Preconditioning of Elliptic Saddle Point Systems by Substructuring and a Penalty Approach

1616thth International Conference on DomainInternational Conference on DomainDecomposition MethodsDecomposition Methods

January 12-15, 2005

Clark R. DohrmannStructural Dynamics Research Department

Sandia National LaboratoriesAlbuquerque, New Mexico

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy’s National Nuclear Security Administration

under contract DE-AC04-94AL85000.

• DD16 scientific & organizing committees

• Sandia– Rich Lehoucq– Pavel Bochev– Kendall Pierson– Garth Reese

• University– Jan Mandel

Thanks

Overview

• Saddle Point Preconditioner– related penalized problem– positive real eigenvalues– conjugate gradients

• BDDC Preconditioner– overview– modified constraints

• Examples– H(grad), H(div), H(curl)

Motivationoriginal system:

penalized (regularized) system: Axelsson (1979)

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

− gf

pu

CBBA T

~ CCC T ~~,0~ =>

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

− gf

pu

CBBA T

A > 0 on kernel of B, C ≥ 0AT = A, CT = C, B full rank

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

− p

u

p

uT

rr

zz

CBBA

~

)~(

)(~

11

1

p

T

uAu

pup

rCBrSz

rBzCz−−

−

+=

−=BCBAS T

A1~−+=

exact solution of penalized system

primal rather than dual Schur complement considered

is

)~( 1~

TC BBACS −+−=

⎟⎠

⎞⎜⎝

⎛=⎟

⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ +⇒

00f

pu

BBBBA TTρ

fuBBA T =+⇒ )( ρ

Some Connections

2/)()(2/ BuBufuAuuG TTT ρ+−=

penalty method for C = 0 and g = 0:

ρ/~ IC =matrix same as SA for

BupGL T+=augmented Lagrangian:

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

− p

u

p

uT

rr

zz

CBBA

~

)(~ 1

pup rBzCz −= −

regularized constraint preconditioning: Benzi survey (2005)

regularized constraint equation satisfied exactly

Penalty Preconditioner

)(~)~(

1

11

pup

pT

uAu

rBzCz

rCBrSz

−=

+=−

−−

BCBAS TA

1~−+=

recall exact solution of penalized system

⇒ preconditioner:

)(~)~(

1

11

pup

pT

uu

rBzCz

rCBrMz

−=

+=−

−−

M preconditioner for SA

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

−⎟⎟⎠

⎞⎜⎜⎝

⎛

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−

−

−

−p

uT

p

u

rr

ICBI

CM

IBCI

zz

1

1

1

1

1 0

~~00

~0

M

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

CBBA T

A

matrix representation:

AM 1−question: what about spectrum of

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

CCMSA ~0

0H

zz MA λ=

zz HAHM λ=−

symmetric

1

eigenproblem:

introduce: Bramble & Pasciak (1988), Klawonn (1998)

eigenvalues same as those for

can show H > 0 ⇒HM-1A > 0

equivalent linear system: HM-1Aw = HM-1b

CG Connection (B&P too)

original linear system: Aw = b

⇒ solve using pcg with H as preconditioner

eigenvalues of preconditioned system same as those of M-1A

H not available, but not a problem ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

CCMSA ~0

0H

kkkk

kkkk

kkkk

kkkk

prrpzz

prrpww

AHMAM

AA

1

1

1

1

1

1

~~ −

−

−

−

−

−

−=−=−=+=

αααα

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

−

−−

−

)(~)~(

1

11

1

pu

p

T

u

p

u

aBdCaCBaM

dd

aM

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−+−

=−

−

ppu

p

T

uuA

CdaBdaCBadS

a)~( 1

1HM

required calculations:

recurrences:

Theory

2

1

1 )( δσδ ≤≤ −AM

mTTTTT

nT

A

TT

ppBBMppCppBBMpuMuuuSuMuu

ℜ∈∀≤≤ℜ∈∀≤≤

−− 1

2

1

1

21

~ γγαα

Theorem (C = 0): Given α1 > 1, γ1 > 0, and

and σ1 > 0, σ2 > 0, σ1 + σ2 = 1

eigenvalues satisfy

where 121222

2212121

/)/2(,2max)/(),/(min

γασσαδγασαασδ

−−==

as ε→ 0

( ) 2/3)(2// 2

1

21 ασαα ≤≤ −AM

Simplification for

CBBS T

A

~1 →−

mTTTTT

nT

A

TT

ppBBMppCppBBMpuMuuuSuMuu

ℜ∈∀≤≤ℜ∈∀≤≤

−− 1

2

1

1

21

~ γγαα

recall with α1 > 1

⇒ γ1 bounded below by 1/α2 and γ2 bounded above by 1/α1

⇒

0

~ CC ε=

two goals:1. Ensure α1 > 1 (i.e. SA – M > 0)2. Minimize α2/α1 (M good preconditioner for SA)

potential issues:1. Scaling preconditioner M to satisfy Goal 12. SA becomes very poorly conditioned as ε→ 0

conclusion: effective preconditioner for SA is essential

Recap of Penalty Preconditioner

• Based on approximate solution of related penalized problem

• Conjugate gradients can be used to solve saddle point system

• Theory provides conditions for scalability

• Effectiveness hinges on preconditioner for primal Schur complement SA

Ref: C. R. Dohrmann and R. B. Lehoucq, “A primal based penalty preconditioner for elliptic saddle point systems,” Sandia National Laboratories, Technical report SAND 2004-5964J.

BDDC Overview

• Primal Substructuring Preconditioner– no coarse triangulation needed– recent theory by Mandel et al– multilevel extensions straightforward

• Numerical Properties– C(1+log(H/h))2 condition number bounds– preconditioned eigenvalues all ≥ 1 (woo hoo!)

• eigenvalues identical to FETI-DP– local and global problems only require sparse

solver for definite systems

M7

FE discretizationof pde

elements partitioned into substructures

solve for unknowns on sub boundaries (Schur complement)

Building Blocks

Additive Schwarz:– Coarse grid correction v1

– Substructure correction v2

– Static condensation correction v3

FETI-DP counterparts: kTIccI rcrc

FKFv λ1*

1

−

↔

∑=

−

↔s T

N

s

ksr

srr

sr BKBv

12

1

λ ↔3v Dirichlet preconditioner

3211 vvvrM ++=−

Coarse Grid and Substructure Problems

⎟⎠

⎞⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ΛΦ

⎟⎟⎠

⎞⎜⎜⎝

⎛IQ

QK

i

i

i

T

ii 00

iiTici KK ΦΦ=⇒

• Kci coarse element matrix• assemble Kci ⇒ Kc

• Kc positive definite

⎟⎠

⎞⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛00 2

2 i

i

i

i

T

ii rvQ

QKλ

substructure problem:

coarse problem:

Mixed Formulation of Linear Elasticity

fuBCBAf

pu

CBBA

T

T

=+⇒⎟⎠

⎞⎜⎝

⎛=⎟

⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛−

− )(0

1

• A + BTC-1B has same sparsity as A (discontinuous pressure p)

• BDD preconditioner with enriched coarse space investigated by Goldfeld (2003)

• related work for incompressible problems in Pavarino and Widlund (2002) and Li (2001), see also M7

BMBABCBA p

TT 1

0

1 −− +=+⇒ λµ

Let A = µA0 and C = (1/λ)Mp where

)21)(1()1(2 νννλ

νµ

−+=

+=

EE

recall

fuBCBA T =+ − )( 1

notice as ν → ½ that λ→∞ ⇒ condition number →∞

2D Plane Strain Example

6.2e30.49999996.2e20.499999

630.499997.20.49992.20.4992.10.491.80.42.00.3κν

condition number estimates for 4 substructure problem with H/h = 8

κ ≈ 1/(1 − 2ν) for ν near ½

Q: why does κ→∞ as ν → ½ for BDDC preconditioner?

of preconditioned system

• Explanation– Movement of substucture boundary nodes is weighted

average of coarse grid and substructure corrections(v1 and v2) from neighboring subs

– If substructure volume changes, then strain energy of substructure →∞ as ν→ ½

– cg step length α→ 0 since cg minimizes energy

• Solution– Modify “standard” BDDC constraint equations to enable

enforcement of zero volume change – Same number of constraints in 2D slightly more in 3D

condition number estimates for 4 substructure problem with H/h = 8

modifiedoriginal

2.32.32.32.32.32.21.82.0

6.2e30.49999996.2e20.499999630.499997.20.49992.20.4992.10.491.80.42.00.3

κν

Recap of BDDC Preconditioner

Ref: C. R. Dohrmann, “A substructuring preconditioner for nearly incompressible elasticity problems,” Sandia National Laboratories, Technical report SAND 2004-5393J.

• Performance sensitive to values of ν near ½ if “standard” constraints used

• Sensitivity caused by substructure volume changes for nearly incompressible materials

• Simple modification of constraints can effectively accommodate problems with ν near ½

Examples

• Problem Types– H(grad): incompressible elasticity– H(div): Darcy’s problem– H(curl): magnetostatics

• Preconditioning Approaches– Penalty/BDDC for H(grad) & H(div) – BDDC for stabilized H(curl) problem

2D Structured Meshes

H(grad): Q2 - P1

H(grad): P2+ - P1H(div) & H(curl): RT0 & Ned

2D Unstructured Mesh

2937 elements

3D Unstructured Mesh

10194 elements

Incompressible Elasticitymixed variational formulation:

qudxqwwdxfwdxpdxvu

∀∫ =⋅∇

∀∫ ∫ ∫ ⋅=⋅∇+

Ω

ΩΩ Ω

0)(:)(2 εεµ

where εi,j(u) = (ui,j + uj,i)/2, u,w ∈ H1(Ω), p,q ∈ L2(Ω), µ = 1

recall:

⎟⎠

⎞⎜⎝

⎛=

0BBA T

A

penalty preconditioner: = (2,2) block of A for ν < ½C~−

2D Incompressible Plane Strain Example

9 (2.6)90.49999

9 (2.7)90.4999

10 (2.7)100.499

11 (3.0)110.49

17 (7.2)150.4

23 (16)190.3

PCGGMRESν

iterations for rtol = 10-6 for 16 substructure problem with H/h = 8

Note: ν is Poisson ratio used to define in penalty preconditioner

C~

Other Saddle Point Preconditioners

block diagonal: Fortin, Silvester, Wathen, Klawonn, …

⎟⎟⎠

⎞⎜⎜⎝

⎛=

p

A

D MM0

0M

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=p

T

A

T MBM

0M

MA: BDDC preconditioner for A, Mp: pressure mass matrix

block triangular: Elman, Silvester, Klawonn, …

⇒ MINRES

⇒ GMRES

2D Structured Mesh Comparison

304711 (3.1)10256

304511 (3.1)10196

294210 (3.1)10144

284010 (3.0)10100

263810 (2.9)964

23359 (2.6)936

20308 (2.1)816

16266 (1.8)64

GMRESGMRESPCGGMRES

MTMDMN

iterations for rtol = 10-6 for N substructure problem with H/h = 4

Note: ν = 0.49999 for penalty preconditioner

Unstructured Mesh Comparisons

9514027 (37)233D

345311 (3.2)112D

GMRESGMRESPCGGMRES

MTMDMdim

Note: ν = 0.49999 in 2D and ν = 0.4999 in 3D for penalty preconditioner

Darcy’s Problemgoverning equations:

compatibility condition:

penalty preconditioner: chosen as εDs and BDDC for SAC~

u = −K∇ p in Ω∇⋅ u = f in Ωu ⋅ n = 0 on ∂Ω

∫Ω fdx = 0

discretization: lowest-order R-T simplicial elements

2D Darcy’s Problem Example

5 (1.5)50.00001

7 (1.9)60.0001

10 (5.8)100.001

20 (34)180.01

47 (2.1e2)380.1

100 (1.9e3)911

PCGGMRESε

iterations for rtol = 10-6 for 16 substructure problem with H/h = 8, K = I

Note: in penalty preconditioner

sDC ε=~

⎟⎟⎠

⎞⎜⎜⎝

⎛=

p

div

D MM

00

1M

⎟⎟⎠

⎞⎜⎜⎝

⎛=

Sp

A

D MM0

02M

Other Saddle Point Preconditioners

block diagonal 1: norm equivalence (Klawonn, 1995)

block diagonal 2:

some others: balancing Neumann-Neumann (BDD), overlapping methods (SPD reduction for div free space)

∫ ∫ ⋅∇⋅∇+⋅=Ω Ω

dxwuwdxuwu div ))((),(

TBBASp 1−=

2D Structured Mesh Comparison

107 (2.0)6256

107 (1.9)6196

107 (1.9)6144

106 (1.8)6100

106 (1.7)664

106 (1.6)536

84 (1.2)416

52 (1.01)24

GMRESPCGGMRES

MD1MN

iterations for rtol = 10-6 for N substructure problem with H/h = 4, K = I

510−=ε

similar results for 3D

2D Structured Mesh (H/h dependence)

6 (2.09)20

5 (1.92)16

5 (1.74)12

5 (1.53)8

4 (1.24)4PCGH/h

iterations for rtol = 10-6 for 9 substructure problem with K = I

similar results for 3D and K ≠ constant

Unstructured Mesh Comparisons

178 (2.5)83D

125 (1.3)52D

GMRESPCGGMRES

MD1Mdimen

Note: ε = 10-5 for penalty preconditioner

Magnetostatics (B = ∇ × u)

mixed variational form with Coulomb gauge (∇ ⋅ u = 0):

∇ ⋅ J = 0 ⇒ ∫J ⋅ ∇p = 0 and w = ∇p ⇒ p = 0 ⇒

)()())(/1( 0 curlHwwdxJpdxwdxwu ∈∀∫ ∫ ⋅=∇⋅+∫ ×∇⋅×∇Ω ΩΩ

µ

∫ ∈∀=∇⋅Ω

1

00 Hqqdxu

Note: ∇ ⋅ u was integrated by parts. Bummer, but …

Acurlx = b with Acurl ≥ 0, but system consistent and B unique

Q: How to solve Acurlx = b with Acurl ≥ 0?

One option: Solve in space restricted to ∇ ⋅ u = 0 ⇒ need basis for this space or back to saddle point system

Another option: Solve Acurl x = b using CG with preconditionerfor Ap

Ap = Acurl + Adiv

Advantage: can apply preconditioners for H(grad) problems!

Ref: C. R. Dohrmann, “Preconditioning of curl-curl equations by a penalty approach,” Sandia National Laboratories, Technical report, in preparation.

Example128 elements, 208 edges, 81 nodes

Acurl Ap

80 zevals before49 zevals after

2 zevals before0 zevals after

where σ > 0 ⇒ (Acurl + σAmass)x = b, σ → 0 equivalance, and precondition:

Some Other Options1. precondition saddle point system: plenary L13 (Zou)

2. introduce regularized problem: Rietzinger & Schöberl (2002)

∫ ∫ ⋅=⋅+∫ ×∇⋅×∇Ω ΩΩ

wdxJwdxudxwu σµ )())(/1(

• multigrid (Hiptmair, Arnold, Falk, Winther, R&S, …)• overlapping (Toselli, Hiptmair)• substructuring (Toselli, Widlund, Wohlmuth, Hu, Zou)• FETI-DP: plenary L11 (Toselli)

Recall Ap = Acurl + Adiv. Apply BCs and consider

)(1 curlcurl

1 ARxxAxxAx

p

T

T

∈∀≤≤α

Note: max α1 is smallest nonzero eig of Acurlw = λ Apw

0.79641/240.79661/200.79701/160.79771/120.79941/80.80641/4α1h

Note: PCG convergence depends on 1/α1 for exact solves w/ Ap

2D Structured MeshBDDC preconditioner for Ap with H/h = 4, rtol = 10-10

2.32.22.12.11.9

cond

1515141312

iter

0.79641/240.79661/200.79701/160.79771/120.79941/80.80641/4α1h

Similar results for 3D problems, OS too

Recap of Talk

• Penalty Preconditioner– approximate solution of related penalized problem– symmetric indefinite, but σ(M-1A) real and positive– CG for saddle point systems

• BDDC Preconditioner– well suited for use with penalty preconditioner– constraint modification for near incompressibility– applicable to problems in H(grad), H(div), and H(curl)

• Divergence Stabilization– permits use of H(grad) preconditioners for curl-curl