Deﬂation acceleration of the PCG method applied to porous...

Delft University of Technology

Deflation acceleration of the PCG methodapplied to porous media flow

Kees Vuik


Delft Institute of Applied Mathematics

SIAM Conference on Mathematical and Computational Issues in the Geosciences

Domain-based and Krylov-based Deflation Methods for Flow in Porous Media

March 20, 2007, Santa Fe, New Mexico

Kees Vuik, March 20, 2007 1 – p.1/29


Contents

1. Introduction

2. IC preconditioned CG

3. Deflated ICCG

4. Physical deflation vectors

5. Conclusions

Kees Vuik, March 20, 2007 2 – p.2/29


1. Introduction

MotivationKnowledge of the fluid pressure in rock layers is important for an oilcompany to predict the presence of oil and gas in reservoirs.

The earth’s crust has a layered structure

Kees Vuik, March 20, 2007 3 – p.3/29


Mathematical model

Computation of fluid pressure −div(σ∇p(x)) = 0 on Ω, p fluid pressure,σ permeability

δ ΩD

δ ΩN

δ ΩN

δ ΩN

Ω4

Ω7

Ω3

Ω5

Ω2

Ω6

Ω1

σh = 1 ( sand) σl = ε = 10−7 (shale)

Kees Vuik, March 20, 2007 4 – p.4/29


Properties and Applications

Ax = b

A is sparse and SPDCondition number of A is O(107), due to large contrast in permeability

Applications- reservoir simulations- porous media flow- electrical power networks- semiconductors- magnetic field simulations

- fictitious domain methods

Kees Vuik, March 20, 2007 5 – p.5/29


2. IC preconditioned CG

Error estimateAx = b

M−1Ax = M−1b

x − xk = (M−1A)−1M−1A(x − xk)

‖x − xk‖2 ≤ 1λmin

‖M−1rk‖2

λmin: smallest eigenvalue of M−1A

Kees Vuik, March 20, 2007 6 – p.6/29


Test problem

earth surface

shale

sandstone

shale

sandstone

shale

sandstone

sandstone

Configuration with 7 straight layers

Kees Vuik, March 20, 2007 7 – p.7/29


Convergence CG

0 5 10 15 20 25 30 35 4010

−6

10−5

10−4

10−3

10−2

10−1

100

101

102

estimate

||rk||

2

λmin

number of iterations

Convergence behavior of CG without preconditioning

Kees Vuik, March 20, 2007 8 – p.8/29


Convergence CG

100

101

102

103

104

10−10

10−5

100

105

estimate

||x−xk||

2

||rk||

2

λmin


Convergence behavior of CG without preconditioning

Kees Vuik, March 20, 2007 9 – p.9/29


Convergence ICCG

10 20 30 40 50 60

10−12

10−10

10−8

10−6

10−4

10−2

100

102

104

estimate

||x−xk||

2

||M−1rk||

2

λmin


Convergence behavior of ICCG

Kees Vuik, March 20, 2007 10 – p.10/29


Spectrum of IC preconditioned matrix

L is the Incomplete Cholesky factor of A

ks is the number of high-permeability domains not connected to aDirichlet boundary

D is a diagonal matrix (dii > 0) and A = D−1

2 AD−1

2

Theorem 1 (scaling invariance)L−1AL−T and L−1AL−T are identical.

Proof:

L = D−1

2 L and L−1AL−T = L−1D1

2 (D−1

2 AD−1

2 )D1

2 L−T = L−1AL−T .

Kees Vuik, March 20, 2007 11 – p.11/29


Spectrum of IC preconditioned matrix

Take D =diag(A)

Theorem 2A has ks eigenvalues of O(ε), where ε is the ratio between high andlow permeability.

Theorem 3The IC preconditioned matrix L−1AL−T has ks eigenvalues of O(ε).

Proof: Scaling invariance (Theorem 1) implies

spectrum(L−1AL−T ) = spectrum(L−1AL−T )

In [Vuik, Segal, Meijerink, Wijma, 2001] we have shown that thenumber and size of small eigenvalues of A and L−1AL−T are thesame. The theorem is proven by using Theorem 2.

Kees Vuik, March 20, 2007 12 – p.12/29


3. Deflated ICCG

Idea: remove the bad eigenvectors from the error/residual.

Krylov Ar

Preconditioned Krylov M−1Ar

Block Preconditioned Krylovm∑

i=1

(M−1i )Ar

Block Preconditioned Deflated Krylovm∑

i=1

(M−1i )PAr

Kees Vuik, March 20, 2007 13 – p.13/29


3. Deflated ICCG

Idea: remove the bad eigenvectors from the error/residual.

Various choices are possible:

• Projection vectorsPhysical vectors, eigenvectors, coarse grid projection vectors(constant, linear, ...)

• Projection methodDeflation, coarse grid projection, balancing, augmented, FETI

• Implementationsparseness, with(out) using projection properties, optimized, ...

Kees Vuik, March 20, 2007 13 – p.13/29


Literature

Deflated CG (start)

Nicolaides 1987, Mansfield 1990

Deflated CG (further development)

Graham and Hagger 1997, 1999, Kolotilina 1998, Vuik, Segal andMeijerink 1999, Saad, Yeung, Erhel and Guyomarc’h 2000, Frank andVuik 2001, Aksoylu, Rodriguez, Klie and Wheeler 2006, Nabben andVuik 2004, 2006, Scheichl and Graham, 2006

Deflation and restarted GMRES

Morgan 1995, Erhel, Burrage and Pohl 1996, Chapman and Saad1997, Morgan 2002, Giraud and Gratton 2005, Kilmer and De Sturler2006

Kees Vuik, March 20, 2007 14 – p.14/29


Deflated ICCG

A is SPD, Conjugate Gradients

P = I − AZE−1ZT with E = ZT AZ

and Z = [z1...zm], where z1, ..., zm are independent deflation vectors.

Properties

1. P T Z = 0 and PAZ = 0

2. P 2 = P

3. AP T = PA

Kees Vuik, March 20, 2007 15 – p.15/29


Deflated ICCG

x = (I − PT )x + PT x

(I − PT )x = ZE−1ZT Ax = ZE−1ZT b, AP T x = PAx = Pb

DICCGk = 0, r0 = Pr0, p1 = z1 = L−T L−1r0;

while ‖rk‖2 > ε dok = k + 1;αk = (rk−1,zk−1)

(pk,PApk) ;xk = xk−1 + αkpk;rk = rk−1 − αkPApk;zk = L−T L−1rk;βk = (rk,zk)

(rk−1,zk−1) ; pk+1 = zk + βkpk;

end while

Kees Vuik, March 20, 2007 16 – p.16/29


Convergence and termination criterion

Choose z1 , z2 , z3 eigenvectors of L−T L−1A

Convergence

‖PT x − PT xk‖2 ≤ 2√

K‖P T x − PT x0‖2

(√K − 1√K + 1

)k

where K = λn

λ4

Termination criterion

‖L−T L−1Pb − L−T L−1PAxk‖2 ≤ δ

λ4implies ‖P T x − PT xk‖2 ≤ δ

‘

Kees Vuik, March 20, 2007 17 – p.17/29


Deflation vectors

Choose eigenvectors of L−T L−1A. Properties of cross sections:• a constant value in sandstone layers• in shale layers their graph is linear

5 10 15 20 25 30 35 40

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Gridpoint in vertical direction

Val

ue

First eigenvector

Kees Vuik, March 20, 2007 18 – p.18/29


Eigenvectors of L−T

L−1

A

5 10 15 20 25 30 35 40

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25


Val

ue

Second eigenvector

5 10 15 20 25 30 35 40

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25


Val

ue

Third eigenvector

Kees Vuik, March 20, 2007 19 – p.19/29


4. Physical deflation vectors

k is number of subdomains

Ωi, i = 1, ..., ks high-permeability subdomains without a Dirichlet B.C.;i = ks + 1, ..., kh remaining high-permeability subdomains

• define zi for i ∈ 1, ..., ks• zi = 1 on Ωi and zi = 0 on Ωj , j 6= i, j ∈ 1, ..., kh• zi satisfies equation:

−div(σj∇zi) = 0 on Ωj , j ∈ kh + 1, ..., k,

with appropriate boundary conditions

Sparse vectors, subproblems are cheap to solve

Kees Vuik, March 20, 2007 20 – p.20/29


Physical deflation vectors

Example with ks = 2, kh = 3, and k = 5The geometry

δ ΩN

δ ΩN

δ ΩD

δ ΩNΩ

1Ω

2Ω

3Ω

4Ω

5

The first projection vector The second projection vector

Kees Vuik, March 20, 2007 21 – p.21/29


Physical deflation vectors

Example with ks = 2, kh = 3, and k = 5The first projection vector The second projection vector

Kees Vuik, March 20, 2007 21 – p.21/29


Properties

Theorem 4The deflation vectors are such that for D = diag(A)

• ‖D−1Azi‖∞ = O(ε)

• ‖L−T L−1Azi‖2 = O(ε)

Define Z = [z1...zks ] and U = [u1...uks ], where ui are ’small’eigenvectors.

Theorem 5

There is a matrix X such that Z = UX + E, with ‖E‖2 = O(√

ε)

Kees Vuik, March 20, 2007 22 – p.22/29


Sensitivity of deflation vectors

• Random vector added in shale layers (amplitude α/2)

α 0 10−1 1 ICCG

λper 0.164 0.164 8.2 · 10−3 1.6 · 10−9

iter 14 15 24 54

• Random vector added to the nonzero parts

α 0 10−3 10−1 ICCG

λper 0.164 9 · 10−4 9 · 10−8 1.6 · 10−9

iter 14 27 56 54

After perturbation the smallest eigenvalues remain exactly zero,however, the smallest non-zero eigenvalue can change considerably.

Kees Vuik, March 20, 2007 23 – p.23/29


Geometry oil flow problem

700

710

720

730

740

750

12601270

12801290

13001310

1320-6

-5

-4

-3

-2

-1

0

10

10-7

10-4

10-7

Permeability

Sandstone

Shale

Shale

Sandstone/Shale

Composition

Kees Vuik, March 20, 2007 24 – p.24/29


Results oil flow problem

Varying σshale

σ ICCG DICCG

λmin iter λmin iter

10−3 1.5 · 10−2 26 6.9 · 10−2 2010−5 2.2 · 10−4 59 7.7 · 10−2 2010−7 2.3 · 10−6 82 7.7 · 10−2 20

Varying accuracy

accuracy ICCG DICCG

iter CPU iter CPU

10−5 82 18.9 20 6.310−3 78 18.0 12 4.110−1 75 17.2 2 1.2

Kees Vuik, March 20, 2007 25 – p.25/29


A groundwater flow problem

The pressure in groundwater satisfies the equation:

−∇ · (A∇u) = F, (1)

where the coefficients and geometry of the problem are:

u=0

u=0

F=100

4

A=10-5

A=102

u=1 u=1A=10

Kees Vuik, March 20, 2007 26 – p.26/29


A groundwater flow problem

The low permeable layer (A = 10−5) and the jump in permeabilitiesbetween the two sand sections lead to a ’small’ eigenvalue.

0 6810

-10

10-8

10-6

10-4

10-2

100

102

104

0 3010

-10

10-8

10-6

10-4

10-2

100

102

104

λ 1

λ 2

error

error

approximatederror

errorapproximated

residualresidual

ICCG DICCG

Kees Vuik, March 20, 2007 27 – p.27/29


5. Conclusions

• DICCG is a robust and efficient method to solve diffusionproblems with discontinuous coefficients.

• The choice of the projection vectors is important for the successof a projection method.

• For layered problems the physical deflation vectors are theoptimal choice for the projection vectors.

• For many problems a second level preconditioner (Deflation)saves a lot of CPU time.

Kees Vuik, March 20, 2007 28 – p.28/29


Further information

• http://ta.twi.tudelft.nl/nw/users/vuik/pub_it_def.html

• C. Vuik, A. Segal and J.A. MeijerinkAn efficient preconditioned CG method for the solution of a classof layered problems with extreme contrasts in the coefficientsJ. Comp. Phys., 152, pp. 385-403, 1999.

• J. Frank and C. VuikOn the construction of deflation-based preconditionersSIAM Journal on Scientific Computing, 23, pp. 442–462, 2001

• R. Nabben and C. VuikA comparison of Deflation and Coarse Grid Correction applied toporous media flowSIAM J. on Numerical Analysis, 42, pp. 1631-1647, 2004

• R. Nabben and C. VuikA comparison of Deflation and the Balancing preconditionerSIAM Journal on Scientific Computing, 27, pp. 1742-1759, 2006

Kees Vuik, March 20, 2007 29 – p.29/29

Deﬂation acceleration of the PCG method applied to porous...

Documents

Transcript of Deﬂation acceleration of the PCG method applied to porous...