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Incomplete Factorization by Local Exact Factorization (ILUE) - - PowerPoint PPT Presentation

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Incomplete Factorization by Local Exact Factorization (ILUE) Johannes Kraus and Maria Lymbery "Modelling 2014" June 26, 2014, Roznov Pod

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

Incomplete Factorization by Local Exact Factorization (ILUE)

Johannes Kraus and Maria Lymbery "Modelling 2014" June 2–6, 2014, Roznov Pod Radhostem In honor of Professor Owe Axelsson’s 80th birthday

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

  • The ILUE preconditioner
  • General setting and definition
  • Properties
  • Application within ASMG preconditioning
  • Problem formulation
  • ASMG method for the weighted H(div)-norm
  • Numerical results

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results General setting and definition

Consider the linear system of algebraic equations

A y = f

where A = N

i=1 RT i AiRi and Ai, i = 1, . . . N, are SP(S)D matrices.

Incomplete factorization using exact local factorization (ILUE) BILUE := LU where U :=

N

  • i=1

RT

i UiRi,

L := UTdiag(U)−1, Ai = LiUi, diag(Li) = Ii.

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Properties

Aim: to estimate κ(B−1

ILUEA)

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Properties

Aim: to estimate κ(B−1

ILUEA)

A − BILUE ≥ 0

  • r

wTBILUEw ≤ wTAw ∀w

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Properties

Aim: to estimate κ(B−1

ILUEA)

A − BILUE ≥ 0

  • r

wTBILUEw ≤ wTAw ∀w wTAw ≤ c wTBILUEw ∀w

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Properties

Aim: to estimate κ(B−1

ILUEA)

A − BILUE ≥ 0

  • r

wTBILUEw ≤ wTAw ∀w wTAw ≤ c wTBILUEw ∀w c := λmax ncolor where

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Properties

Aim: to estimate κ(B−1

ILUEA)

A − BILUE ≥ 0

  • r

wTBILUEw ≤ wTAw ∀w wTAw ≤ c wTBILUEw ∀w c := λmax ncolor where ncolor is the coloring constant of the adjacency graph of subgraphs Gi color(vi) = color(vj) ⇔ G(Ai) ∩ G(Aj) = ∅

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Properties

Aim: to estimate κ(B−1

ILUEA)

A − BILUE ≥ 0

  • r

wTBILUEw ≤ wTAw ∀w wTAw ≤ c wTBILUEw ∀w c := λmax ncolor where ncolor is the coloring constant of the adjacency graph of subgraphs Gi color(vi) = color(vj) ⇔ G(Ai) ∩ G(Aj) = ∅ λmax := max

1≤i≤N{λi,max}

RT

i AiRiv = λiUTdiag(U)−1Uv

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation

Second order elliptic boundary value problem in mixed form u + K(x)∇p = 0 in Ω, div u = f in Ω, p = 0

  • n ΓD ,

u · n = 0

  • n ΓN.

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation

Second order elliptic boundary value problem in mixed form u + K(x)∇p = 0 in Ω, div u = f in Ω, p = 0

  • n ΓD ,

u · n = 0

  • n ΓN.
  • Ω is a domain in R2;

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation

Second order elliptic boundary value problem in mixed form u + K(x)∇p = 0 in Ω, div u = f in Ω, p = 0

  • n ΓD ,

u · n = 0

  • n ΓN.
  • n is the outward unit vector normal to the boundary ∂Ω;

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation

Second order elliptic boundary value problem in mixed form u + K(x)∇p = 0 in Ω, div u = f in Ω, p = 0

  • n ΓD ,

u · n = 0

  • n ΓN.
  • ∂Ω = ΓD ∪ ΓN;

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation

Second order elliptic boundary value problem in mixed form u + K(x)∇p = 0 in Ω, div u = f in Ω, p = 0

  • n ΓD ,

u · n = 0

  • n ΓN.
  • f ∈ L2 is the forcing term;

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation

Second order elliptic boundary value problem in mixed form u + K(x)∇p = 0 in Ω, div u = f in Ω, p = 0

  • n ΓD ,

u · n = 0

  • n ΓN.
  • K(x) : R2 → R2×2

SPD is the permeability tensor;

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation

Second order elliptic boundary value problem in mixed form u + K(x)∇p = 0 in Ω, div u = f in Ω, p = 0

  • n ΓD ,

u · n = 0

  • n ΓN.
  • p ∈ H1

0 is the fluid pressure;

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation

Second order elliptic boundary value problem in mixed form u + K(x)∇p = 0 in Ω, div u = f in Ω, p = 0

  • n ΓD ,

u · n = 0

  • n ΓN.
  • u ∈ H(div) is the velocity.

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation

Dual mixed weak form: Find u ∈ V and p ∈ W such that ADM(u, p; v, q) = −(f, q), for all (v, q) ∈ V × W, where ADM(u, p; v, q) : (V, W) × (V, W) → R is defined as ADM(u, p; v, q) := (αu, v) − (p, div v) − (div u, q), where α(x) = K−1(x) and V ≡ HN(div; Ω) = {v ∈ L2(Ω) : div v ∈ L2(Ω), and v · n = 0 on ΓN} W ≡ {q ∈ L2(Ω) and

  • Ω

q dx = 0 if ΓN = ∂Ω}.

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation

  • piecewise constant functions for the pressure variable
  • lowest order Raviart-Thomas functions for the velocity

Discrete system

  • Mα

Bdiv BT

div

u p

  • =
  • f
  • where

vTMαu = (αuh, vh) vTBT

divp = (ph, div vh)

Arnold-Falk-Winther Bh =   A I   uTAv = (α uh, vh) + (∇ · uh, ∇ · vh) = Λα(uh, vh) Efficient preconditioning of the system Au = b, u, b ∈ RN.

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results ASMG method for the weighted H(div)-norm

Ω =

n

  • i=1

Ωi A =

n

  • i=1

RT

i AiRi,

D = Df ⊕ Dc, A = A11 A12 A21 A22

  • ,

Ai = Ai:11 Ai:12 Ai:21 Ai:22

  • ,

i = 1, . . . , n.

  • A =

           A1:11 A1:12R1:2 A2:11 A2:12R2:2 ... . . . An:11 An:12Rn:2 RT

1:2A1:21

RT

2:2A2:21

. . . RT

n:2An:21 n

  • i=1

RT

i:2Ai:22Ri:2

           Setting A11 = diag{A1:11, . . . , An:11}, A22 = n

i=1 RT i:2Ai:22Ri:2 we have

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results ASMG method for the weighted H(div)-norm

Ω =

n

  • i=1

Ωi A =

n

  • i=1

RT

i AiRi,

D = Df ⊕ Dc, A = A11 A12 A21 A22

  • ,

Ai = Ai:11 Ai:12 Ai:21 Ai:22

  • ,

i = 1, . . . , n.

  • A =

           A1:11 A1:12R1:2 A2:11 A2:12R2:2 ... . . . An:11 An:12Rn:2 RT

1:2A1:21

RT

2:2A2:21

. . . RT

n:2An:21 n

  • i=1

RT

i:2Ai:22Ri:2

           Setting A11 = diag{A1:11, . . . , An:11}, A22 = n

i=1 RT i:2Ai:22Ri:2 we have

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results ASMG method for the weighted H(div)-norm

Ω =

n

  • i=1

Ωi A =

n

  • i=1

RT

i AiRi,

D = Df ⊕ Dc, A = A11 A12 A21 A22

  • ,

Ai = Ai:11 Ai:12 Ai:21 Ai:22

  • ,

i = 1, . . . , n.

  • A =

           A1:11 A1:12R1:2 A2:11 A2:12R2:2 ... . . . An:11 An:12Rn:2 RT

1:2A1:21

RT

2:2A2:21

. . . RT

n:2An:21 n

  • i=1

RT

i:2Ai:22Ri:2

           Setting A11 = diag{A1:11, . . . , An:11}, A22 = n

i=1 RT i:2Ai:22Ri:2 we have

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results ASMG method for the weighted H(div)-norm

  • A =

A11

  • A12
  • A21
  • A22
  • Incomplete Factorization by Local Exact Factorization (ILUE)
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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results ASMG method for the weighted H(div)-norm

A = R ART, R = R1 I2

  • ,

R1 =

  • RT

1:1

RT

2:1

. . . RT

n:1

  • Incomplete Factorization by Local Exact Factorization (ILUE)
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SLIDE 25

Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results ASMG method for the weighted H(div)-norm

  • A =

A11

  • A12
  • A21
  • A22
  • Incomplete Factorization by Local Exact Factorization (ILUE)
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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results ASMG method for the weighted H(div)-norm

  • A =

A11

  • A12
  • A21
  • A22
  • Additive Schur complement approximation (ASCA):

Q := A22 − A21 A−1

11

A12 =

n

  • i=1

RT

i:2(Ai:22 − Ai:21A−1 i:11Ai:12)Ri:2.

Let V = I RN and V = I R

N.

Π

D :

V → V, Π

D := (R

DRT)−1R D,

  • D =

D11 I

  • ,
  • D11 =

A11 or D11 = diag( A11)

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results ASMG method for the weighted H(div)-norm

Two-level auxiliary space preconditioner B−1 := M

−1 + (I − M−TA)C−1(I − AM−1)

where

  • C−1 = Π

D

A−1ΠT

  • D is the fictitious space preconditioner;
  • M is an A-norm convergent smoother, i.e., I − M−1AA < 1;
  • M = M(M + MT − A)−1MT is the corresponding symmetrized

smoother.

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

Subject to numerical testing: ∗ a randomly distributed coefficient α which is constant on each element, i.e. τ ∈ Th, α = ατ = 10−qrand, qrand ∈ {0, 1, 2, . . . , q}. Discretization parameters: ∗ a uniform mesh consisting of N×N squares, N = 8, . . . , 128; ∗ subdomains composed of 8 × 8 elements overlapping with half of their width and/or height. Solver parameters: ∗ a random start vector; ∗ a zero right-hand side; ∗ a residual reduction factor of 108.

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

Aim: Compare the performance of:

  • (scaled) one-level additive Schwarz preconditioner BAS,
  • ILU(0) preconditioner BILU ,
  • ILUE preconditioner BILUE

for

  • R1

A11RT

1 (respectively SR1

A11RT

1S)

  • and their efficiency within the ASMG algorithm.

BAS = SR1 S−1( S A11 S)−1 S−1RT

1S

S = [diag(A11)]−1/2

  • S = [diag(

A11)]−1/2

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

Scaled one-level additive Schwarz preconditioner BAS size of contrast A11 1 2 3 4 5 6 104 1 1 1 1 1 1 1 400 28 49 76 131 220 394 621 1568 62 130 251 543 >1000 >1000 >1000 6208 124 276 602 >1000 >1000 >1000 >1000 24704 250 557 >1000 >1000 >1000 >1000 >1000

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

ILU(0) preconditioner BILU size of non-zeros contrast A11 in U 1 2 3 4 5 6 104 424 21 34 52 52 66 70 77 400 1680 24 54 92 152 193 181 221 1568 6688 23 60 117 250 321 353 363 6208 26688 25 54 124 180 237 315 304 24704 106624 21 52 101 138 158 158 158

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

ILUE preconditioner BILUE size of non-zeros contrast A11 in U 1 2 3 4 5 6 104 630 1 1 1 1 1 1 1 400 2656 3 3 3 3 3 3 3 1568 10884 3 3 3 3 3 3 3 6208 44044 3 2 2 2 2 2 2 24704 177180 2 2 2 2 2 2 2

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

ASMG W-cycle: BAS contrast 1 2 3 4 5 6 ℓ=3 4 4 4 7 >100 >100 >100 ℓ=4 4 5 >100 >100 >100 >100 >100 ℓ=5 4 >100 >100 >100 >100 >100 >100 ℓ=6 >100 >100 >100 >100 >100 >100 >100

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

ASMG W-cycle: BILU contrast 1 2 3 4 5 6 ℓ=3 4 4 4 4 36 67 >100 ℓ=4 4 5 5 32 79 >100 >100 ℓ=5 4 4 5 >100 >100 >100 >100 ℓ=6 4 4 4 >100 >100 >100 >100

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

ASMG W-cycle: BILUE contrast 1 2 3 4 5 6 ℓ=3 4 4 4 4 4 4 4 ℓ=4 4 5 5 5 5 5 5 ℓ=5 4 4 5 5 5 5 5 ℓ=6 4 4 4 5 5 5 5

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

Summary: We have

Incomplete Factorization by Local Exact Factorization (ILUE)

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SLIDE 37

Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

Summary: We have

  • proposed an incomplete factorization preconditioner based on

exact local factorization;

Incomplete Factorization by Local Exact Factorization (ILUE)

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SLIDE 38

Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

Summary: We have

  • proposed an incomplete factorization preconditioner based on

exact local factorization;

  • estimated its relative condition number;

Incomplete Factorization by Local Exact Factorization (ILUE)

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SLIDE 39

Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

Summary: We have

  • proposed an incomplete factorization preconditioner based on

exact local factorization;

  • estimated its relative condition number;
  • tested it on linear systems arising from modelling of flows in

highly heterogeneous media;

Incomplete Factorization by Local Exact Factorization (ILUE)

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SLIDE 40

Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

Summary: We have

  • proposed an incomplete factorization preconditioner based on

exact local factorization;

  • estimated its relative condition number;
  • tested it on linear systems arising from modelling of flows in

highly heterogeneous media;

  • compared its performance against one-level additive Schwarz

and standard ILU preconditioners;

Incomplete Factorization by Local Exact Factorization (ILUE)

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SLIDE 41

Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

Summary: We have

  • proposed an incomplete factorization preconditioner based on

exact local factorization;

  • estimated its relative condition number;
  • tested it on linear systems arising from modelling of flows in

highly heterogeneous media;

  • compared its performance against one-level additive Schwarz

and standard ILU preconditioners;

  • demonstrated its robustness and uniformity within auxiliary

space multigrid method (ASMG).

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

Acknowledgements: Advanced Computing for Innovation, FP7 Capa- city Programme, Research Potential of Convergence Regions.

Incomplete Factorization by Local Exact Factorization (ILUE)

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Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results

Thank You! Thank You! Thank You! Thank You! Thank You! Thank You!

Incomplete Factorization by Local Exact Factorization (ILUE)