fast solvers and practical implementation in pde-constrained … · 2016. 6. 9. · outline i...

Lawrence Livermore National Laboratory

Fast solvers and practical implementation inPDE-constrained optimization

Andrew T. Barker

LLNL-PRES-693611. This work was performed under the auspices of the

U.S. Department of Energy under Contract DE-AC52-07NA27344. Lawrence

Livermore National Security, LLC

Outline

I Solvers for fully coupled time-dependent optimizationproblems. (with M. Stoll)

I H1 regularization in PDE-constrained optimization. (withT. Rees and M. Stoll)

I Optimization of Navier slip boundary conditions. (withH. Antil and S. W. Walker)

I Algebraic multigrid optimization solvers. (with A. Draganescu)

PART ONE

Solvers for fully coupled time-dependentoptimization problems

Joint work with:

I M. Stoll, Max Planck Institute for Dynamics of ComplexTechnical Systems, Magdeburg.

Time-dependent optimization

We are interested in finding a state y and a control u to minimizethe functional

J(y, u) =

∫ T

0‖y − y‖2 dt+

ω

2

∫ T

0‖u‖2 dt

subject to some time–dependent partial differential equation

L(y) = u

where y is a given desired state.

Examples

I Heat equation: L =∂

∂t−∆

I Convection–diffusion: L =∂

∂t− ε∆ + β · ∇

I Stokes: L ≈(∂/∂t

0

)+

(−∆ ∇∇· 0

)(with appropriate boundary conditions)

All-at-once approach

−∂λ∂t−A∗λ = −(y − y) (with final condition)

∂y

∂t−Ay = (1/ω)λ (with initial condition)

Taking liberties with boundary conditions and regularity in spaceand time, we can (formally) obtain the normal equations

−ω∂2y

∂t2+ (ωA+ ωA∗)

∂y

∂t+ (ωA∗A+ 1)y = y

(with boundary conditions in time)


We have been solving this problem with an “all-at-once” approach.If y is a vector that represents a spatial state, then we will solve asystem for the unknown block vector

[y1, y2, y3, . . . , u1, u2, u3, . . . , λ1, λ2, λ3]

So

number of unknowns = 3× (spatial degrees of freedom)

× (number of timesteps)

Memory use and parallel computing

N NT Nx iterations time (sec)

2.11e+08 256 274 625 13 39624.22e+08 512 15 63558.44e+08 1024 16 108791.69e+09 2048 19 21890

Heat equation

We want to minimize

J(y, u) =

∫ T

0‖y − y‖2L2(Ω) dt+

ω

2

∫ T

0‖u‖2L2(Ω) dt

subject to the heat equation

∂y

∂t−∆y = u

with some boundary conditions.

Discretization and Optimization of Heat Equation

I We discretize in space with quadrilateral finite elements.

I We discretize the time derivative with backward Euler.

I We integrate the functional in time with the trapezoid rule.

I We follow the discretize then optimize approach.

(Stoll and Wathen 2010)


The optimality system can be written τM1/2 0 −KT

0 ωτM1/2 τM−K τM 0

yuλ

=

τM1/2y

0d

Some of the pieces

K =

M + τK−M M + τK

−M M + τK. . .

. . .

−M M + τK

Some of the pieces

M =

M

MM

. . .

M


The optimality system can be written τM1/2 0 −KT


yuλ

=

τM1/2y

0d

This system is symmetric but indefinite and in serial it iswell-suited to preconditioned Minres.

Schur complement preconditioning

(A BT

B 0

)=

M1/2 0 −KT



P =

(A 00 BA−1BT

)=

τM1/2

ωτM1/2

S

where

S = KM−1KT +τ

ωMM−1

u MT


True Schur complement:

S = KM−1KT +τ

ωMM−1

u MT

[Dollar, Rees, Wathen 2010] (our heat and Stokes problems)

S = −KM−1KT

[Pearson, Wathen 2010] (our convection–diffusion problem)

S =

(K +

1√ωM)M−1

(KT +

1√ωM)


We need to approximate the inverse of the Schur complementBA−1BT ≈ KM−1KT . Recall

K =

M + τK−M M + τK

−M M + τK. . .

We can do a sweep through time with the help of an algebraicmultigrid preconditioner for M + τK.(Rees et. al. 2010, Stoll and Wathen 2010)


K =

M + τK−M M + τK

−M M + τK. . .

I This works fine, but it does not parallelize well.

I Our problems are too big to solve in serial.

Schwarz preconditioning

T1T2

δ

Schwarz preconditioning

If A is the large system matrix, then we define a one-level additiveSchwarz preconditioner

B−1 =

Np∑j=1

IjA−1j ITj

where Ij extends a function on a sub-timedomain to the wholetimedomain, and Aj represents an optimization problem with zeroinitial and final condition on the sub-timedomain.

Local subproblems

Aj =

(Aj BT

j

Bj 0

)Within the Schwarz preconditioner, we approximate the inverse ofAj using Minres, preconditioned by using the diagonal Schurcomplement approach discussed earlier.

Other approaches to this problem

I (Yang, Prudencio, Cai 2012) do parallel domaindecomposition in space.

I (Heinkenschloss 2005) breaks time into subdomains, but focusis not on parallel computation.

I (Du et. al. 2013) uses the Parareal method to precondition(only) the forward problem.

One–level Schwarz preconditioner

B−1 =∑j

IjA−1j ITj

I This preconditioner is in general indefinite.

I Minres requires a symmetric and positive definitepreconditioner, so will instead use the quasi-minimal residualmethod (QMR).

Symmetric, flexible QMR

I There is a special case of QMR with a symmetric matrix anda symmetric (but not necessarily positive definite!)preconditioner which reduces the cost to be comparable toMinres. (Freund and Nachtigal 1994)

I Another variation of QMR (flexible QMR) allows for thepreconditioner to change (a little) at each iteration. (Szyldand Vogel 2001)

I We will use both of these variations together.

Our approach in one slide

I Solve the global optimization system with flexible, symmetricQMR.

I The global preconditioner is a one–level additive Schwarzpreconditioner in time:

I Divide the time domain into a bunch of slices, give each sliceto a processor.

I Solve each subdomain problem with Minres.I Precondition the local subdomain problems with the Schur

complement approach.

Numerical results

I Running on a standard Linux cluster with 24 GB of memoryper node.

I Using deal.II and MPI.

I Key issue is memory use—we use 4 processes per node eventhough each node has 12 cores.

Numerical results

Slice of a 3D solution with ω = 10−6, desired state on the left,achieved solution on the right.

Numerical results

Slice of a 3D solution with ω = 10−6, control is shown.

[scale is from 0 to 2.01e+4]

Numerical results

Large problems running on 64 processors (16 nodes), overlap of 2timesteps, and ω = 10−4:

N NT Nx iterations time (sec)

2.11e+08 256 274 625 13 39624.22e+08 512 15 63558.44e+08 1024 16 108791.69e+09 2048 19 21890

Strong scaling

With 274625 spatial degrees of freedom and 256 timesteps, overlapof 2 timesteps, ω = 10−4:

cores iterations time (sec) time/iteration

8 8 10904.2 1363.016 8 6152.1 769.032 9 4035.8 448.464 13 3956.2 304.3

Weak scaling

With 274625 spatial degrees of freedom and overlap of 2:

NT cores iterations total time (sec) time/iteration

64 4 6 4096.2 682.7128 8 7 4850.3 692.9256 16 9 5532.8 614.8512 32 17 9952.0 585.4

Stokes equations

We want to minimize

J(y, u) =

∫ T

0‖y − y‖2L2(Ω) dt+

ω

2

∫ T

0‖u‖2L2(Ω) dt

subject to the Stokes system

∂y

∂t− ν∆y +∇p = u

∇ · y = 0,

with some boundary conditions.

Stokes equations

Stokes problem—strong scaling

In two dimensions, with 37507 spatial degrees of freedom,NT = 256 timesteps, overlap of 2, and ω = 10−4.


8 6 9580.1 1596.716 6 5480.9 913.532 6 3435.3 572.664 10 3206.3 320.6

Stokes problem—weak scaling

In two dimensions, with 37507 spatial degrees of freedom,NT = 256 timesteps, overlap of 2, and ω = 10−4.

NT cores iterations time (sec) time/iteration

33 2 4 2208.1 552.065 4 4 3125.3 781.3129 8 5 4331.2 866.2257 16 6 5541.2 923.5513 32 11 11321.4 1029.21025 64 18 17496.6 972.0

Convection–diffusion

Strong scaling, convection–diffusion

ε = 0.1, ω = 10−3, 32768 spatial degrees of freedom, overlap of 2timesteps:


4 6 5577.2 929.58 6 2946.2 491.016 6 1663.9 277.332 6 985.7 164.364 8 915.1 114.4

Weak scaling, convection–diffusion

ε = 0.1, ω = 10−3, 32768 spatial degrees of freedom, overlap of 2timesteps:

NT cores iterations time (sec) time/iteration

63 2 4 1542.3 385.6127 4 5 2570.2 514.0255 8 6 2974.8 495.8511 16 7 3333.1 476.21023 32 9 4099.9 455.52047 64 13 5760.5 443.1


Time per iteration.

Robustness with ω for convection–diffusion

32768 spatial degrees of freedom, 32 cores, ε = 0.1.

ωNT 10−2 10−3 10−4 10−6

127 11 7 4 3255 12 8 7 3511 14 10 6 31023 14 11 7 3

Interplay of ω and ε for convection–diffusion

NT = 127, 32768 spatial degrees of freedom, running on 32 cores.

ωε 0.1 0.01 10−3 10−4 10−6

1.0 7 7 6 4 30.1 33 11 7 4 310−2 18 14 7 5 310−4 32 13 7 4 3

Reference

A. T. Barker and M. Stoll, Domain decomposition in time forPDE–constrained optimization, Computer Physics Communications197 (2015) pp. 136–143.

PART TWO

H1 regularization in PDE-constrained optimization

Joint work with:

I T. Rees, Rutherford Appleton Laboratory, Oxford.

I M. Stoll, Max Planck Institute for Dynamics of ComplexTechnical Systems, Magdeburg.

H1 regularization in PDE-constrained optimization

Now we minimize

J(y, u) =

∫ T

0‖y − y‖2L2

dt+ω

2

∫ T

0‖u‖2H1 dt

subject to the heat equation.

All at once system

τM 0 −KT

0 ωτ(Mu +Ku) τM−K τM 0

yuλ

=

τMy0d

(being a bit sloppy with some timestep scaling)


The true Schur complement is

S =1

τKM−1K +

τ

ωM(Mu +Ku)−1MT

For large ω we can just use the first term

S =1

τKM−1K

But this is not robust with respect to ω.


Following the ideas of (Pearson, Stoll, Wathen 2012).The true Schur complement is

S =1

τKM−1K +

τ

ωM(Mu +Ku)−1MT

So we will look for matrices X1,X2 so that

S =1

τ(K + X1)M−1(K + X2)

is close to S.


Skipping all the details, we end up with

S =1

τ

(K +

τ√ωM(Mu +Ku)−1M

)M−1

(K +

τ√ωM)T

We have to approximate the inverse of this.

A few results

Iteration counts for H1 regularized heat equation optimization:

dofs ω = 10−2 ω = 10−4 ω = 10−6

1089 13 13 224225 13 15 2216641 15 15 2566049 17 20 31263169 19 22 34

Reference

A. T. Barker, T. Rees and M. Stoll, A fast solver for an H1

regularized PDE–constrained optimization problem,Communications in Computational Physics, 19 (2016) pp.143–167.

PART THREE

Optimization of Navier slip boundary conditions

Joint work with:

I H. Antil, George Mason University.

I S. W. Walker, Louisiana State University.

Superhydrophobic materials

490 A. Busse, N. D. Sandham, G. McHale and M. I. Newton

Trappedgas

Flowing liquid

Interface Surface structure

FIGURE 1. Illustration of an idealised superhydrophobic surface.

hh

RGas

Gas

Gas Gas Gas

Liquid

U

2h LiquidLiquid Liquid

(a) (b) (c) (d)

FIGURE 2. Basic flow geometries: (a) Couette flow, (b) symmetric pressure-driven channelflow, (c) one-sided pressure-driven channel flow, (d) pipe flow.

previous approaches (Joseph, Nguyen & Beavers 1984; Than, Rosso & Joseph 1987;Vinogradova 1999) where a finite mass flow rate is allowed in the air layer.

2. Basic assumptionsA superhydrophobic surface may be modelled as a continuous air layer of constant

thickness δ superimposed on a wall (see figure 1). In the following analysis thesupporting structure of a superhydrophobic surface is neglected; only its beneficialeffects are kept, i.e. retaining an air layer at the surface which is undeformable, andthus suppressing instabilities at the air–water interface. The potentially drag-increasingproperties of the surface structure due to its roughness are neglected. By neglectingall potentially adverse effects this acts as a model for an ‘optimal’ superhydrophobicsurface.

While the present investigation is mainly aimed at superhydrophobic surfaces,similar problems occur in other contexts as discussed in § 1. Therefore, the generalproblem of the laminar flow of two immiscible fluids is investigated. The first fluidflows over an infinite layer of constant thickness of the second fluid, which has alower dynamic viscosity and acts as a lubricant for the flow of the first fluid. In thefollowing, the first fluid will be called ‘liquid’. The second fluid will be referred toas ‘gas’. The names ‘liquid’ and ‘gas’ are adapted only for ease of nomenclature. Thesecond fluid need not be a gas; in the case of the oil-water problem both the first andthe second fluid would be liquids.

Four different basic flow configurations, illustrated in figure 2, will be studied:

(i) Couette flow with the lower wall covered by a gas layer;

I Among many applications is drag reduction.

I They are nanoscale materials and are extremely expensive toproduce.

Model

I Flow past superhydrophobic materials can be modeled with aNavier boundary condition.

u · ν = 0

1

γTu+ Tσ(u, p)ν = 0

I These kind of boundary conditions are fairly well studiedmathematically, but very little has been done computationally

I We aim to apply PDE-constrained optimization to determineif using these materials can be economically beneficial, and ifso, where to use these materials and how much to use.

Optimization problem

We aim to minimize

J(u, p, γ) = drag(u, p) + ωcost(γ)

subject to

steady-state Stokes equations

u · ν = 0

1

γTu+ Tσ(u, p)ν = 0

Regularization parameter ω = 0.01

Regularization parameter ω = 1.0e+ 4

PART FOUR

Algebraic multigrid for PDE-constrainedoptimization

Joint work with:

I A. Draganescu, University of Maryland, Baltimore County.

The problem

Given a desired state y, find a (distributed) control u to minimizethe functional

J =1

2‖y − y‖2Y +

ω

2‖u‖2U

subject to the constraint

Ay = Myuu,

where A is an (elliptic) spatial operator on the state space.

The reduced space problem

The solution operator for the state equation is K = A−1Myu.The reduced space problem is then

Gu = (K∗K + ωI)u = K∗y,

WhereK∗ = M−1

u KTMy

We are interested in preconditioning the Hessian G.

Multigrid for the Hessian

I We would like to form a coarse Hessian GH .

I We cannot apply algebraic multigrid directly to G, because wedon’t want to form it.

I We follow the geometric multigrid of (Draganescu andDupont, 2008).

AMG for the Hessian

Given an AMG prolongator P , we can coarsen:

I AH = P TAP

I MH = P TMP

I KH = A−1H MH

I GH = (K∗HK + ωI)

And define a two-level preconditioner by

M−1h = P (GH)−1Π +

1

ω(I − PΠ),

whereΠ = M−1

H P TM.

Implementation

I We use a multilevel (W-cycle based) version of this algorithm.

I The discretization is standard low order Lagrange finiteelements.

I The AMG is Hypre BoomerAMG with default parameters.

I We cheat by using (almost) the same space for state andcontrol, so we can reuse mass matrices.

I We solve the Poisson problem on a 3D cube with anunstructured mesh on 8 processors.

Results

Iteration counts for unpreconditioned CG for the Hessian

ωrefine 0.0001 0.01 1.0 100.0

2 60 53 53 533 62 60 60 604 61 65 65 655 57 66 67 67

Results

Iteration counts for CG preconditioned with our multilevelpreconditioner

ωrefine 0.0001 0.01 1.0 100.0

2 11 4 2 23 12 4 2 24 12 4 2 25 12 4 2 2

Results

Solve times for unpreconditioned CG

ωrefine 0.0001 0.01 1.0 100.0

2 1.4565 1.3022 1.2594 1.26543 8.3637 8.0912 8.2210 8.09704 98.7817 119.7712 105.5309 119.51205 1114.9886 1526.1242 1513.2152 1325.6787

Results

Solve times for CG preconditioned with our multilevelpreconditioner

ωrefine 0.0001 0.01 1.0 100.0

2 0.4549 0.1830 0.1019 0.10423 2.4991 0.9040 0.5052 0.55594 27.7249 9.6749 5.0136 5.10485 287.4749 104.0656 59.8052 58.5670

Conclusions

Solvers in PDE-constrained optimization are important for

I Large 3D problems

I Time-dependent problems

I Robustness with respect to parameters

Solvers for the forward problem are not enough. For realisticoptimization problems you need more.

fast solvers and practical implementation in pde-constrained … · 2016. 6. 9. · outline i...

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