 2.22710e-04 3.17 4913 1.43171e-03 2.01 5832 2.70486e-05 3.04 35937 3.57195e-04 2.00 39304 3.35780e-06 3.00 p = 3 p = 4 Dofs u − uhL2(Q) Rate Dofs u − uhL2(Q) Rate 64 2.39325e-02 125 4.75391e-04 125 2.03108e-03 3.56 216 4.73425e-04 0.01 343 2.68174e-04 2.92 512 3.34666e-05 3.82 1331 1.41715e-05 4.24 1728 8.74291e-07 5.26 6859 8.42223e-07 4.07 8000 2.60544e-08 5.07 42875 5.19528e-08 4.01 46656 8.07120e-10 5.01

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 13 / 29

Institute of Computational Mathematics

Outline

Model problem Space-time method Numerical analysis Numerical examples Solvers Standard solvers Space-time multigrid method Conclusions and outlook

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 14 / 29

Institute of Computational Mathematics

Algebraic multigrid solver

◮ Setting: Q = (0, 1)4 and p = 1 (FEM case) ◮ Solver:

GMRES with AMG prec. with rel. tol. 10−6

◮ Supercomputer:

Vulcan BlueGene/Q in Livermore, California U.S.A (thanks to Panayot Vassilevski)

◮ Software:

MFEM, AMG library hypre dofs u − uhL2(Q) rate iter time [s] cores 16 2.61353e−01

• 1

0.01 1 81 7.24784e−02 1.85 2 0.01 1 625 1.75301e−02 2.05 6 0.02 16 6 561 4.32537e−03 2.02 8 0.06 16 83 521 1.07679e−03 2.01 10 0.61 512 1 185 921 2.68823e−04 2.00 12 2.25 512 17 850 625 6.71720e−05 2.00 15 15.92 16 384 276 922 881 1.67895e−05 2.00 21 53.78 16 384 4 362 470 401 4.19714e−06 2.00 30 186.42 65 536

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 15 / 29

Institute of Computational Mathematics

Outline

Model problem Space-time method Numerical analysis Numerical examples Solvers Standard solvers Space-time multigrid method Conclusions and outlook

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 16 / 29

Institute of Computational Mathematics

Motivation

M.Gander and M.N.: Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems (2016)

Heat equation: ∂tu − ∆u = f in Q := Ω × (0, T), u = 0

• n Σ := ∂Ω × (0, T),

u = u0 in Ω.

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 17 / 29

Institute of Computational Mathematics

Motivation

M.Gander and M.N.: Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems (2016)

Heat equation: ∂tu − ∆u = f in Q := Ω × (0, T), u = 0

• n Σ := ∂Ω × (0, T),

u = u0 in Ω. FEM: Mhu′

h(t) + Khuh(t) = f h(t)

for t ∈ (0, T), uh(0) = u0.

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 17 / 29

Institute of Computational Mathematics

Motivation

M.Gander and M.N.: Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems (2016)

Heat equation: ∂tu − ∆u = f in Q := Ω × (0, T), u = 0

• n Σ := ∂Ω × (0, T),

u = u0 in Ω. FEM: Mhu′

h(t) + Khuh(t) = f h(t)

for t ∈ (0, T), uh(0) = u0. Approximate: u′

h(tn+1) ≈ 1

τ

• un+1 − un

.

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 17 / 29

Institute of Computational Mathematics

Motivation

M.Gander and M.N.: Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems (2016)

Heat equation: ∂tu − ∆u = f in Q := Ω × (0, T), u = 0

• n Σ := ∂Ω × (0, T),

u = u0 in Ω. FEM: Mhu′

h(t) + Khuh(t) = f h(t)

for t ∈ (0, T), uh(0) = u0. Approximate: u′

h(tn+1) ≈ 1

τ

• un+1 − un

. Implicit Euler: [Mh + τKh] un+1 = f n+1 + Mhun for n = 0, . . . , N − 1.

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 17 / 29

Institute of Computational Mathematics

Time stepping

Implicit Euler: [Mh + τKh] un+1 = f n+1 + Mhun for n = 0, . . . , N − 1. Sequential approach: t t0 t1 t2 t3 · · · · · · tN−1 tN

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 18 / 29

Institute of Computational Mathematics

Time stepping

Implicit Euler: [Mh + τKh] un+1 = f n+1 + Mhun for n = 0, . . . , N − 1. Sequential approach: t t0 t1 t2 t3 · · · · · · tN−1 tN Global linear system:        Aτ,h Bτ,h Aτ,h Bτ,h Aτ,h ... ... Bτ,h Aτ,h               u1 u2 u3 . . . uN        =        f 1 + Mhu0 f 2 f 3 . . . f N        with Aτ,h = Mh + τKh and Bτ,h = −Mh.

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 18 / 29

Institute of Computational Mathematics

Motivation

Example:

◮ Space: Ω = (0, 1)3 with 25 165 824 elements ◮ Time: τ = 10−2 with 4 096 time steps

◮ Space dofs:

4 243 841

◮ Space-time dofs: 17 382 772 736

◮ CG method with MG preconditioner (rel. tol. 10−10) ◮ Software: MFEM (hypre) ◮ computed on VULCAN (IBM Blue Gene/Q), LLNL, USA

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 19 / 29

Institute of Computational Mathematics

Motivation

Example:

◮ Space: Ω = (0, 1)3 with 25 165 824 elements ◮ Time: τ = 10−2 with 4 096 time steps

◮ Space dofs:

4 243 841

◮ Space-time dofs: 17 382 772 736

◮ CG method with MG preconditioner (rel. tol. 10−10) ◮ Software: MFEM (hypre) ◮ computed on VULCAN (IBM Blue Gene/Q), LLNL, USA

procs iter per time step

• verall

256 17 6.9 28 139.8 512 17 3.0 12 363.2 1 024 17 1.6 6 727.0 2 048 17 1.1 4 691.2 4 096 18 1.0 3 894.5 8 192 18 0.9 3 672.4 16 384 18 0.9 3 680.9

Table: Solving times in [s]

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 19 / 29

Institute of Computational Mathematics

Multigrid solver for the heat equation

Heat equation: ∂tu − ∆u = f in Q, u = 0

• n Σ,

u = u0

• n Σ0.

Linear system:        Aτ,h Bτ,h Aτ,h Bτ,h Aτ,h ... ... Bτ,h Aτ,h               u1 u2 u3 . . . uN        =        f 1 f 2 f 3 . . . f N        , Aτ,h := Mh ⊗ Kτ + Kh ⊗ Mτ, Bτ,h := −Mh ⊗ Nτ.

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 20 / 29

Institute of Computational Mathematics

Multigrid solver for the heat equation

Heat equation: ∂tu − ∆u = f in Q, u = 0

• n Σ,

u = u0

• n Σ0.

Linear system:        Aτ,h Bτ,h Aτ,h Bτ,h Aτ,h ... ... Bτ,h Aτ,h               u1 u2 u3 . . . uN        =        f 1 f 2 f 3 . . . f N        , Aτ,h := Mh ⊗ Kτ + Kh ⊗ Mτ, Bτ,h := −Mh ⊗ Nτ.

◮ Multigrid method in space and time [Hackbusch, 1984], [Horton, 1992], [Horton, Vandewalle, 1995], [Weinzierl, K¨

• ppl, 2012],

[R. D. Falgout, et al, 2014], [Gander, M.N., 2016],. . .

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 20 / 29

Institute of Computational Mathematics

Multigrid method

MGCycle(uL, f L) if L = 0 then Coarse grid solver: uL = A−1

L f L

else Pre-smoothing: uL = SL(uL, f L) Compute defect: dL = f L − ALuL Restriction: dL−1 = RLdL Initialize: w L−1 = 0 for i = 1, . . . , γ do MGCycle(w L−1,dL−1) Prolongation: w L = R⊤

L w L−1

Correction: uL = uL + w L Post-smoothing: uL = SL(uL, f L)

L = 0 L = 1 L = 2 L = 3

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 21 / 29

Institute of Computational Mathematics

Smoother

Damped Block Jacobi xn+1 = xn + ωtD−1

τ,h [f − Lτ,hxn] ,

Dτ,h := diag{Aτ,h}N

i=1

and ωt = 1 2.        Aτ,h Bτ,h Aτ,h Bτ,h Aτ,h ... ... Bτ,h Aτ,h               u1 u2 u3 . . . uN        =        f 1 f 2 f 3 . . . f N       

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 22 / 29

Institute of Computational Mathematics

Smoother

Damped Block Jacobi xn+1 = xn + ωtD−1

τ,h [f − Lτ,hxn] ,

Dτ,h := diag{Aτ,h}N

i=1

and ωt = 1 2.        Aτ,h Bτ,h Aτ,h Bτ,h Aτ,h ... ... Bτ,h Aτ,h               u1 u2 u3 . . . uN        =        f 1 f 2 f 3 . . . f N        → D−1

τ,h approximated by 1 space MG iteration.

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 22 / 29

Institute of Computational Mathematics

Smoother

Damped Block Jacobi xn+1 = xn + ωtD−1

τ,h [f − Lτ,hxn] ,

Dτ,h := diag{Aτ,h}N

i=1

and ωt = 1 2.        Aτ,h Bτ,h Aτ,h Bτ,h Aτ,h ... ... Bτ,h Aτ,h               u1 u2 u3 . . . uN        =        f 1 f 2 f 3 . . . f N        → D−1

τ,h approximated by 1 space MG iteration.

Fourier mode analysis:

◮ Smoother ◮ Two-Grid Operator

◮ Semi coarsening in time ◮ Coarsening in space and time

With respect to the discretization parameter µ := τ

h2 .

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 22 / 29

Institute of Computational Mathematics

Two-grid analysis

Semi coarsening in time: pt = 1 10−6 10−4 10−2 1 102 104 106 0.2 0.4 0.6 discretization parameter µ convergence factor ̺

ν = 1 ν = 2 ν = 5

• exp. two-grid ν = 1
• exp. two-grid ν = 2
• exp. two-grid ν = 5
• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 23 / 29

Institute of Computational Mathematics

Two-grid analysis

Space-time coarsening: pt = 1 10−6 10−4 10−2 1 102 104 106 0.2 0.4 0.6 0.8 1 discretization parameter µ convergence factor ̺

ν = 1 ν = 2 ν = 5

• exp. two-grid ν = 1
• exp. two-grid ν = 2
• exp. two-grid ν = 5
• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 24 / 29

Institute of Computational Mathematics

Space-time restriction and prolongation

Discretization parameter: µ = τ h2 .

◮ If µ < µ∗ only semi coarsening in time ◮ If µ ≥ µ∗ apply space-time coarsening

7 6 5 4 3 2 1 1 2 3 time levels space levels

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 25 / 29

Institute of Computational Mathematics

Parallelization

Communication pattern: x t

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 26 / 29

Institute of Computational Mathematics

Parallelization

Communication pattern: x t

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 26 / 29

Institute of Computational Mathematics

Example:

◮ Space: Ω = (0, 1)3 with 25 165 824 elements ◮ Time: τ = 10−2 with 4 096 time steps ◮ DOFs:

◮ Space:

4 243 841

◮ Space-time: 17 382 772 736

◮ computed on VULCAN (IBM Blue Gene/Q), LLNL, USA

processors space 256 512 1 024 2 048 4 096 8 192 16 384 processors time 2 − 22 544.2 15 251.3 13 034.1 11 901.9 11 533.7 12 687.4 4 18 934.3 10 983.6 7 328.4 6 198.5 5 407.3 5 363.6 5 748.3 8 9 497.6 5 415.3 3 600.4 3 034.0 2 580.4 2 589.4 2 748.6 16 4 747.0 2 688.0 1 787.1 1 503.1 1 266.0 1 280.1 32 2 386.8 1 345.8 892.2 752.5 630.5 64 1 190.6 673.5 449.2 379.1 128 598.9 342.5 228.4 256 302.5 173.3 512 153.8 256 512 1 024 2 048 4 096 8 192 16 384 28 139.8 12 363.2 6 727.0 4 691.2 3 894.5 3 672.4 3 680.9

Table: Solving times in [s]

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 27 / 29

Institute of Computational Mathematics

Parallelization

◮ Ω = (0, 1)3, τ = 10−1, pt = 3, space level 4 (49 152 elements) ◮ space multigrid settings: Gauß-Seidel ωx = 1.285, sx1 = sx2 = 1, γx = 1 ◮ time multigrid settings: ωt = 1, st1 = st2 = 1, γt = 1 ◮ computed on VULCAN (IBM Blue Gene/Q), LLNL, USA

Scaling results:

cores time steps dof iter time

• fwd. sub.

1 2 59 768 7 28.8 19.0 2 4 119 536 7 29.8 37.9 4 8 239 072 7 29.8 75.9 8 16 478 144 7 29.9 152.2 16 32 956 288 7 29.9 305.4 32 64 1 912 576 7 29.9 613.6 64 128 3 825 152 7 29.9 1 220.7 128 256 7 650 304 7 29.9 2 448.4 256 512 15 300 608 7 30.0 4 882.4 512 1 024 30 601 216 7 29.9 9 744.2 1 024 2 048 61 202 432 7 30.0 19 636.9 2 048 4 096 122 404 864 7 29.9 38 993.1 4 096 8 192 244 809 728 7 30.0 81 219.6 8 192 16 384 489 619 456 7 30.0 162 551.0 16 384 32 768 979 238 912 7 30.0 313 122.0 32 768 65 536 1 958 477 824 7 30.0 625 686.0 65 536 131 072 3 916 955 648 7 30.0 1 250 210.0 131 072 262 144 7 833 911 296 7 30.0 2 500 350.0 262 144 524 288 15 667 822 592 7 30.0 4 988 060.0

Table: Weak scaling.

cores time steps dof iter time 1 512 15 300 608 7 7 635.2 2 512 15 300 608 7 3 821.7 4 512 15 300 608 7 1 909.9 8 512 15 300 608 7 954.2 16 512 15 300 608 7 477.2 32 512 15 300 608 7 238.9 64 512 15 300 608 7 119.5 128 512 15 300 608 7 59.7 256 512 15 300 608 7 30.0 512 524 288 15 667 822 592 7 15 205.9 1 024 524 288 15 667 822 592 7 7 651.5 2 048 524 288 15 667 822 592 7 3 825.3 4 096 524 288 15 667 822 592 7 1 913.4 8 192 524 288 15 667 822 592 7 956.6 16 384 524 288 15 667 822 592 7 478.1 32 768 524 288 15 667 822 592 7 239.3 65 536 524 288 15 667 822 592 7 119.6 131 072 524 288 15 667 822 592 7 59.8 262 144 524 288 15 667 822 592 7 30.0

Table: Strong scaling.

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 28 / 29

Institute of Computational Mathematics

Summary and outlook

◮ Space-time isogeometric analysis

◮ Stability analysis ◮ Error analysis ◮ Standard solvers work quite well but are not robust

◮ Space-time multigrid method

◮ Robust ◮ Fully parallel method in space and time

• U. Langer, S.E. Moore and M.N., Space–time isogeometric analysis of parabolic

evolution problems (2016)

• C. Koutschan, M.N. and C.S. Radu: Inverse Inequality Estimates with Symbolic

Computation (2016) M.Gander and M.N.: Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems (2016)

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 29 / 29

Institute of Computational Mathematics

Summary and outlook

◮ Space-time isogeometric analysis

◮ Stability analysis ◮ Error analysis ◮ Standard solvers work quite well but are not robust

◮ Space-time multigrid method

◮ Robust ◮ Fully parallel method in space and time

• U. Langer, S.E. Moore and M.N., Space–time isogeometric analysis of parabolic

evolution problems (2016)

• C. Koutschan, M.N. and C.S. Radu: Inverse Inequality Estimates with Symbolic

Computation (2016) M.Gander and M.N.: Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems (2016)

Outlook:

◮ Space-time multigrid method for IgA ◮ Multilevel Monte Carlo in space and time

• M. Neum¨

uller Parallel Space-Time Methods Linz, Nov. 7, 2016 29 / 29