preconditioning of elliptic saddle point systems by ... · preconditioning of elliptic saddle point...
TRANSCRIPT
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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.
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• DD16 scientific & organizing committees
• Sandia– Rich Lehoucq– Pavel Bochev– Kendall Pierson– Garth Reese
• University– Jan Mandel
Thanks
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Overview
• Saddle Point Preconditioner– related penalized problem– positive real eigenvalues– conjugate gradients
• BDDC Preconditioner– overview– modified constraints
• Examples– H(grad), H(div), H(curl)
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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
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⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
− 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 −+−=
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⎟⎠
⎞⎜⎝
⎛=⎟
⎠
⎞⎜⎝
⎛⎟⎠
⎞⎜⎝
⎛ +⇒
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:
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⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
− p
u
p
uT
rr
zz
CBBA
~
)(~ 1
pup rBzCz −= −
regularized constraint preconditioning: Benzi survey (2005)
regularized constraint equation satisfied exactly
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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
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⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
−⎟⎟⎠
⎞⎜⎜⎝
⎛
−⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⎟⎟⎠
⎞⎜⎜⎝
⎛
−
−
−
−
−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
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⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
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
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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
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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:
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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
−−==
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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=
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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
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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.
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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
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FE discretizationof pde
elements partitioned into substructures
solve for unknowns on sub boundaries (Schur complement)
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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 ++=−
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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:
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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
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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 →∞
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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
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• 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
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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
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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
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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
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2D Structured Meshes
H(grad): Q2 - P1
H(grad): P2+ - P1H(div) & H(curl): RT0 & Ned
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2D Unstructured Mesh
2937 elements
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3D Unstructured Mesh
10194 elements
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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~−
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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~
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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
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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
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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
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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
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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=~
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⎟⎟⎠
⎞⎜⎜⎝
⎛=
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−=
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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
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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
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Unstructured Mesh Comparisons
178 (2.5)83D
125 (1.3)52D
GMRESPCGGMRES
MD1Mdimen
Note: ε = 10-5 for penalty preconditioner
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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
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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.
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Example128 elements, 208 edges, 81 nodes
Acurl Ap
80 zevals before49 zevals after
2 zevals before0 zevals after
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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)
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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
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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
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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