Preconditioned Inverse Iteration for Hierarchical Matrices
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Transcript of Preconditioned Inverse Iteration for Hierarchical Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Preconditioned Inverse Iteration for H-Matrices
Peter Benner and Thomas Mach
Mathematics in Industry and TechnologyDepartment of Mathematics
TU Chemnitz
23rd Chemnitz FEM Symposium 2010
1/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
FÜR DYNAMIK KOMPLEXERTECHNISCHER SYSTEME
MAGDEBURG
MAX−PLANCK−INSTITUT
Hierarchical (H-)Matrices PINVIT Numerical Results
Preconditioned Inverse Iteration for H-Matrices
Peter Benner and Thomas Mach
Computational Methods in Systems and Control TheoryMax Planck Institute for Dynamics of Complex Technical Systems
Magdeburg
23rd Chemnitz FEM Symposium 2010
1/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Outline
1 Hierarchical (H-)Matrices
2 PINVIT
3 Numerical Results
2/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
H-Matrices [Hackbusch 1998]
Some dense matrices, e.g. BEM or FEM, can be approximated byH-matrices in a data-sparse manner.
hierarchical tree TI block H-tree TI× I
I = {1, 2, 3, 4, 5, 6, 7, 8}
{1, 2, 3, 4} {5, 6, 7, 8}
{1, 2} {3, 4} {5, 6} {7, 8}
{1}{2}{3}{4}{5}{6}{7}{8}
1 1 1 12 2 2 23 3 3 34 4 4 45 5 5 56 6 6 67 7 7 78 8 8 8
1 1 1 12 2 2 23 3 3 34 4 4 45 5 5 56 6 6 67 7 7 78 8 8 8
dense matrices, rank-k-matrices
rank-k-matrix: Ms×t = ABT , A ∈ Rn×k ,B ∈ Rm×k (k � n,m)
3/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
H-Matrices [Hackbusch 1998]
Hierarchical matrices
H(TI× I, k) ={
M ∈ RI× I∣∣ rank (Ma×b) ≤ k ∀a× b admissible
}22
3 3
7 10
3 7
3 10
19 10
10 31
14 8
11 11
14 118 11
19 10
10 31 11
11 31
11 9
9 16 12
11 1611 8
9 16 11
11 16
619 73 3
8
11 11
9 3
7 3
11
8 11
25 10
10 19 11
11 31
8 5
11 8
8 15 8 12
116 515 6
13
13
7 5
13 8
8 11 8 11
116 5
15 6
8
11 8
8 15
5 8 11
126 15
5 6 13
137
13 8
8 11
5 8 11
116 15
5 6
6110 103
6 14
10 3 6
10 14
25 10 6
10
6
19 10
10 31
15 9
9 11 10
10 1615 9
8 11 10
10 16
5110 107 9 7
3 3
10 7
109 3
7 3
25 11
1125 10
10 19
11 811 8
8 15 9
8 15 12 13
1310 7
13 8
8 11 9
9 15 1119
2010 7
13 8
9
11 8
8 15 11 12
139 7
13 8
9
13 8
8 1111
1011 8
8 15 8
9 15
7 12 13
1310
13 8
8 11 8
9 15
7 1120
199
13 9
8
11 8
8 15
7 11 12
129
13 9
813 8
8 11
7 11
39 10
10 25
3 7
3 10 107 10 6
3 3
7 10 7
1010
6
22
3 3
7 10
3 7
3 10
19 10
10 31
15 10
11
11 9
9 16
15 11
1011 8
9 16
34 10
10 25
13 10
7 1113 7
10 11 61
6 513 6 11
12
8 5
11 8
8 15 812
136 5
13 6 11
11 236 13
5 6 12
118
11 8
8 15
5 813
12
6 13
5 6 10
11 23
20 9
9 39
9 7
103 7
3 10
9 107
3 3
7 10 61
15 10
1015 9
9 11
15 10
1015 9
8 11
20 9
9 34
9 7
10 13
9 107 13 51
adaptive rank k(ε)
storage NSt,H(T , k) = O(n log n k(ε))
complexity of approximate arithmetic
MHv O(n log n k(ε))+H,−H O(n log n k(ε)2)
∗H,HLU(·), (·)−1H O(n (log n)2 k(ε)2)
4/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Goal: Eigenvalues of symmetric H-Matrices
M = MT ∈ H(T , k),
Λ(M) = {λ1, λ2, . . . , λn},λ1 ≥ λ2 ≥ · · · ≥ λn > 0
⇓How to find λi in O(n (log n)α kβ)?
X
5/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Goal: Eigenvalues of symmetric H-Matrices
M = MT ∈ H(T , k),
Λ(M) = {λ1, λ2, . . . , λn},λ1 ≥ λ2 ≥ · · · ≥ λn > 0
⇓How to find λn in O(n (log n)α kβ)?
X
5/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Preconditioned Inverse Iteration[Knyazev, Neymeyr, et al.]
Definition
The function
µ(x) = µ(x ,M) =xTMx
xT x
is called the Rayleigh quotient.
Minimize the Rayleigh quotient by a gradient method:
xi+1 := xi − α∇µ(xi ), ∇µ(x) =2
xT x(Mx − xµ(x)) ,
+ preconditioning ⇒ update equation:
xi+1 := xi − B−1 (Mxi − xiµ(xi )) .
Residual r(x) = Mx − xµ(x).
Preconditioned residual B−1r(x) = B−1 (Mx − xµ(x)).
6/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Preconditioned Inverse Iteration[Knyazev, Neymeyr, et al.]
Definition
The function
µ(x) = µ(x ,M) =xTMx
xT x
is called the Rayleigh quotient.
Minimize the Rayleigh quotient by a gradient method:
xi+1 := xi − α∇µ(xi ), ∇µ(x) =2
xT x(Mx − xµ(x)) ,
+ preconditioning ⇒ update equation:
xi+1 := xi − B−1 (Mxi − xiµ(xi )) .
Residual r(x) = Mx − xµ(x).
Preconditioned residual B−1r(x) = B−1 (Mx − xµ(x)).
6/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Preconditioned Inverse Iteration[Knyazev, Neymeyr, et al.]
Definition
The function
µ(x) = µ(x ,M) =xTMx
xT x
is called the Rayleigh quotient.
Minimize the Rayleigh quotient by a gradient method:
xi+1 := xi − α∇µ(xi ), ∇µ(x) =2
xT x(Mx − xµ(x)) ,
+ preconditioning ⇒ update equation:
xi+1 := xi − B−1 (Mxi − xiµ(xi )) .
Residual r(x) = Mx − xµ(x).
Preconditioned residual B−1r(x) = B−1 (Mx − xµ(x)).
6/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Preconditioned Inverse Iteration[Knyazev, Neymeyr, et al.]
Definition
The function
µ(x) = µ(x ,M) =xTMx
xT x
is called the Rayleigh quotient.
Minimize the Rayleigh quotient by a gradient method:
xi+1 := xi − α∇µ(xi ), ∇µ(x) =2
xT x(Mx − xµ(x)) ,
+ preconditioning ⇒ update equation:
xi+1 := xi − B−1 (Mxi − xiµ(xi )) .
Residual r(x) = Mx − xµ(x).
Preconditioned residual B−1r(x) = B−1 (Mx − xµ(x)).
6/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Preconditioned Inverse Iteration[Knyazev, Neymeyr, et al.]
Definition
The function
µ(x) = µ(x ,M) =xTMx
xT x
is called the Rayleigh quotient.
Minimize the Rayleigh quotient by a gradient method:
xi+1 := xi − α∇µ(xi ), ∇µ(x) =2
xT x(Mx − xµ(x)) ,
+ preconditioning ⇒ update equation:
xi+1 := xi − B−1 (Mxi − xiµ(xi )) .
Residual r(x) = Mx − xµ(x).
Preconditioned residual B−1r(x) = B−1 (Mx − xµ(x)).
6/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Preconditioned Inverse Iteration[Knyazev, Neymeyr 2009]
xi+1 := xi − B−1 (Mxi − xiµ(xi ))
If
M symmetric positive definite and
B−1 approximates the inverse of M, so that∥∥I − B−1M∥∥
M≤ c < 1,
then Preconditioned INVerse ITeration (PINVIT) converges.
7/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Preconditioned Inverse Iteration[Knyazev, Neymeyr 2009]
xi+1 := xi − B−1 (Mxi − xiµ(xi ))
If
M symmetric positive definite and
B−1 approximates the inverse of M, so that∥∥I − B−1M∥∥
M≤ c < 1,
then Preconditioned INVerse ITeration (PINVIT) converges.
7/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Preconditioned Inverse Iteration[Knyazev, Neymeyr, et al.]
The residual
ri = Mxi − xiµ(xi )
converges to 0, so that
‖ri‖2 < ε
is a useful termination criterion.
8/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Why is it called Preconditioned Inverse Iteration?
Inverse Iteration:
xi+1 = µ(xi )A−1xi ,
xi+1 = xi − A−1Axi︸ ︷︷ ︸=0
+µ(xi )A−1xi ,
xi+1 = xi − A−1 (Axi + µ(xi )xi ) .
Replace A−1 with the inexact solution by the preconditioner B:
xi+1 = xi − B−1 (Axi + µ(xi )xi ) .
9/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Variants of PINVIT[Neymeyr 2001: A Hierarchy of Preconditioned Eigensolvers for Elliptic Diff. Operators]
Classification by Neymeyr:
PINVIT(1): xi+1 := xi − B−1ri .
PINVIT(2): xi+1 := arg minv∈span{xi ,B−1ri} µ(v).
PINVIT(3): xi+1 := arg minv∈span{xi−1,xi ,B−1ri} µ(v).
PINVIT(n): Analogously.
PINVIT(·,d): Replacing x by a rectangular full rank matrixX ∈ Rn×d one gets the subspace version of PINVIT(·).
10/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Algorithm
H-PINVIT(1,d)
Input: M ∈ Rn×n, X0 ∈ Rn×d (e.g. randomly chosen)Output: Xp ∈ Rn×d , µ ∈ Rd×d , with ‖MXp − Xpµ‖ ≤ εB−1 = (M)−1
HOrthogonalize X0
R := MX0 − X0µ, µ = XT0 MX0
for (i := 1;‖R‖F > ε; i + +) doXi := Xi−1 − B−1ROrthogonalize Xi
R := MXi − Xiµ, µ = XTi MXi
end
11/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Complexity
The number of iterations is independent of matrix size n.
Once:
adaptive H-matrix inversion: O(n (log n)2 k (c)2).
Per iteration:
H-matrix-vector products with M and B−1: O(n (log n) k (c))and O(n (log n) k (c)2), and
some dense arithmetic handling vectors of length n.
The complexity of the algorithm is determined by the H-matrixinversion: ⇒ O(n (log n)2 k (c)2).
Competitive to MATLAB ® eigs.
Expensive.
12/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Goal: Eigenvalues of symmetric H-Matrices
M = MT ∈ H(T , k),
Λ(M) = {λ1, λ2, . . . , λn},λ1 ≥ λ2 ≥ · · · ≥ λn > 0
⇓How to find λn in O(n (log n)α kβ)?
X
13/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Goal: Eigenvalues of symmetric H-Matrices
M = MT ∈ H(T , k),
Λ(M) = {λ1, λ2, . . . , λn},λ1 ≥ λ2 ≥ · · · ≥ λn > 0
⇓How to find λn in O(n (log n)α kβ)? X
13/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Goal: Eigenvalues of symmetric H-Matrices
M = MT ∈ H(T , k),
Λ(M) = {λ1, λ2, . . . , λn},λ1 ≥ λ2 ≥ · · · ≥ λn > 0
⇓How to find λi in O(n (log n)α kβ)?
X
14/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Folded Spectrum Method
How to find λi in O(n (log n)α kβ)?
If i = n − d with d < O(log n),
use subspace version PINVIT(·,d).
0λn λn−1λn−2. . .
λ1
But (M − σI) is not positive definite!
15/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Folded Spectrum Method
How to find λi in O(n (log n)α kβ)?
If i = n − d with d � log n?
shift with σ near λi .
0λn. . .
λi+1 λi λi−1. . .
λ1
But (M − σI) is not positive definite!
15/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Folded Spectrum Method
How to find λi in O(n (log n)α kβ)?
If i = n − d with d � log n,
shift with σ near λi .
0λn. . .
λi+1 λi λi−1. . .
λ1σ
But (M − σI) is not positive definite!
15/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Folded Spectrum Method
How to find λi in O(n (log n)α kβ)?
If i = n − d with d � log n,
shift with σ near λi .
0λn. . .
λi+1 λi λi−1. . .
λ1σ
But (M − σI) is not positive definite!
15/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Folded Spectrum Method
Folded Spectrum Method [Wang, Zunger 1994]
Mσ = (M − σI)2
Mσ is s.p.d., if M is s.p.d. and σ 6= λi .
Assume all eigenvalues of Mσ are simple.
Mv = λv ⇔ Mσv = (M − σI)2 v
= M2v − 2σMv + σ2v
= λ2v − 2σλv + σ2v
= (λ− σ)2v
16/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Folded Spectrum Method
1 Choose σ.2 Compute
a) Mσ = (M − σI)2 andb) M−1
σ .
3 Use PINVIT to find the smallest eigenpair (µσ, v) of Mσ.4 Compute µ = vTMv/vTv .
(µ, v) is the nearest eigenpair to σ.
H-arithmetic enables us to shift, square and invert M resp.Mσ in linear-polylogarithmic complexity.
17/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Folded Spectrum Method
1 Choose σ.2 Compute
a) Mσ = (M − σI)2 andb) M−1
σ .
3 Use PINVIT to find the smallest eigenpair (µσ, v) of Mσ.4 Compute µ = vTMv/vTv .
(µ, v) is the nearest eigenpair to σ.
H-arithmetic enables us to shift, square and invert M resp.Mσ in linear-polylogarithmic complexity.
17/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Folded Spectrum Method
1 Choose σ.2 Compute
a) Mσ = (M − σI)2 andb) M−1
σ .
3 Use PINVIT to find the smallest eigenpair (µσ, v) of Mσ.4 Compute µ = vTMv/vTv .
(µ, v) is the nearest eigenpair to σ.
H-arithmetic enables us to shift, square and invert M resp.Mσ in linear-polylogarithmic complexity.
17/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Folded Spectrum Method
1 Choose σ.2 Compute
a) Mσ = (M − σI)2 andb) M−1
σ .
3 Use PINVIT to find the smallest eigenpair (µσ, v) of Mσ.4 Compute µ = vTMv/vTv .
(µ, v) is the nearest eigenpair to σ.
H-arithmetic enables us to shift, square and invert M resp.Mσ in linear-polylogarithmic complexity.
17/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Goal: Eigenvalues of symmetric H-Matrices
M = MT ∈ H(T , k),
Λ(M) = {λ1, λ2, . . . , λn},λ1 ≥ λ2 ≥ · · · ≥ λn > 0
⇓How to find λi in O(n (log n)α kβ)?
X
18/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Goal: Eigenvalues of symmetric H-Matrices
M = MT ∈ H(T , k),
Λ(M) = {λ1, λ2, . . . , λn},λ1 ≥ λ2 ≥ · · · ≥ λn > 0
⇓How to find λi in O(n (log n)α kβ)? X
18/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Numerical Results
Numerical Results
19/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Hlib
Hlib [Borm, Grasedyck, et al.]
We use the Hlib1.3 (www.hlib.org) for theH-arithmetic operations and some examples out ofthe library for testing the eigenvalue algorithm.
20/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, smallest eigenvalues
2D Laplace over [−1, 1]× [−1, 1]
Name ni error titi
ti−1
N(ni )N(ni−1)
FEM8 64 4.3466E-08 0.006
FEM16 256 1.2389E-07 0.014 2.33 80.00
FEM32 1 024 4.4308E-07 0.036 2.57 21.67
FEM64 4 096 1.8003E-06 0.228 6.33 6.72
FEM128 16 384 6.7069E-06 0.992 4.35 4.67
FEM256 65 536 1.6631E-05 2.397 2.42 4.57
FEM512 262 144 4.4015E-05 8.120 3.39 5.79
d = 4, c = 0.2, εeig = 10−4, N(ni ) = ni (log2 ni ) CspCid
Time t only PINVIT (without H-inversion)
21/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, smallest eigenvalues
FEM8
FEM16
FEM32
FEM64
FEM12
8
FEM25
6
FEM51
210−3
10−2
10−1
100
101
102
103
CP
Uti
me
ins
PINVIT(1,4)
PINVIT(3,4)
O(N(ni ))H-inversion
O(N(ni ) log ni )
22/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, smallest eigenvalues
FEM8
FEM16
FEM32
FEM64
FEM12
8
FEM25
6
FEM51
210−3
10−2
10−1
100
101
102
103
CP
Uti
me
ins
PINVIT(1,4)
PINVIT(3,4)
O(N(ni ))H-inversion
O(N(ni ) log ni )eigs
22/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Complexity
The number of iterations is independent of matrix size n.
Once:
adaptive H-matrix inversion: O(n (log n)2 k (c)2).
Per iteration:
H-matrix-vector products with M and B−1: O(n (log n) k (c))and O(n (log n) k (c)2), and
some dense arithmetic handling vectors of length n.
The complexity of the algorithm is determined by the H-matrixinversion: ⇒ O(n (log n)2 k (c)2).
Competitive to MATLAB eigs.
Expensive.
23/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Complexity
The number of iterations is independent of matrix size n.
Once:
adaptive H-matrix inversion: O(n (log n)2 k (c)2).
Per iteration:
H-matrix-vector products with M and B−1: O(n (log n) k (c))and O(n (log n) k (c)2), and
some dense arithmetic handling vectors of length n.
The complexity of the algorithm is determined by the H-matrixinversion: ⇒ O(n (log n)2 k (c)2).
Competitive to MATLAB eigs.
Expensive.
23/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Complexity
The number of iterations is independent of matrix size n.
Once:
adaptive H-matrix inversion: O(n (log n)2 k (c)2).
Per iteration:
H-matrix-vector products with M and B−1: O(n (log n) k (c))and O(n (log n) k (c)2), and
some dense arithmetic handling vectors of length n.
The complexity of the algorithm is determined by the H-matrixinversion: ⇒ O(n (log n)2 k (c)2).
Competitive to MATLAB eigs.
Expensive.
23/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, vn, FEM64
−1 −0.5 00.5 1−1
0
1−5
0
5
·10−2
−6
−4
−2
0
2
4
6·10−2
24/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, vn−1, FEM64
−1 −0.5 00.5 1−1
0
1−5
0
5
·10−2
−6
−4
−2
0
2
4
6·10−2
24/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, vn−2, FEM64
−1 −0.5 00.5 1−1
0
1−5
0
5
·10−2
−6
−4
−2
0
2
4
6·10−2
24/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, vn−3, FEM64
−1 −0.5 00.5 1−1
0
1−5
0
5
·10−2
−6
−4
−2
0
2
4
6·10−2
24/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, inner eigenvalues
2D Laplace over [−1, 1]× [−1, 1]
Name ni error titi
ti−1
N(ni )N(ni−1)
FEM8 64 2.2560E-08 <0.01
FEM16 256 3.4681E-06 0.01 — 80.00
FEM32 1 024 1.1875E-04 0.04 4.00 21.67
FEM64 4 096 7.2115E-04 0.20 5.00 6.72
FEM128 16 384 2.0747E-05 0.93 4.65 4.67
FEM256 65 536 9.2370E-05 3.59 3.86 4.57
d = 4, c = 0.2, εeig = 10−4, N(ni ) = ni (log2 ni ) CspCid
Time t only PINVIT (without H-inversion)
25/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, L-shape, smallest eigenvalues
2D Laplace over [−1, 1]× [−1, 1] \ [0, 1]× [0, 1]
Name n error t titi−1
N(ni )N(ni−1)
LFEM8 48 6.0999E-07 0.023
LFEM16 192 5.7426E-07 0.037 1.57 10.86
LFEM32 768 1.4644E-06 0.180 4.91 10.11
LFEM64 3 072 4.4753E-06 0.497 2.76 108.78
LFEM128 12 288 1.8770E-04 0.770 1.55 9.49
LFEM256 49 152 — 2.133 2.77 4.59
d = 4, c = 0.2, εeig = 10−4, N(ni ) = ni (log2 ni ) CspCid
Time t only PINVIT (without H-inversion)
26/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, L-shape, smallest eigenvalues
LFEM8
LFEM16
LFEM32
LFEM64
LFEM12
8
LFEM25
610−3
10−2
10−1
100
101
102
103
CP
Uti
me
ins
PINVIT(1,4)
PINVIT(3,4)
O(N(ni ))H-inversion
O(N(ni ) log ni )
27/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, L-shape, smallest eigenvalues
LFEM8
LFEM16
LFEM32
LFEM64
LFEM12
8
LFEM25
610−3
10−2
10−1
100
101
102
103
CP
Uti
me
ins
PINVIT(1,4)
PINVIT(3,4)
O(N(ni ))H-inversion
O(N(ni ) log ni )eigs
27/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, L-shape, vn, LFEM64
−1 −0.5 00.5 1−1
0
1−5
0
5
·10−2
−6
−4
−2
0
2
4
6·10−2
28/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, L-shape, vn−1, LFEM64
−1 −0.5 00.5 1−1
0
1−5
0
5
·10−2
−6
−4
−2
0
2
4
6·10−2
28/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, L-shape, vn−2, LFEM64
−1 −0.5 00.5 1−1
0
1−5
0
5
·10−2
−6
−4
−2
0
2
4
6·10−2
28/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
2D Laplace, L-shape, vn−3, LFEM64
−1 −0.5 00.5 1−1
0
1−5
0
5
·10−2
−6
−4
−2
0
2
4
6·10−2
28/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
BEM example, smallest eigenvalues
dense matrix, but approximable by an H-matrix ⇒ MATLAB eigsis expensive
Name ni error titi
ti−1
N(ni )N(ni−1)
BEM8 258 2.0454E-05 0.01
BEM16 1 026 3.5696E-05 0.03 3.00 26.81
BEM32 4 098 7.1833E-05 0.14 4.67 14.42
BEM64 16 386 — 0.46 3.29 16.51
BEM128 65 538 — 2.00 4.36 26.35
d = 4, c = 0.2, εeig = 10−4, N(ni ) = ni (log2 ni ) CspCid
Time t only PINVIT (without H-inversion)
29/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
BEM example, smallest eigenvalues
BEM8
BEM16
BEM32
BEM64
BEM12
8
10−2
100
102
104
CP
Uti
me
ins
PINVIT(1,4)
PINVIT(3,4)
O(N(ni ))H-inversion
O(N(ni ) log ni )
30/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
BEM example, smallest eigenvalues
BEM8
BEM16
BEM32
BEM64
BEM12
8
10−2
100
102
104
CP
Uti
me
ins
PINVIT(1,4)
PINVIT(3,4)
O(N(ni ))H-inversion
O(N(ni ) log ni )eigs
30/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Conclusions
Once we have inverted M ∈ H(T , k) finding the eigenvaluesby PINVIT is cheap and storage efficient.
For dense, data-sparse matrices, H-PINVIT can solveproblems where eigs (implicitly, restarted Arnoldi/Lanczositeration) is not applicable.
The folded spectrum method enables us to compute innereigenvalues, too.
We will investigate the use of H-Cholesky factorizationsinstead of the H-inversion.
Thank you for your attention.
31/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Conclusions
Once we have inverted M ∈ H(T , k) finding the eigenvaluesby PINVIT is cheap and storage efficient.
For dense, data-sparse matrices, H-PINVIT can solveproblems where eigs (implicitly, restarted Arnoldi/Lanczositeration) is not applicable.
The folded spectrum method enables us to compute innereigenvalues, too.
We will investigate the use of H-Cholesky factorizationsinstead of the H-inversion.
Thank you for your attention.
31/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Conclusions
Once we have inverted M ∈ H(T , k) finding the eigenvaluesby PINVIT is cheap and storage efficient.
For dense, data-sparse matrices, H-PINVIT can solveproblems where eigs (implicitly, restarted Arnoldi/Lanczositeration) is not applicable.
The folded spectrum method enables us to compute innereigenvalues, too.
We will investigate the use of H-Cholesky factorizationsinstead of the H-inversion.
Thank you for your attention.
31/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Conclusions
Once we have inverted M ∈ H(T , k) finding the eigenvaluesby PINVIT is cheap and storage efficient.
For dense, data-sparse matrices, H-PINVIT can solveproblems where eigs (implicitly, restarted Arnoldi/Lanczositeration) is not applicable.
The folded spectrum method enables us to compute innereigenvalues, too.
We will investigate the use of H-Cholesky factorizationsinstead of the H-inversion.
Thank you for your attention.
31/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Hierarchical (H-)Matrices PINVIT Numerical Results
Conclusions
Once we have inverted M ∈ H(T , k) finding the eigenvaluesby PINVIT is cheap and storage efficient.
For dense, data-sparse matrices, H-PINVIT can solveproblems where eigs (implicitly, restarted Arnoldi/Lanczositeration) is not applicable.
The folded spectrum method enables us to compute innereigenvalues, too.
We will investigate the use of H-Cholesky factorizationsinstead of the H-inversion.
Thank you for your attention.
31/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices
Appendix
Adaptive H-Inversion [Grasedyck 2001]
Adaptive H-Inversion
Input: M ∈ H (TI×I ), c = c√‖M‖2
∈ R
Output: M−1H , with
∥∥I −M−1H M
∥∥2< c
Compute M−1H with εlocal = c/(Csp(M) ‖M‖H2 )
δ0M :=∥∥I −M−1
H M∥∥H
2/c
while δiM > 1 doCompute M−1
H with εlocal = εlocal/δiM
δiM :=∥∥I −M−1
H M∥∥H
2/c
end
∥∥I −M−1H M
∥∥2< c =
c√‖M‖2
⇔∥∥I −M−1
H M∥∥
M≤ c
31/31 Thomas Mach Preconditioned Inverse Iteration for H-Matrices