€¦ · Concept Application Eigenvalues H-Matrices [Hackbusch ’98] 22 3 3 7 10 3 7 3 10 19 10 10 31 14 8 11 11 14 11 8 11 19 10 10 31 11 31 11 9 9 16 12 11 16 11 8 9 16 11 11 16

HIERARCHICAL STRUCTURES

MAX PLANCK INSTITUTE

FOR DYNAMICS OF COMPLEX

TECHNICAL SYSTEMS

MAGDEBURG

November 5, 2010

Hierarchical Matrices:Concept, Applications and Eigenvalues

Thomas Mach

Max Planck Institute for Dynamics of Complex Technical SystemsComputational Methods in Systems and Control Theory

Max Planck Institute Magdeburg Thomas Mach, Hierarchical Matrices 1/28




TECHNICAL SYSTEMS

MAGDEBURG

November 5, 2010


Thomas Mach






TECHNICAL SYSTEMS

MAGDEBURG

November 5, 2010


Thomas Mach






TECHNICAL SYSTEMS

MAGDEBURG

November 5, 2010


Thomas Mach



Concept Application Eigenvalues

Householder Notation

λ ∈ R

Scalar




x =

x1

x2

x3...xn

∈ Rn

Vector




A =

a11 a12 a13 . . . a1n

a21 a22 a23 . . . a2n

a31 a32 a33 . . . a3n...

......

. . ....

an1 an2 an3 . . . ann

∈ Rn×n

Matrix



Dense Matrix Format

Store n vectors (Fortran):

a11

a21

a31...

an1

a12

a22

a32...

an2

a13

a23

a33...

an3

· · ·a1n

a2n

a3n...

ann

n2 entries in the storage

Ax costs O(n2) flops

AB costs O(nδ) flops, δ ≥ 2.376 usually δ = 3

A−1 costs O(n3) flops

There is a constant c , so that Ax costs not more than cn2 flops

Landau Symbol

1 flop = (αβ + γ → γ)

Flops

expensive!



Dense Matrix Format


a11

a21

a31...

an1

a12

a22

a32...

an2

a13

a23

a33...

an3

· · ·a1n

a2n

a3n...

ann






Landau Symbol

1 flop = (αβ + γ → γ)

Flops

expensive!



Dense Matrix Format


a11

a21

a31...

an1

a12

a22

a32...

an2

a13

a23

a33...

an3

· · ·a1n

a2n

a3n...

ann






Landau Symbol

1 flop = (αβ + γ → γ)

Flops

expensive!



Sparse Matrix Format

4 −1 0 −1 0 0 0 0 0−1 4 −1 0 −1 0 0 0 00 −1 4 0 0 −1 0 0 0−1 0 0 4 −1 0 −1 0 00 −1 0 −1 4 −1 0 −1 00 0 −1 0 −1 4 0 0 −10 0 0 −1 0 0 4 −1 00 0 0 0 −1 0 −1 4 −10 0 0 0 0 −1 0 −1 4

non-zero entries per row/column = c ≤ 5




4 −1 0 −1 0 0 0 0 0−1 4 −1 0 −1 0 0 0 00 −1 4 0 0 −1 0 0 0−1 0 0 4 −1 0 −1 0 00 −1 0 −1 4 −1 0 −1 00 0 −1 0 −1 4 0 0 −10 0 0 −1 0 0 4 −1 00 0 0 0 −1 0 −1 4 −10 0 0 0 0 −1 0 −1 4





4 −1 −1 −1−1 −1 −1 4−1 4 −1 −1−1 −1 4 −14 −1 −1 −1−1 −1 −1 4−1 −1 4−1 4 −14 −1 −1




Sparse Matrix Format (Matlab)

Store vector of tripel: (1, 1, 4)(2, 1,−1)(4, 1,−1)(1, 2,−1)

...

O(n) entries in the storage (if #non-zeros/column < c n)

Ax costs O(n) flops

AB may be much denser ⇒ better use A(Bx)

A−1 not possible, only solve Ax = b



Is there something in between?

Yes, e.g.

Low rank matrices and Tucker tensor format

Toeplitz/Hankel matrices

Semiseparable matrices [Gantmacher, Krein 1937]

Cauchy matrices

Fast Multipole Method (FMM) [Greengard, Rokhlin ’87]

Mosaic-skeleton matrices [Tyrtyshnikov ’96]

...




Yes, e.g.




Cauchy matrices



...




Yes, e.g.




Cauchy matrices



Hierarchical (H-) Matrices [Hackbusch ’98]



Integral Equation

Fredholm equation of first kind:∫Ω g(x , y)u(y)dy = f (x)

electrostatic problem:

g(x , y) = 14πε‖x−y‖

u(y) charge densityf (x) electrostatic potential

inverse of elliptic differential operators

population dynamics [Koch, Hackbusch, Sundmacher]

unknown



Integral Equation



g(x , y) = 14πε‖x−y‖




unknown



Integral Equation



g(x , y) = 14πε‖x−y‖




unknown



H-Matrices [Hackbusch ’98]

∫Ω g(x , y)u(y)dy = f (x)

Ritz-Galerkin method:

u(y) =∑

i uiψi (y)

⇒∑i

∫Ω

∫Ωg(x , y)ψi (y)dyψj(x)dx︸︷︷︸

:=Mji

ui=

∫Ωf (x)ψj(x)dx︸︷︷︸

=fj

x ∈ Ωs , y ∈ Ωt : g(x , y) ≈∑kp=1 g

x ,sp (x)g y ,t

p (y)

Mji =

∫Ω

∫Ωg(x , y)ψj(x)ψi (y)dydx

≈k∑

p=1

∫Ωg xp (x)ψj(x)dx

∫Ωg yp (y)ψi (y)dy = Aj ·B

Ti ·

discretization error ε ∼ 1nκ , 0 < κ < 1

Mu = f






u(y) =∑

i uiψi (y)

⇒∑i

∫Ω


:=Mji

ui=


=fj

x ∈ Ωs , y ∈ Ωt : g(x , y) ≈∑kp=1 g

x ,sp (x)g y ,t

p (y)

Mji =

∫Ω


≈k∑

p=1



Ti ·


Mu = f






u(y) =∑

i uiψi (y)

⇒∑i

∫Ω


:=Mji

ui=


=fj

x ∈ Ωs , y ∈ Ωt : g(x , y) ≈∑kp=1 g

x ,sp (x)g y ,t

p (y)

Mji =

∫Ω


≈k∑

p=1



Ti ·


Mu = f






u(y) =∑

i uiψi (y)

⇒∑i

∫Ω


:=Mji

ui=


=fj

x ∈ Ωs , y ∈ Ωt : g(x , y) ≈∑kp=1 g

x ,sp (x)g y ,t

p (y)

Mji =

∫Ω


≈k∑

p=1



Ti ·


Mu = f






u(y) =∑

i uiψi (y)

⇒∑i

∫Ω


:=Mji

ui=


=fj

x ∈ Ωs , y ∈ Ωt : g(x , y) ≈∑kp=1 g

x ,sp (x)g y ,t

p (y)

Mji =

∫Ω


≈k∑

p=1



Ti ·


Mu = f




22

3 3

7 10

3 7

3 10

19 10

10 31

14 8

11 11

14 11

8 11

19 10

10 31 11

11 31

11 9

9 16 12

11 1611 8

9 16 11

11 16

61

9 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 5

15 613

13

7 5

13 8

8 11 8 11

11

6 5

15 6

8

11 8

8 15

5 8 11

12

6 15

5 613

13

7

13 8

8 11

5 811

11

6 15

5 6

6110 103

6 14

10 3 6

10 14

25 10 6

10

6

19 10

10 31

15 9

9 1110

10 16

15 9

8 1110

10 16

5110 10

7 9 73 3

10 7

10

9 3

7 3

25 11

11

25 10

10 19

11 811 8

8 15 9

8 1512 13

13

10 7

13 8

8 11 9

9 15 1119

20

10 7

13 8

9

11 8

8 15 11 12

13

9 7

13 8

9

13 8

8 1111

10

11 8

8 15 8

9 15

7 12 13

13

10

13 8

8 11 8

9 15

7 1120

19

9

13 9

8

11 8

8 15

7 11 12

129

13 9

8

13 8

8 11

7 11

39 10

10 25

3 7

3 10 10

7 10 63 3

7 10 7

10

10

6

22

3 3

7 10

3 7

3 10

19 10

10 31

15 10

11

11 9

9 16

15 11

10

11 8

9 16

34 10

10 25

13 10

7 11

13 7

10 11 61

6 5

13 6 11

12

8 5

11 8

8 15 812

13

6 5

13 6 11

11 23

6 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

10

3 7

3 10

9 10

7

3 3

7 1061

15 10

1015 9

9 11

15 10

1015 9

8 11

20 9

9 34

9 7

10 13

9 10

7 1351

x ∈ Ωs , y ∈ Ωt : g(x, y) ≈k∑

p=1

gx,sp (x)gy,tp (y)

Mji =

∫Ω

∫Ωg(x, y)ψj (x)ψi (y)dydx

≈k∑

p=1

∫Ωgxp (x)ψj (x)dx

∫Ωgyp (y)ψi (y)dy = Aj·B

Ti·

rank (Ms×t) = 19store:

A ∈ R|s|×19,

B ∈ R|t|×19

with Ms×t = ABT

⇒ only 19(|s|+ |t|) instead of |s| |t| storage⇒ reduces the required storage to 15%

s

t

=

if x ≈ y then 1‖x−y‖ is large

small matrices on the diag-onal have no low rank ap-proximation⇒ use dense matrix format

non-admissible block

admissible block




22

3 3

7 10

3 7

3 10

19 10

10 31

14 8

11 11

14 11

8 11

19 10

10 31 11

11 31

11 9

9 16 12

11 1611 8

9 16 11

11 16

61

9 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 5

15 613

13

7 5

13 8

8 11 8 11

11

6 5

15 6

8

11 8

8 15

5 8 11

12

6 15

5 613

13

7

13 8

8 11

5 811

11

6 15

5 6

6110 103

6 14

10 3 6

10 14

25 10 6

10

6

19 10

10 31

15 9

9 1110

10 16

15 9

8 1110

10 16

5110 10

7 9 73 3

10 7

10

9 3

7 3

25 11

11

25 10

10 19

11 811 8

8 15 9

8 1512 13

13

10 7

13 8

8 11 9

9 15 1119

20

10 7

13 8

9

11 8

8 15 11 12

13

9 7

13 8

9

13 8

8 1111

10

11 8

8 15 8

9 15

7 12 13

13

10

13 8

8 11 8

9 15

7 1120

19

9

13 9

8

11 8

8 15

7 11 12

129

13 9

8

13 8

8 11

7 11

39 10

10 25

3 7

3 10 10

7 10 63 3

7 10 7

10

10

6

22

3 3

7 10

3 7

3 10

19 10

10 31

15 10

11

11 9

9 16

15 11

10

11 8

9 16

34 10

10 25

13 10

7 11

13 7

10 11 61

6 5

13 6 11

12

8 5

11 8

8 15 812

13

6 5

13 6 11

11 23

6 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

10

3 7

3 10

9 10

7

3 3

7 1061

15 10

1015 9

9 11

15 10

1015 9

8 11

20 9

9 34

9 7

10 13

9 10

7 1351

x ∈ Ωs , y ∈ Ωt : g(x, y) ≈k∑

p=1

gx,sp (x)gy,tp (y)

Mji =

∫Ω


≈k∑

p=1



Ti·


A ∈ R|s|×19,

B ∈ R|t|×19

with Ms×t = ABT


s

t

=if x ≈ y then 1

‖x−y‖ is largesmall matrices on the diag-onal have no low rank ap-proximation⇒ use dense matrix format


admissible block




22

3 3

7 10

3 7

3 10

19 10

10 31

14 8

11 11

14 11

8 11

19 10

10 31 11

11 31

11 9

9 16 12

11 1611 8

9 16 11

11 16

61

9 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 5

15 613

13

7 5

13 8

8 11 8 11

11

6 5

15 6

8

11 8

8 15

5 8 11

12

6 15

5 613

13

7

13 8

8 11

5 811

11

6 15

5 6

6110 103

6 14

10 3 6

10 14

25 10 6

10

6

19 10

10 31

15 9

9 1110

10 16

15 9

8 1110

10 16

5110 10

7 9 73 3

10 7

10

9 3

7 3

25 11

11

25 10

10 19

11 811 8

8 15 9

8 1512 13

13

10 7

13 8

8 11 9

9 15 1119

20

10 7

13 8

9

11 8

8 15 11 12

13

9 7

13 8

9

13 8

8 1111

10

11 8

8 15 8

9 15

7 12 13

13

10

13 8

8 11 8

9 15

7 1120

19

9

13 9

8

11 8

8 15

7 11 12

129

13 9

8

13 8

8 11

7 11

39 10

10 25

3 7

3 10 10

7 10 63 3

7 10 7

10

10

6

22

3 3

7 10

3 7

3 10

19 10

10 31

15 10

11

11 9

9 16

15 11

10

11 8

9 16

34 10

10 25

13 10

7 11

13 7

10 11 61

6 5

13 6 11

12

8 5

11 8

8 15 812

13

6 5

13 6 11

11 23

6 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

10

3 7

3 10

9 10

7

3 3

7 1061

15 10

1015 9

9 11

15 10

1015 9

8 11

20 9

9 34

9 7

10 13

9 10

7 1351

x ∈ Ωs , y ∈ Ωt : g(x, y) ≈k∑

p=1

gx,sp (x)gy,tp (y)

Mji =

∫Ω


≈k∑

p=1



Ti·


A ∈ R|s|×19,

B ∈ R|t|×19

with Ms×t = ABT


s

t

=




admissible block




22

3 3

7 10

3 7

3 10

19 10

10 31

14 8

11 11

14 11

8 11

19 10

10 31 11

11 31

11 9

9 16 12

11 1611 8

9 16 11

11 16

61

9 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 5

15 613

13

7 5

13 8

8 11 8 11

11

6 5

15 6

8

11 8

8 15

5 8 11

12

6 15

5 613

13

7

13 8

8 11

5 811

11

6 15

5 6

6110 103

6 14

10 3 6

10 14

25 10 6

10

6

19 10

10 31

15 9

9 1110

10 16

15 9

8 1110

10 16

5110 10

7 9 73 3

10 7

10

9 3

7 3

25 11

11

25 10

10 19

11 811 8

8 15 9

8 1512 13

13

10 7

13 8

8 11 9

9 15 1119

20

10 7

13 8

9

11 8

8 15 11 12

13

9 7

13 8

9

13 8

8 1111

10

11 8

8 15 8

9 15

7 12 13

13

10

13 8

8 11 8

9 15

7 1120

19

9

13 9

8

11 8

8 15

7 11 12

129

13 9

8

13 8

8 11

7 11

39 10

10 25

3 7

3 10 10

7 10 63 3

7 10 7

10

10

6

22

3 3

7 10

3 7

3 10

19 10

10 31

15 10

11

11 9

9 16

15 11

10

11 8

9 16

34 10

10 25

13 10

7 11

13 7

10 11 61

6 5

13 6 11

12

8 5

11 8

8 15 812

13

6 5

13 6 11

11 23

6 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

10

3 7

3 10

9 10

7

3 3

7 1061

15 10

1015 9

9 11

15 10

1015 9

8 11

20 9

9 34

9 7

10 13

9 10

7 1351

x ∈ Ωs , y ∈ Ωt : g(x, y) ≈k∑

p=1

gx,sp (x)gy,tp (y)

Mji =

∫Ω


≈k∑

p=1



Ti·


A ∈ R|s|×19,

B ∈ R|t|×19

with Ms×t = ABT


s

t

=




admissible block




22

3 3

7 10

3 7

3 10

19 10

10 31

14 8

11 11

14 11

8 11

19 10

10 31 11

11 31

11 9

9 16 12

11 1611 8

9 16 11

11 16

61

9 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 5

15 613

13

7 5

13 8

8 11 8 11

11

6 5

15 6

8

11 8

8 15

5 8 11

12

6 15

5 613

13

7

13 8

8 11

5 811

11

6 15

5 6

6110 103

6 14

10 3 6

10 14

25 10 6

10

6

19 10

10 31

15 9

9 1110

10 16

15 9

8 1110

10 16

5110 10

7 9 73 3

10 7

10

9 3

7 3

25 11

11

25 10

10 19

11 811 8

8 15 9

8 1512 13

13

10 7

13 8

8 11 9

9 15 1119

20

10 7

13 8

9

11 8

8 15 11 12

13

9 7

13 8

9

13 8

8 1111

10

11 8

8 15 8

9 15

7 12 13

13

10

13 8

8 11 8

9 15

7 1120

19

9

13 9

8

11 8

8 15

7 11 12

129

13 9

8

13 8

8 11

7 11

39 10

10 25

3 7

3 10 10

7 10 63 3

7 10 7

10

10

6

22

3 3

7 10

3 7

3 10

19 10

10 31

15 10

11

11 9

9 16

15 11

10

11 8

9 16

34 10

10 25

13 10

7 11

13 7

10 11 61

6 5

13 6 11

12

8 5

11 8

8 15 812

13

6 5

13 6 11

11 23

6 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

10

3 7

3 10

9 10

7

3 3

7 1061

15 10

1015 9

9 11

15 10

1015 9

8 11

20 9

9 34

9 7

10 13

9 10

7 1351

x ∈ Ωs , y ∈ Ωt : g(x, y) ≈k∑

p=1

gx,sp (x)gy,tp (y)

Mji =

∫Ω


≈k∑

p=1



Ti·


A ∈ R|s|×19,

B ∈ R|t|×19

with Ms×t = ABT


s

t

=




admissible block




rank-k-matrix: Ms×t = ABT , A ∈ Rn×k ,B ∈ Rm×k (k n,m)

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

12345678

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




Hierarchical matrices

H(TI× I, k) =M ∈ RI× I

∣∣ rank (Ms×t) ≤ k ∀s × t admissible

22

3 3

7 10

3 7

3 10

19 10

10 31

14 8

11 11

14 11

8 11

19 10

10 31 11

11 31

11 9

9 16 12

11 1611 8

9 16 11

11 16

61

9 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 5

15 613

13

7 5

13 8

8 11 8 11

11

6 5

15 6

8

11 8

8 15

5 8 11

12

6 15

5 613

13

7

13 8

8 11

5 811

11

6 15

5 6

6110 103

6 14

10 3 6

10 14

25 10 6

10

6

19 10

10 31

15 9

9 1110

10 16

15 9

8 1110

10 16

5110 10

7 9 73 3

10 7

10

9 3

7 3

25 11

11

25 10

10 19

11 811 8

8 15 9

8 1512 13

13

10 7

13 8

8 11 9

9 15 1119

20

10 7

13 8

9

11 8

8 15 11 12

13

9 7

13 8

9

13 8

8 1111

10

11 8

8 15 8

9 15

7 12 13

13

10

13 8

8 11 8

9 15

7 1120

19

9

13 9

8

11 8

8 15

7 11 12

129

13 9

8

13 8

8 11

7 11

39 10

10 25

3 7

3 10 10

7 10 63 3

7 10 7

10

10

6

22

3 3

7 10

3 7

3 10

19 10

10 31

15 10

11

11 9

9 16

15 11

10

11 8

9 16

34 10

10 25

13 10

7 11

13 7

10 11 61

6 5

13 6 11

12

8 5

11 8

8 15 812

13

6 5

13 6 11

11 23

6 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

10

3 7

3 10

9 10

7

3 3

7 1061

15 10

1015 9

9 11

15 10

1015 9

8 11

20 9

9 34

9 7

10 13

9 10

7 1351

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)




Hierarchical matrices

H(TI× I, k) =M ∈ RI× I

∣∣ rank (Ms×t) ≤ k ∀s × t admissible

22

3 3

7 10

3 7

3 10

19 10

10 31

14 8

11 11

14 11

8 11

19 10

10 31 11

11 31

11 9

9 16 12

11 1611 8

9 16 11

11 16

61

9 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 5

15 613

13

7 5

13 8

8 11 8 11

11

6 5

15 6

8

11 8

8 15

5 8 11

12

6 15

5 613

13

7

13 8

8 11

5 811

11

6 15

5 6

6110 103

6 14

10 3 6

10 14

25 10 6

10

6

19 10

10 31

15 9

9 1110

10 16

15 9

8 1110

10 16

5110 10

7 9 73 3

10 7

10

9 3

7 3

25 11

11

25 10

10 19

11 811 8

8 15 9

8 1512 13

13

10 7

13 8

8 11 9

9 15 1119

20

10 7

13 8

9

11 8

8 15 11 12

13

9 7

13 8

9

13 8

8 1111

10

11 8

8 15 8

9 15

7 12 13

13

10

13 8

8 11 8

9 15

7 1120

19

9

13 9

8

11 8

8 15

7 11 12

129

13 9

8

13 8

8 11

7 11

39 10

10 25

3 7

3 10 10

7 10 63 3

7 10 7

10

10

6

22

3 3

7 10

3 7

3 10

19 10

10 31

15 10

11

11 9

9 16

15 11

10

11 8

9 16

34 10

10 25

13 10

7 11

13 7

10 11 61

6 5

13 6 11

12

8 5

11 8

8 15 812

13

6 5

13 6 11

11 23

6 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

10

3 7

3 10

9 10

7

3 3

7 1061

15 10

1015 9

9 11

15 10

1015 9

8 11

20 9

9 34

9 7

10 13

9 10

7 1351

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)

A1

BT1

+ A2

BT2

= A1 A2

BT1

BT2



Special Case: H`-Matrices

A8BT8

B8AT8

A4BT4

B4AT4

A12BT12

B12AT12

A2BT2

A6BT6

A10BT10

A14BT14

B2AT2

B6AT6

B10AT10

B14AT14

F1

F3

F5

F7

F9

F11

F13

F15

Figure: Structure of an H3(k)-matrix



Applications



Applications

Finite-Element-Method (FEM)

Inverse of FEM matrices

Integral equations

Solving matrix equations for model order reduction

AX + XAT + BBT = 0, A ∈ H,B ∈ Rn×k

Sign function iteration[Baur ’07, Grasedyck ’03, et al.]

Boundary-Element-Method (BEM)



BEM for Electrostatic Computations

−4 −20

24

·103

−20

2·103

−4

−2

0

2

4

·103




−4 −20

24

·103

−20

2·103

−4

−2

0

2

4

·103Electrostatic problem:∫

Ω g(x , y)u(y)dy = f (x)

g(x , y) = 14πε‖x−y‖

unknown: u(y) charge density

given: f (x) electrostatic potential




−4 −20

24

·103

−20

2·103

−4

−2

0

2

4

·103










−4 −20

24

·103

−20

2·103

−4

−2

0

2

4

·103



BEM for Electrostatic Computations26 14

14 20 19

1925 14

14 20

23 16

17 23

23 17

16 23

22 20

20 22 22

22 36

13

15

17

18

13 15 17 18

19 15

15 21

20 16

19 25 8

11 18 8

20 19

16 25

11

188 8

25 14

14 20 16

16 26

22 15

13 15

22 13

15 15

36 19

19 288

8 8

17 12

27 19

19 26 20 2513 23 10 11 23 9

25 14

16 10 13

3016 10

26 16

16 18 17 16

8 8

1727 19

19 26

12 20

13

23

10

11

2523

9

30

25

16

10

14 13

1626 16

16 18

10 17 8

16 8

22 20

20 22 22

22 36

28 20

20 28

28 20

20 28

21 11

11 12 20

20 22 29

2928 18

18 18

18

36 22

32

21

28 22

17 18

18 36 32

22 21

28 17

22 18

19 13

13 15 23

2323 13

13 1525

25

20 12

12 15 23

2320 12

12 15

50

19 12

15 20

24 20

17 25

20 14

13 20

50

24 17

20 2519 15

12 20

20 13

14 20

36 19

19 28 40

40

26 18

18 18 23

2321 13

13 14

20 14

17 20

20 17

14 20

20 12

12 20

22

22 11

6 18 10 8

20

13 22

13

17

18 12 21

17 2224

8 6

22

19 8

23 9

17 10 9 7

20 43

19

23 10

6 16 10

6 11 9

15

10

27 19

16 19

6 19 15

17 8

15

34 17

13

25

16

20 19

16 17

20 18

17 24 19

20

20 17

18 20

48 12

27 13 18 12

18 12 27 13 10

23 14

19 24

20 13

14 20

36 26

20 24 33

4644 29

23 3024

20 16

17 20 11

22 20

22

6

18

10

11 8

13 13

22

17 18

12

17

21 228

24 6

19 15

23

6 6

16

10 11

10 9

10 6

27 16

19 1919 17

15 8

2219

8 20

23

17

10

9

9 7

43

15 34 13 25

17 1620 16

19 17

20 17

18 24 20

1920 18

17 20

23 19

14 24

20 14

13 20

48

27

13

18

12

18

12

27

13

12 10

36 20

26 2446

33

44 23

29 30

20 17

16 20

24 11

22 18

18

19 13

13 18 9 17

9

17

31 19

19 27 12

12 12

27

27

18 18

18

24 18

18

21 17

17

24 19

19 19

19 12

13 17 14

9

16

11

12

18 13

16 15

19 13

12 17 9 16

14 11 12

18 16

13 15

28 14

14 17 14

1418 12

12 15

15 15

18

8

10 6 6

35 25

24

13 16

17

10 10

12 13

12 27 17 19 19

8 9 6 6

11 19 9 14

15 18 8

15

10

35 24

25

13 17

16

10

12 12

13 27

10 17

6

619

8

9

11

19

6

69

19 14

20 17

17 24 19

1920 17

17 20

29 12 17 10

16 10 26 11 11

29

12

16

10

17

10

26

11

11

20 15

15 15 27

2723 15

15 15

15 17

15 18

15 15

17 18

15 14

14 18

49

49

31 31

31

15 15

15

27 20

20 2037

37

18 17

17

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15 24

14 14

14 21

14 14

14 21

14 13

13 21

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25 49

39 25

27 49

20 15

15 15 28

2820 15

15 1530

30 3045

45

17 16

16 18 35

35

15 15

15

28 16

16 16

13 13

12 2913 12

13 29

15 15

15

27 18

18 18

26 17

18 26

26 18

17 26

26 13

13 26

35

12 138 6 7

7

27 8

10 117 6 7 7

9

19 11

22

12

11 10

116 7

8 9 6

12 12 10

32 13

49

8 6

17 11

11 17 15

8

15

11

40

18 12

12 17 15 9

8

15

12 713

20 15

15 25 12

18 12

12 16 15

1318 14

12 15

24 13

13 17 15 9

1318 14

12 15 10

10 339 5

9

13

24

6

6

9

10

8

926

6

614

17 18

8

8

8

11 7

15

45

12

19

12

22

12

36

15 11

46 16 23

25

12 11

38 1726 15

14 26

26 16

16 26

14

35 4912 8

13

6

727

7 8

22

10

7

6

11

7

7

19

9 11

12 12

11 11

10

6 8

7

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6

12 32

10 13

8

17 11

11 17 8 15

15 11 40

18 12

12 16 13

15

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14 15

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12 17 8 15

15

9

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20 15

15 25

7 12

24 13

13 17 13

15

9

18 12

14 15

10

6 10

9

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13 24 9 10 17

8 9 6 6

26 14 18

5 8 8 8

33

11

15 45 12 22

12 19 12 36

26 14

15 26

15 46

11 16 25

23

12 38

11 17

26 16

16 26

7 14

26 16

16

20 14

14 25

15 12

15 23

15 15

12 23

28 19

19 3640

40

20 14

14 25 18

18

20 14

14 26

22 11 11

16 24

22 16

11

11 24

36 22

22

22 20

20 22

16 1118 16

16 26 17 24

25

18 1126 19

19 27 20

16

18 16

16 26

11 17 25

24

18

26 19

19 27

11 20

28 19

19 36 40

40

21 12

12 14 23

23

28 18

18 18

50

50

36 22

22

22 20

20 22

28 20

20 29

28 20

20 29

26 18

18 18 29

29

12 12

12 21 21

21 22

34

13 20

13

20 16

18 27

11

17

2413

26

23 36 21

13 23 26

41 18

23

23

14

34 20

26 38

23 19

24 36 31

41

27 22

24 4245

19 47

36 23

30 40

34

13 13

20

20 18

16 27

24

11 17 13 26

41 23

18 23 14

23

36

13

23

21 26

34 26

20 38

23 24

19 3641

31

27 24

22 42

19

47

45

36 30

23 40

19 12

12 17 16

16

21 16

16

31 13

13 13

19 13

13 17 17 9

16

21 16

16

27 15

13 1318

19 13

13 17 16

17

9

21 16

16

27 13

15 13

18

28 15

15 18 16 9

16

9

21 16

16

27 16

16 2624

24

32 17

17 21 25

25 25

9

6

11

27

9 6 11 27

10 10

10

29 17

17 23

9 5

30 10

9 5 6

30 18 10 10

8 5

29

17 12

10 5 6

32 19

21 41 18 2610 12 23 11 34 11

9 30

5 10

8 29 17

5 12

9

30

18

5

6

10

10

10

32 21

19 41

5

618

10

12

23

11

2634

11

32 13

13 13 29

29

16 16

16

12 12

12 21 14

14 14

31

31

30 13

13 15 29

2930 14

14 15

27 15

13 13 29

27

16

12 12

14 20 16

14 14

32 29 19

32 18

30 13

14 15 27

2830 13

14 15

3127 13

15 13 27

2916

12 14

12 20 14

16 14

32

32

18

29

19

30 14

13 15 28

2730 14

13 15

31

27 15

15 17 29

29

17 17

17 23 16

16 1937

37

20 13

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17 12

12 13

17 12

12 13

21 13

13 15

25 23

20 30

25 20

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25 20

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31 22

19 45

31 19

22 45

30 14

14 15 30

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14 15

31

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13 15 28

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13 15

25 21

22 25

25 22

21 25

25 20

20 25

18

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11

27

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25 12 45

20 20

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20 17

17 24 29

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19 2937

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20 20

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17 25 29

29

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19 1018 11

30 21

22 29 19 29

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25 1032

33

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42 16 18

18 14

19

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22

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10 9 8 7

18 12 10 9 8

21 46 14

14 924 22 21

26

10

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7

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12

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41 1519

14 8

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17 1413 13

13 29 1720 19

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21 11 13

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11 9

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16 19

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12 9

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22 30 1818

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22 2814

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20 1332 18

18 46 20

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8

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8 10 8

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9 12 16

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10 13 107 7 6 8 6 8

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20 22 22

22 36

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21 2630

23

19

18 18

17 18

21

18 21

17 26 23

30 19

18 17

18 18

20 12

12 15 23

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12 1525

25

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13 15 23

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13 15

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20 15

13 19

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21 24

20 13

14 20

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16 2420 13

15 19

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13 20

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13 2140

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12 20

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25 25

16

9

16 9

18 15

15 28

9

11

27

6

9 11 27 6

10 10

10

29 18

18 23

==

fuM




−4 −20

24

·103

−20

2·103

−4

−2

0

2

4

·103

−5V

+5V+2V

−1

−0.5

0

0.5

1



Eigenvalues



Definition

Mx = λx

M ∈ Rn×n

eigenvectorx ∈ Cn, x 6= 0

eigenvalueλ ∈ C



Definition

Mx = λx

M ∈ Rn×n


eigenvalueλ ∈ C



Definition

Mx = λx

M ∈ Rn×n


eigenvalueλ ∈ C



Definition

Mx = λx

M ∈ Rn×n


eigenvalueλ ∈ C



Definition

Mx = λx

M = MT ∈ Rn×n

eigenvectorx ∈ Rn, x 6= 0

eigenvalueλ ∈ R



Application

Eigenvalue problems occur in many different applications:

vibration analysis,

molecular and quantum dynamics(e.g. Schrodinger or Kohn-Sham equations),

finite state of a Markov chain,

numerical mathematics (e.g. MOR, eM , convergence theory),

...



Definition

Mx = λx

M = MT ∈ H(T , k)

eigenvectorx ∈ Rn, x 6= 0

eigenvalueλ ∈ R



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 − T−1 (Mxi − xiµ(xi )).

Residual r(x) = Mx − xµ(x).




Definition

The function


xT x



xi+1 := xi − α∇µ(xi ), ∇µ(x) =2



xi+1 := xi − T−1 (Mxi − xiµ(xi )).





Definition

The function


xT x



xi+1 := xi − α∇µ(xi ), ∇µ(x) =2



xi+1 := xi − T−1 (Mxi − xiµ(xi )).





Definition

The function


xT x



xi+1 := xi − α∇µ(xi ), ∇µ(x) =2



xi+1 := xi − T−1 (Mxi − xiµ(xi )).




Preconditioned Inverse Iteration[Knyazev, Neymeyr ’09]

xi+1 := xi − T−1 (Mxi − xiµ(xi ))

If

M symmetric positive definite and

T−1 approximates the inverse of M, so that∥∥I − T−1M∥∥M≤ c < 1,

then Preconditioned INVerse ITeration (PINVIT) converges.



Preconditioned Inverse Iteration[Knyazev, Neymeyr ’09]

xi+1 := xi − T−1 (Mxi − xiµ(xi ))

If

M symmetric positive definite and

T−1 approximates the inverse of M, so that∥∥I − T−1M∥∥M≤ c < 1,

then Preconditioned INVerse ITeration (PINVIT) converges.



BEM, smallest eigenvalues

dense matrix, but approximable by an H-matrix

d = 4, c = 0.2, εeig = 10−4

258 1026 4098 16386 6553810−3

10−2

10−1

100

101

102

103

104

CP

Uti

me

ins

PINVIT

O(ni (log2 ni )CH)H-Cholesky factor.

O(ni (log2 ni )2 CH)





ARPACK / eigs

d = 4, c = 0.2, εeig = 10−4

258 1026 4098 16386 6553810−3

10−2

10−1

100

101

102

103

104

CP

Uti

me

ins

PINVIT





Concluding Remarks

H-arithmetic is cheap and applicable too many problems.

Computing eigenvalues of H-matrices in linear-polylogarithmiccomplexity is possible.

See poster: “Slicing the Spectrum of Symmetric H`-Matrices”.

Thank you for your attention.



Concluding Remarks







Concluding Remarks







Concluding Remarks






Appendix



d = 4, c = 0.2, εeig = 10−4

258 1026 4098 16386 6553810−3

10−2

10−1

100

101

102

103

104

CP

Uti

me

ins

PINVIT




Appendix



eig ∈ O(n3)

d = 4, c = 0.2, εeig = 10−4

258 1026 4098 16386 6553810−3

10−2

10−1

100

101

102

103

104

CP

Uti

me

ins

PINVIT




€¦ · Concept Application Eigenvalues H-Matrices [Hackbusch ’98] 22 3 3 7 10 3 7 3 10 19 10 10 31 14 8 11 11 14 11 8 11 19 10 10 31 11 31 11 9 9 16 12 11 16 11 8 9 16 11 11 16

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