Te hnis he Universit�at Chemnitz

Sonderfors hungsberei h 393

Numeris he Simulation auf massiv parallelen Re hnern

Matthias Bollh�ofer

A Robust ILU Based on

Monitoring the Growth of the

Inverse Fa tors

Preprint SFB393/00-31

Abstra t

An in omplete LU de omposition with pivoting is presented that progressively

monitors the growth of the inverse fa tors of L;U . The information on the growth of

the inverse fa tors is used as feedba k for dropping entries in L and U . This method

yields a robust pre onditioner in many ases and is often e�e tive espe ially when

the system is highly inde�nite. Numeri al examples demonstrate the e�e tiveness of

this approa h.

Keywords: sparse matri es, ILU , sparse approximate inverse, AINV , pivoting,

ondition estimator.

AMS subje t lassi� ation: 65F05, 65F10, 65F50.

Preprint-Reihe des Chemnitzer SFB 393

SFB393/00-31 July 2000

Contents

1 Introdu tion 1

2 A simple ILU approa h 1

3 Stabilized ILU 3

4 Numeri al Results 8

5 Con lusions 16

Author's addresses:

Matthias Bollh�ofer

Fakult�at f�ur Mathematik

TU Chemnitz

D{09107 Chemnitz

http://www.tu- hemnitz.de/�bolle/

1 Introdu tion

We onsider problems of the form

Ax = b;(1)

with A 2 R

n;n

nonsingular and b 2 R

n

. We fo us on problems where A is sparse and where

we do not have mu h information about the system beforehand. These systems might be

highly inde�nite or ill{ onditioned. Sin e often these systems are very large solving them

is a hallenge for numeri al algorithms. Sometimes it is ex eedingly diÆ ult to solve them

by iterative te hniques and in these ases dire t solvers might be preferred. However, there

are situations in whi h `general purpose' or `bla k{box' iterative solvers are required. The

most popular and promising iterative te hniques so far are pre onditioned Krylov{subspa e

solvers, see, e.g., [15, 22, 12℄. Among many te hniques, pre onditioners based on in omplete

LU{fa torizations, see e.g., [17, 18, 19℄ are known to give ex ellent results for many im-

portant lasses of problems, su h as those arising from the dis retization of ellipti partial

di�erential equations.

Nevertheless, there are still many situations where in omplete LU de omposition give poor

results. One often has to play around with the parameters, e.g., to adapt a drop toleran e

in the in omplete LU de omposition to obtain a su essful pre onditioner. This is time{

onsuming sin e for any problem one has to sele t the orre t values. This redu es the

exibility as a `bla k{box' solver. In addition by de reasing parameters to obtain a su -

essful pre onditioner we might get enormous �ll{in or an una eptable omputational

time. In this ase dire t solvers are the only alternative.

The intention of this paper is to take a loser look at in omplete LU de ompositions and

espe ially on how entries are dropped. The main key used here for analyzing dropping in

the in omplete LU de omposition is its strong relation [7, 8℄ to fa tored sparse approximate

inverse methods [3, 4, 2, 16, 21℄. In an earlier paper [8℄ omparisons between an in om-

plete LU de ompositions with pivoting and a fa tored approximate inverse with pivoting

have shown several examples where the approximate inverse was superior to the ILU . So

apparently ILUs may gain more stability from approximate inverses by taking a lose look

at their relations and espe ially at the way how entries are dropped.

The main idea is to monitor the growth of the inverse fa tors of L; U while omputing L,

U and to use this information as feedba k for a re�ned dropping strategy for the entries of

L and U .

2 A simple ILU approa h

We start with a simple des ription of a lass of in omplete LU fa torizations. For the

solution of (1) we onstru t an approximate de omposition

A � LDU;

1

where L; U

>

are lower triangular matri es with unit diagonal and D is diagonal. One way

to onstru t these de ompositions is to partition A as

A =

�

B F

E C

�

2 R

n;n

with B 2 R and the other blo ks have orresponding size. Then A is fa tored as

�

B F

E C

�

=

�

1 0

L

E

I

�

| {z }

L

�

D

B

0

0 S

�

| {z }

D

�

1 U

F

0 I

�

| {z }

U

;(2)

where

S = C � L

E

D

B

U

F

2 R

n�k;n�k

(3)

denotes the so{ alled S hur{ omplement. The exa t LU{de omposition of A (if it exists)

an be obtained by su essively applying (2) to the S hur{ omplement S. Even if there

exists a de omposition (2) for A and for S, there is no need to ompute L

E

; U

F

; S exa tly

when onstru ting a pre onditioner. A ommon approa h for redu ing �ll{in onsists of

dis arding entries in L

E

; U

F

of small size and de�ning the approximate S hur{ omplement

only with these sparsi�ed ve tors

~

L

E

;

~

U

F

. Here we will on entrate on

~

S = B �

~

L

E

F �

�

E �

~

L

E

B

�

~

U

F

(4)

as one possible de�nition of an approximate S hur{ omplement. Equation (4) an be ob-

tained from the lower right blo k of

~

L

�1

A

~

U

�1

.

We use the MATLAB notation [1℄ for onvenien e. For two integers k; l, k : l denotes the

sequen e (k; k + 1; : : : ; l) with the onvention that whenever k > l the set is empty. For a

matrix A = (A

ij

)

i=1;:::;m; j=1;:::;n

, we de�ne

A

k:l;q:r

:= (A

ij

)

i=k;:::;l; j=q;:::;r

:

The notation : as a subs ript indi ates that all olumns/rows entries are taken. Thus, A

:;2

denotes the se ond olumn of A and A

2;:

denotes its se ond row. Similarly for a nonempty

set s � f1; : : : ; mg we denote by A

s;:

the matrix (A

ij

)

i2s; j=1;:::;n

. With this notation the

asso iated ILU algorithm is roughly as follows.

Algorithm 1 (In omplete LU fa torization (ILU))

Given A = (A

ij

)

ij

2 R

n;n

and a drop toleran e � 2 [0; 1℄. Compute A � LDU .

L = U = I; S = A;D

11

= S

11

.

for i = 1 : n� 1

p

i+1:n

= S

>

i+1:n;i

=S

ii

, q

i+1:n

= S

i;i+1:n

=S

ii

Drop all entries jp

i

j; jq

i

j if they are less than � .

L

i+1:n;i

= p

>

i+1:n

, U

i;i+1:n

= q

i+1:n

.

S

i+1:n;i+1:n

= S

i+1:n;i+1:n

� L

i+1:n;i

D

i;i+1:n

� (S

i+1:n;i

� L

i+1:n;i

S

ii

)U

i;i+1:n

D

i+1;i+1

= S

i+1;i+1

end

2

Pra ti al versions of in omplete LU de ompositions are typi ally implemented in a slightly

di�erent way. It is usually not advisable to update the whole S

i+1:n;i+1:n

by a rank{one or

rank{two modi� ation. Instead, typi ally only the leading row of S

i+1:n;i+1:n

is omputed,

and the transformations on the other rows are post{poned. This orresponds to the so{

alled I,K,J version of Gaussian elimination[21℄. Besides saving memory, this approa h is

easier to implement sin e all updates and modi� ations are performed only on e for ea h

row. Thus one an use simple sparse row storage s hemes, e.g. the Compressed Sparse Row

(CSR) format [21℄.

Algorithm 2 (In omplete LU fa torization, I;K; J version)

Given A = (A

ij

)

ij

2 R

n;n

and a drop toleran e � 2 [0; 1℄. Compute A � LDU .

L = U = I; S = A; C = R = ;.

for i = 1 : n

w = A

i;:

for j = 1; : : : ; i� 1 and when w

k

6= 0

w

j

= w

j

=D

jj

if jw

j

j 6 � , w

j

= 0, else w

j+1:n

:= w

j+1:n

� w

j

U

j;j+1:n

end

for all j > i: if jw

j

=w

i

j 6 � , w

j

= 0

De�ne D

ii

= w

i

; U

i;i:n

= w

i:n

=D

ii

; L

i;1:i�1

= w

1:i�1

end

Mathemati ally Algorithm 2 an be read as a spe ial version of Algorithm 1, if the approx-

imate S hur{ omplement is repla ed by

S

i+1:n;i+1:n

= S

i+1:n;i+1:n

� L

i+1:n;i

D

i;i

U

i;i+1:n

:

Clearly this repla ement would also end up in an exa t LU de omposition on e we do not

drop entries anymore.

3 Stabilized ILU

One problem in dropping entries in Algorithm 1 or Algorithm 2 is that we do not have

ontrol of the hanges whi h are a�e ted by dropping. One way to get a more reliable

dropping riterion is to take the norm of the i{th row of A into a ount, e.g. repla e � by

� �kA

i;:

k

1

. This is essentially what the ILUT{Algorithm [19℄ does. A slightly re�ned version

of this strategy, at least if the information on the S hur{ omplement is available, ould be

to onsider the norm of the i{th row of the S hur{ omplement as well. This makes sense

espe ially when the orresponding row of the S hur{ omplement has signi� antly smaller

entries. I.e., instead of dropping entries that are less than � or � � kA

i;:

k

1

in absolute value,

we ould drop entries that are less than � �minfkA

i;:

k

1

; kS

i;i:n

k

1

g in absolute value. Often

both hoi es are a very good ompromise but learly there may still be ases where we

ould end up in a poor pre onditioner.

3

Algorithm 1, 2 an be supplemented with pivoting. When olumn pivoting is added to Al-

gorithm 2 it essentially orresponds to the ILUTP{Algorithm whi h is part of SPARSKIT,

see e.g. [21, 20℄. So far we have ignored this option to have more lear presentation. Later

on, we will return to this point and �nally in lude pivoting. For simpli ity let us onsider

the algorithms without pivoting at this stage.

Re ently it has been shown in [7℄ that Algorithm 1 has a strong relation to sparse approx-

imate inverse pre onditioners. Without going into the details, we will roughly des ribe the

idea of AINV{type algorithms [3, 4, 2, 8℄. The idea is to dire tly ompute upper triangular

matri es W;Z su h that W

>

AZ = D, with a diagonal matrix D. The version whi h we

will fo us on is the so{ alled right looking AINV, where W and Z are updated by a rank{1

update. Essentially a biorthogonalization pro ess forW and Z is performed, in whi hW

>

A

and Z

>

A

>

are transformed step by step to upper triangular form. Clearly this only holds

if no dropping is applied to W;Z.

Algorithm 3 (Fa tored Approximate INVerse, rank{1 update version)

Given A = (A

ij

)

ij

2 R

n;n

and a drop toleran e � 2 [0; 1℄. Compute A

�1

� ZD

�1

W

>

.

p = q = (0; : : : ; 0) 2 R

n

; Z = W = I

n

; C = R = ;.

for i = 1 : n

p

i:n

= Z

>

:;i

A

>

i:n;:

, q

i:n

= W

>

:;i

A

:;i:n

Set p

i+1:n

:= p

i+1:n

=p

i

; q

i+1:n

:= q

i+1:n

=q

i

W

:;i+1:n

= W

:;i+1:n

�W

:;i

p

i+1:n

, Z

:;i+1:n

= Z

:;i+1:n

� Z

:;i

q

i+1:n

Drop entries W

kl

of W

1:i;i+1:n

, if jW

kl

j 6 �

Drop entries Z

kl

of Z

1:i;i+1:n

, if jZ

kl

j 6 �

end

Choose diagonal entries of D as the omponents of p (or equivalently of q).

In prin iple we ould modify Algorithm 1 su h that the inverses of its triangular fa tors L; U

are omputed on the y. For this purpose we supplement Algorithm 1 with a progressive

inversion of L; U . At step i� 1, U is of the form

U =

�

U

1:i�1;1:i�1

U

1:i�1;i:n

O I

�

and the i-th step will ompute the entries U

i;i+1:n

and add them to the urrent U to get

U

new

. Let q

>

be the row ve tor q

>

= U

i;:

� e

>

i

. Note that the 'diagonal' element q

i

of q is

zero. Then,

U

new

= U + e

i

q

>

= (I + e

i

q

>

)U:

It follows that

U

�1

new

= U

�1

(I + e

i

q

>

)

�1

= U

�1

(I � e

i

q

>

) :

Of ourse analogous arguments hold for L. This provides a formula for progressively om-

puting L

�>

; U

�1

throughout the algorithm.We all the inverse fa tors Z;W as in Algorithm

3. With these additional fa tors Z;W and a modi�ed S hur{ omplement it was shown in

[7℄ that the supplemented version of Algorithm 1 is essentially equivalent to Algorithm 3.

4

Theorem 4 Suppose that Algorithm 1 is supplemented with a progressive inversion of

L; U . Suppose in addition that in step i of Algorithm 1 an entry L

ji

is dis arded only if

jL

ji

jmaxf1g [ fjW

ki

j : k < ig 6 � , i = 1; : : : ; n. Suppose that in Algorithm 1 and 3 W

kl

is dropped from W

1:i;i+1:n

if jW

kl

j 6 � . If the (modi�ed) S hur{ omplement S

i+1:n;i+1:n

is

de�ned via

S

i+1:n;i+1:n

=W

>

:;i+1:n

A

:;i+1:n

;

then we have for any k > l:

j(L

�>

)

kl

�W

kl

j 6 �(2(k � l)� 1)

and the diagonal entries of D are those of p.

Proof. See [7℄. 2

The most interesting point about this relation is that Theorem 4 requires to modify the

dropping strategy for L (and similarly for U). Now typi ally applying dropping to sparse

approximate inverse fa tors is less harmful than for in omplete LU de ompositions, be ause

in dropping small entries of size � inW;Z the e�e tive error inW

>

AZ is only between linear

and quadrati with respe t to � . And W

>

AZ is the matrix whi h needs to be transformed

to an approximately diagonal matrix D. On the other hand if we apply dropping to the

fa tors L; U of an ILU the related e�e t is rational sin e we do not know in advan e the

e�e t for L

�1

AU

�1

. But for pre onditioning, this is pre isely what we need to know. So if

we an onstru t an ILU that is somehow almost equivalent to an approximate inverse,

then we might hope that dropping is more reliable and the resulting pre onditioner is mu h

more eÆ ient for those situations where dropping has a serious impa t on the quality of

the pre onditioner. Numeri al results in [8℄ illustrate that for some extremely inde�nite

and ill{ onditioned problems the approximate inverse behaves better than an ILU .

To turn the result of Theorem 4 into an algorithm we will ertainly not invert L; U in

Algorithm 1. Let us take a look at the riterion for dropping entries in L. We need to

know maxf1g [ fjW

ki

j : k < ig, whi h means we need to know the i{th row of L

�1

, i.e.,

W

1:i�1;i

= (L

�1

)

i;1:i�1

. At least it would be onvenient to have an estimate for k(L

�1

)

i;1:i�1

k

1

whi h ould serve as a substitute for fjW

ki

j : k < ig. To do this we use a general ondition

estimator for triangular matri es from [14, 9℄ as a helpful estimate for k(L

�1

)

i;1:i�1

k

1

.

This ondition estimator is based on solving a system with an upper triangular matrix U

where the right hand side y only onsists of �1 and the signs are hosen to su essively

maximize the solution x of Ux = y. Another look at this ondition estimator shows that

the omponents of x = U

�1

y pre isely estimate k(U

�1

)

i;i:i:n

k

1

. To adapt this estimator to

our problem we will onsider Lx

L

= y

L

and U

>

x

U

= y

U

to get estimates for k(L

�1

)

i;1:i�1

k

1

and k(U

�1

)

1:i�1;i

k

1

.

Algorithm 5 (Condition Estimator for (L

�1

) adapted from [14, 9℄ )

Let L = (L

ij

)

ij

2 R

n;n

be unit lower triangular. Compute Lx = y, where y

>

2 (�1; � � � ;�1).

p = p

+

= p

�

= x = (0; : : : ; 0)

>

2 R

n

, and let � = 0 be the asso iated 1{norm of p

5

for i = 1 : n

x

+

= 1� p

i

, x

�

= �1� p

i

Let s be the set of nonzero omponents of L

i+1:n;i

p

+;s

= p

s

+ L

s;i

x

+

, p

�;s

= p

s

+ L

s;i

x

�

�

+

= � � kp

s

k

1

+ kp

+;s

k

1

, �

�

= � � kp

s

k

1

+ kp

�;s

k

1

if jx

+

j+ �

+

> jx

�

j+ �

�

x

i

= x

+

, � = �

+

p

s

= p

+;s

, p

�;s

= p

+;s

else

x

i

= x

�

, � = �

�

p

s

= p

�;s

, p

+;s

= p

�;s

end

end

In prin iple one ould also use di�erent ondition estimators, e.g. [5, 6℄. But what we really

need is not an estimate for the norm of L

�1

but an estimate for the norm of ea h row of

L

�1

. From this point of view to take as right hand side a ve tor y whi h only onsists of

�1 is reasonable and is more attra tive for this problem.

Now we an supplement Algorithm 1 with the ondition estimator Algorithm 5 applied to

the L and U

>

fa tors of the ILU and the omponents of x

L

= L

�1

y

L

; x

U

= U

�1

y

u

are

serving as estimates for (L

�1

)

i;1:i�1

, (U

�1

)

1:i�1;i

. We still have to dis uss the hoi e of the

approximate S hur{ omplement. Although Theorem 4 is based on de�ning the approximate

S hur{ omplement via S

i+1:n;i+1:n

= W

>

:;i+1:n

A

:;i+1:n

, it an be seen in the proof of this

Theorem that an analogous relation will hold for the ase where p; q of Algorithm 3 are

de�ned via

p

i:n

= Z

>

:;i

A

>

W

i:n;:

; q

i:n

=W

>

:;i

AZ

:;i:n

:(5)

In this ase the related S hur{ omplement for Theorem 4 is

S

i+1:n;i+1:n

=W

>

:;i+1:n

AZ

:;i+1:n

:(6)

In fa t, when Algorithm 3 is supplemented with pivoting, (5) is used to ensure that p

i

=

q

i

6= 0. Furthermore (6) is related to the hoi e of S in Algorithm 1 sin e it onsists of

taking the lower right blo k of

~

L

�1

A

~

U

�1

. Clearly the de�nition of the approximate S hur{

omplement in Algorithm 1 is not pre isely the same as (6). But a ording to [7℄, one has

a lose onne tion to Algorithm 3 with this hoi e of an approximate S hur{ omplement

if dropping is applied in slightly di�erent way.

As a next step to de�ne the ILU we introdu e pivoting. We de�ne permutation ve tors

�; �, su h that A(�; �) = LD(�; �)U provided that no dropping is applied. In prin iple,

applying permutation matri es �;� to (2), hanges this equation to

�

I O

O �

>

��

B F

E C

��

I O

O �

�

=

�

1 0

�

>

L

E

I

� �

D

B

0

0 �

>

S�

� �

1 U

F

�

0 I

�

:

This illustrates how S; L and U have to be adapted. It is lear that if we in lude the

ondition estimator, analogous hanges are made. It should also be obvious that in pra ti e

6

one will not physi ally inter hange rows of L and olumns of U but instead one uses index

ve tors.

In prin iple we an introdu e a pivoting pro ess to Algorithm 1 whi h ensures that in the

permuted matrix jp

i

j > �max

j=i+1;:::;n

jp

j

j and jq

i

j > �max

j=i+1;:::;n

jq

j

j. This guarantees

that after the division by p

i

; q

i

the entries of p

j

=p

i

, q

j

=q

i

are less than 1=� in absolute value.

Here the parameter � 2 [0; 1℄ is hosen a priori. The hoi e � = 1 refers to stri t pivoting,

i.e. the maximum entry in absolute value will be ome p

i

or q

i

, while any smaller hoi e of

� auses only pivoting if the diagonal entry is mu h smaller than the maximum entry of

jp

i+1:n

j; jq

i+1:n

j. Now we an go one step further and use the freedom in the hoi e of pivots

to add a strategy of Markowitz type [10℄, i.e., we onsider the set of pivots jp

k

j that are

larger than �max

j=i+1;:::;n

jp

j

j and among these we take the one with the minimum �ll{in.

This is a typi al strategy to maintain sparsity in the S hur{ omplement when using dire t

methods [10℄. To do this, repla e max

j=i+1;:::;n

jp

j

j by z and de�ne a set piv(p) by

piv(p; z) = fk : jp

k

j > �zg:(7)

For any k, let nnz

i

(k) denote the number of nonzeros of S

i:n;k

and let nnzr

i

(k) denote the

number of nonzeros of S

k;i:n

. As pivot we will hoose j 2 piv(p; z) su h that

nnz

i

(j) = min

k2piv(p;z)

nnz

i

(k):(8)

I.e., among all admissible pivots hoose the one whi h lo ally minimizes the �ll{in. The

same pro ess needs to be repeated for q. In theory this pro ess needs to be alternated

between p and q be ause we have to make sure the diagonal pivots are not getting smaller.

For this reason we always in rease z. A lo al pivoting step an then look as follows.

Algorithm 6 (Lo al pivoting with respe t to �ll{in)

Given A = (A

ij

)

ij

2 R

n;n

and a pivoting toleran e � 2 [0; 1℄.

Let S

i:n;i:n

denote the S hur{ omplement on entry to step i of Algorithm 1.

z = 0

while pivots not satisfa tory

p

i:n

= S

i;i:n

, z = maxfz;max

j=i;:::;n

jp

j

jg

Choose � 2 piv(p; z) su h that nnz

i

(�) is minimal.

Inter hange olumns/ omponents i; � of p; �; S

i:n;:

; U

1:i�1;:

q

i:n

= S

i:n;i

, z = maxfz;max

j=i;:::;n

jq

j

jg

Choose � 2 piv(q; z) su h that nnzr

i

(�) is minimal.

Inter hange olumns/ omponents i; � of q; �; S

:;i:n

; L

:;1:i�1

end

The while loop only terminates if no more inter hanges are performed.

Together with the ondition estimator in Algorithm 5, Algorithm 6 is used to stabilize the

in omplete LU de omposition from Algorithm 1. We summarize these hanges to a new

ILU algorithm.

7

Algorithm 7 (Stabilized In omplete LU fa torization (ILUSTAB))

Given A = (A

ij

)

ij

2 R

n;n

, a drop toleran e � 2 [0; 1℄ and a pivoting toleran e � 2 [0; 1℄.

Compute A(�; �) � LDU .

L = U = I; S = A;D

11

= S

11

; � = � = (1; : : : ; n).

x

L

= p

L

= p

+;L

= p

�;L

= x

U

= p

U

= p

+;U

= p

�;U

= (0; : : : ; 0)

>

2 R

n

, �

L

= �

U

= 0.

for i = 1 : n� 1

Apply Algorithm 6.

Whenever p requires pivoting, inter hange p

U

; p

+;U

; p

�;U

; x

U

as well

Whenever q requires pivoting, inter hange p

L

; p

+;L

; p

�;L

; x

L

as well

L

i+1:n;i

= p

>

i+1:n

, U

i;i+1:n

= q

i+1:n

.

Apply step i of Algorithm 5 for L with

�; x; p; p

+

; p

�

repla ed by �

L

; x

L

; p

L

; p

+;L

; p

�;L

Apply step i of Algorithm 5 for U

>

with

�; x; p; p

+

; p

�

repla ed by �

U

; x

U

; p

U

; p

+;U

; p

�;U

drop all entries jL

ji

j of L

i+1:n;i

, if jL

ji

jmaxf1; jx

L;i

jg 6 � minfkA

i;:

k

1

; kS

i;i:n

k

1

g

drop all entries jU

ji

j of U

i;i+1:n

, if jU

ij

jmaxf1; jx

U;i

jg 6 � minfkA

i;:

k

1

; kS

i;i:n

k

1

g

S

i+1:n;i+1:n

= S

i+1:n;i+1:n

� L

i+1:n;i

S

i;i+1:n

� (S

i+1:n;i

� L

i+1:n;i

S

ii

)U

i;i+1:n

D

i+1;i+1

= S

i+1;i+1

end

The two major di�eren es between Algorithm 1 and Algorithm 7 are the appli ation of

pivoting and the in lusion of a ondition estimator. The latter is motivated by the strong

relations between in omplete LU fa torizations and fa tored approximate inverse pre on-

ditioners.

4 Numeri al Results

This se tion presents numeri al experiments to validate the algorithms. So far, Algorithm

7 is implemented in MATLAB [1℄.

� The matri es are initially reordered using the symmetri minimum degree ordering

[13℄.

� An a priori s aling is used su h any row of the given matrix has unit 1{norm.

� For the pivoting pro ess � = 0:1 is used.

� Di�erent values were used for the drop toleran e � = 0:1; 0:3.

For the numeri al experiments several unsymmetri matri es from the Harwell{Boeing

Colle tion [11℄ were hosen.

8

The result are ompared with

� LU from MATLAB also with pivoting toleran e � = 0:1

� LUINC from MATLAB with � = 0:1 and drop toleran es � = 0:1, 0:01, 10

�3

,

10

�4

, 10

�5

� ILUTP from SPARSKIT using the same toleran e � = 0:1 for pivoting but � = 0:1,

0:01, 10

�3

, 10

�4

, 10

�5

for dropping.

The numeri al results for ILUTP [21℄ were performed on an SGI workstation with two

190 MHz R10000 (IP25) pro essors under IRIX 6.2 and 512 MB memory.

As iterative solvers GMRES(30) [22℄ is used. The iteration was stopped after the residual

norm was less than

p

eps times the initial residual norm, where eps � 2:2204 � 10

�16

denotes the ma hine pre ision. The iteration was stopped after 500 steps. Every iterative

solution whi h broke down or did not onverge within the number of steps was noted as a

failure.

We brie y des ribe the results for several matri es and then give detailed numeri al results

for several sele ted examples.

To give a rough idea on how the method performed on the Harwell{Boeing olle tion we

simply summarize in Table 1 whi h method su essfully solved how many problems with

respe t to the drop toleran e � . The tests were done on 94 matri es from the Harwell-Boeing

olle tion.

Table 1: Summary of results | Su essful Computation

Harwell{Boeing Colle tion (94 test matri es)

Pre onditioner Drop toleran e �

0:3 0:1 0:01 10

�3

10

�4

10

�5

ILUSTAB 89 92

LUINC 31 52 68 79 87

ILUTP 53 69 78 84 90

Note that there were only two matri es whi h ould not be solved with ILUSTAB for

� 2 f0:3; 0:1g. These are the matri es fa simile/fs7603, grenoble/gre216b. These matri es

ould be solved with � = 0:01. But LUINC ould also not solve fa simile/fs7603 and

for fa simile/fs7603 ILUTP needed � = 10

�4

. For grenoble/gre216b LUINC and ILUTP

needed � = 10

�5

.

We now omment on several matri es from the Harwell{Boeing{Colle tion. This olle tion

onsists of many matri es from di�erent areas. Related matri es are put together in a group

and omments are done with respe t to these groups. For some sele ted examples we will

9

show separate tables. In ea h table (e.g., Table 2) we will present the the hoi e of the

drop toleran e � and the related �ll{in fa tor (that is the ratio of the number of nonzeros

of L + U divided by the number of nonzeros of A). Next the number of iteration steps

using GMRES(30) is shown. For the MATLAB algorithms LU, ILUSTAB and LUINC

we use the op ount as measure for the number of operations. The op ount is split

into the ops required for the de omposition and the ops to solve a linear system using

GMRES(30).

� CHEMWEST: These matri es are some of those for whi h LUINC and ILUTP needed

smallest drop toleran es to be su essful while ILUSTAB was able to solve all of

them already for � = 0:3. Detailed results for the three biggest WEST{matri es are

given in Table 2, 3, 4.

Table 2: Matrix CHEMWEST/WEST0989

Method / � �ll{in # it. ops

fa tor steps de . solve

sparse LU 3.2 1 1:2�10

5

9:9�10

4

0:3 1.3 20 1:8�10

5

1:4�10

6

ILUSTAB

0:1 1.5 14 1:7�10

5

8:3�10

5

10

�1

0.7 | 1:1�10

4

|

10

�2

1.0 | 1:4�10

4

|

LUINC 10

�3

1.2 | 2:0�10

4

|

10

�4

1.6 | 3:8�10

4

|

10

�5

1.9 6 4:7�10

4

2:7�10

5

10

�1

1.0 |

10

�2

1.4 |

ILUTP 10

�3

1.9 |

10

�4

2.4 309

10

�5

2.7 10

� FACSIMILE: LUINC from MATLAB ould not solve most of these matri es for

� = 0:1; 0:01. For � = 10

�3

it was able to solve 50% of them and for � = 10

�4

; 10

�5

only fs1836, fs7602, fs7603 ould not be solved. For those problems that ould be

solved, the �ll{in was moderate and the number of iteration steps was small.

In ontrast to this ILUSTAB ould solve all of these matri es already for � = 0:3

ex ept fs7603 whi h ould not be solved. The �ll{in was small as well. The number

of iteration steps was small ex ept for fs7602 whi h required 60 steps for � = 0:3 and

31 for � = 0:1.

ILUTP solved most of these problems for � = 0:1. All problems in luding fs7603 were

solved for � = 10

�4

; 10

�5

.

10

Table 3: Matrix CHEMWEST/WEST1505

Method / � �ll{in # it. ops

fa tor steps de . solve

sparse LU 4.2 1 4:0�10

5

1:7�10

5

0:3 1.4 22 3:5�10

5

2:5�10

6

ILUSTAB

0:1 1.7 17 4:0�10

5

1:7�10

6

10

�1

0.7 | 1:7�10

4

|

10

�2

1.0 | 2:2�10

4

|

LUINC 10

�3

1.2 | 3:2�10

4

|

10

�4

1.7 16 6:8�10

4

1:5�10

6

10

�5

2.0 6 8:6�10

4

4:1�10

5

10

�1

1.0 |

10

�2

1.4 |

ILUTP 10

�3

1.9 |

10

�4

2.4 |

10

�5

2.7 |

Table 4: Matrix CHEMWEST/WEST2021

Method / � �ll{in # it. ops

fa tor steps de . solve

sparse LU 5.6 1 1:2�10

6

2:7�10

5

0:3 1.6 20 6:8�10

5

2:9�10

6

ILUSTAB

0:1 1.7 14 6:7�10

5

1:7�10

6

10

�1

0.7 | 2:2�10

4

|

10

�2

0.9 | 3:0�10

4

|

LUINC 10

�3

1.2 | 4:6�10

4

|

10

�4

1.6 | 8:7�10

4

|

10

�5

1.9 6 1:2�10

5

5:5�10

5

10

�1

1.0 |

10

�2

1.4 |

ILUTP 10

�3

1.9 |

10

�4

2.5 |

10

�5

3.1 14

11

For those problems that ould be solved the �ll{in was small. The largest number of

iterations were 155 for fs7602 and � = 0:1, 62 for fs7602 and � = 10

�3

. For all other

methods it was less, if they ould be solved at all.

� GEMAT: ILUSTAB ould not solve these matri es for � = 0:3 but for � = 0:1.

LUINC ould solve these matri es for � = 10

�4

but with roughly four times of the

�ll{in of ILUSTAB.

ILUTP ould solve these matri es for � = 10

�3

but with more than twi e as mu h

�ll{in as ILUSTAB. For these matri es the LU de omposition needed more than 70

times of �ll than the initial matrix.

For gemat12 see Table 5. The results for gemat11 are similar.

Table 5: Matrix GEMAT/GEMAT12

Method / � �ll{in # it. ops

fa tor steps de . solve

sparse LU 73.5 1 1:7�10

9

1:0�10

7

0:3 1.0 | 1:4�10

6

|

ILUSTAB

0:1 1.3 67 2:0�10

6

3:5�10

7

10

�1

0.6 | 1:8�10

5

|

10

�2

1.4 | 1:6�10

5

|

LUINC 10

�3

2.7 | 1:3�10

7

|

10

�4

5.2 10 5:4�10

7

5:7�10

6

10

�5

9.2 5 1:3�10

8

3:9�10

6

10

�1

1.0 |

10

�2

2.0 |

ILUTP 10

�3

3.4 17

10

�4

5.2 7

10

�5

7.4 4

� GRENOBLE: for � = 0:1 LUINC ould only solve gre115, gre216a, gre343, gre512.

But even for some of those the �ll{in fa tor was already enormous ( e.g. 5:9 for

gre216a, 7:7 for gre343, 11:3 for gre512). The same problem o urred for the other

matri es that ould only be solved for smaller � . All matri es ould �nally be solved

with � = 10

�5

.

ILUSTAB solved all matri es ex ept gre216b, gre1107 for � = 0:3. The �ll{in was

slightly better (e.g. i.e. 3:8 for gre216a, 4:9 for gre343, 7:8 for gre512). gre1107 ould

be solved with � = 0:1 but with a �ll{in fa tor 7:4. This was still better than LUINC,

whi h needed � = 10

�3

and produ ed a �ll{in fa tor 23:0!

For � = 0:1, ILUTP ould solve gre115, gre185, gre216a. But even then the �ll{in

fa tor was sometimes large ( i.e. 7:0 for gre216a, 12:6 for gre512). The same problem

12

o urred for the other matri es that ould only be solved for smaller � . For example

gre1107 ould be solved with � = 10

�3

and a �ll{in fa tor 21:3. All matri es ould

�nally be solved with � = 10

�5

.

The problem with the �ll{in also extremely a�e ts the sparse LU de omposition. For

example gre1107 required a �ll{in fa tor 44:1!

For those problems that ould be solved by one of these methods the number of

iteration steps was moderate.

� LNS: ILUSTAB solved them all for � = 0:3. The �ll{in was moderate (3:6 for lns3937

was already maximum) an so was the number of iteration steps (at most 29).

LUINC ould not solve any of these matri es for � = 0:1; 0:01 but lns511, lnsp511,

lns3937, lnsp3937 for � = 10

�3

. The biggest matri es required twi e as mu h �ll{in

as ILUSTAB.

ILUTP ould solve the two smallest matri es for � = 0:1 and the medium size matri es

for � = 10

�3

.

The two biggest matri es ould only be solved for � = 10

�5

. For lns3937 see Table

6. The results for lnsp3937 were quite similar.

Table 6: Matrix LNS/LNS3937

Method / � �ll{in # it. ops

fa tor steps de . solve

sparse LU 46.1 1 2:9�10

8

4:9�10

6

0:3 3.6 28 2:3�10

7

1:4�10

7

ILUSTAB

0:1 4.9 16 4:1�10

7

7:7�10

6

10

�1

1.0 | 6:4�10

5

|

10

�2

3.7 | 1:0�10

7

|

LUINC 10

�3

7.4 29 3:2�10

7

2:0�10

7

10

�4

12.3 9 6:6�10

7

7:1�10

6

10

�5

17.0 9 1:0�10

8

5:0�10

6

10

�1

0.8 |

10

�2

1.4 |

ILUTP 10

�3

2.5 |

10

�4

3.5 |

10

�5

4.4 |

� NUCL: ILUSTAB ould solve all matri es for � = 0:3 but the �ll{in was poor, e.g.,

28.6 for nn 1374. The number of iteration steps was at most 28.

LUINC did not solve any of these matri es for � = 10

�1

; : : : ; 10

�5

.

ILUTP ould solve all the problem for � = 10

�3

and a better �ll{in fa tor than

ILUSTAB (e.g. 6.6 for nn 1374 but 463 iteration steps).

13

Here the dire t solver produ ed signi� antly less �ll{in for nn 1374 (fa tor 14.6) than

ILUSTAB.

� PORES: PORES1, PORES3 ould be solved by ILUSTAB for � = 0:3 and ILUTP

for � = 0:1. LUINC needed � = 0:01 for PORES3. The number of iteration steps

was small ex ept for PORES3, � = 0:1 and ILUTP whi h needed 248 steps, but for

� = 0:01 the number of steps were small while the �ll{in was still below the �ll{in of

the original matrix. For matrix PORES2 see Table 7.

Table 7: Matrix PORES/PORES2

Method / � �ll{in # it. ops

fa tor steps de . solve

sparse LU 5.1 1 2:5�10

6

2:9�10

5

0:3 1.0 54 6:8�10

5

6:9�10

6

ILUSTAB

0:1 1.2 15 9:6�10

5

1:5�10

6

10

�1

0.5 | 4:4�10

4

|

10

�2

0.6 113 6:0�10

4

1:4�10

7

LUINC 10

�3

1.1 26 1:6�10

5

3:1�10

6

10

�4

1.8 9 4:1�10

5

8:3�10

5

10

�5

2.4 5 6:7�10

5

4:7�10

5

10

�1

0.4 |

10

�2

0.8 |

ILUTP 10

�3

1.7 30

10

�4

3.2 13

10

�5

4.6 9

� SAYLOR: SAYLR1/SAYLR3 were solved by ILUSTAB for � = 0:3 and ILUTP for

� = 0:1. For SAYLR3, LUINC failed for all � . For SAYLR4 see Table 8.

� SHERMAN: ILUSTAB solved all the matri es for � = 0:3, but for sherman3 it

needed 138 iteration steps. For the other matri es the iteration ount was less than

half as mu h. The �ll{in was less than twi e as mu h as the initial �ll. The number of

iterations was mu h lower for � = 0:1 but with more �ll{in. LUINC ould only solve

sherman4, sherman5 for � = 0:1 and it needed 123 iteration steps for sherman5. For

� = 0:01 it needed only a moderate number of iteration steps, but sherman1 still

ould not be solved for � = 10

�2

. ILUTP ould solve all matri es but for sherman2

it needed � = 10

�5

(see Table 9). For sherman4 and � = 0:1 the number of iteration

steps (449) was still big. This hanged when using � = 0:01.

14

Table 8: Matrix SAYLOR/SAYLR4

Method / � �ll{in # it. ops

fa tor steps de . solve

sparse LU 14.7 1 3:7�10

7

1:5�10

6

0:3 2.6 44 1:6�10

7

1:8�10

7

ILUSTAB

0:1 3.1 15 2:0�10

7

5:2�10

6

10

�1

0.6 | 1:1�10

5

|

10

�2

0.6 | 1:1�10

5

|

LUINC 10

�3

0.6 | 1:1�10

5

|

10

�4

1.6 33 9:5�10

5

1:2�10

7

10

�5

2.7 11 3:5�10

6

3:1�10

6

10

�1

0.6 352

10

�2

0.6 155

ILUTP 10

�3

0.6 153

10

�4

2.4 18

10

�5

3.5 8

Table 9: Matrix SHERMAN/SHERMAN2

Method / � �ll{in # it. ops

fa tor steps de . solve

sparse LU 14.0 1 8:1�10

7

1:4�10

6

0:3 0.4 30 1:6�10

6

4:4�10

6

ILUSTAB

0:1 0.6 14 2:6�10

6

1:7�10

6

10

�1

0.2 | 9:7�10

4

|

10

�2

0.4 21 2:6�10

5

2:6�10

6

LUINC 10

�3

0.7 7 7:7�10

5

7:5�10

5

10

�4

1.1 5 2:4�10

6

6:2�10

5

10

�5

1.7 4 5:1�10

6

5:9�10

5

10

�1

0.3 |

10

�2

0.6 |

ILUTP 10

�3

1.0 |

10

�4

1.6 |

10

�5

2.1 61

15

The numeri al examples have illustrated the robustness of taking the growth of the in-

verse triangular fa tors into a ount when omputing an in omplete LU de omposition.

Of ourse ILUSTAB is neither always the most eÆ ient nor always the fastest (with respe t

to the ops ) nor always the ILU with the smallest amount of �ll{in. But in many ases

it is a pretty good ompromise between standard in omplete LU de ompositions and the

full sparse LU de omposition. In many examples it is not ne essary to use a trial{and{

error strategy for hoosing the drop toleran e. The drop toleran e is automati ally adapted

with respe t to the growth of the inverse fa tors. In several ases where a dire t solver is

superior to iterative method ( f.Table 2), 3, 4 with respe t to the number of ops, the

�ll{in for ILUSTAB is still moderate and often even less less than that for LUINC, ILUTP.

Conversely on some problems whi h ause trouble to dire t solvers ( f. Table 5) ILUSTAB

gains from its sparsity and being used as iterative solver.

The drawba k of this algorithm is of ourse that it is more omparable with sparse dire t

solvers be ause it requires expli it knowledge of the S hur{ omplement. Clearly there are

several problems where standard in omplete LU de ompositions used as pre onditioners

give powerful iterative solvers. In these ases apparently ILUSTAB will be slower be ause

one has a ertain time onsuming overhead for omputing and administrating the approx-

imate S hur{ omplement.

5 Con lusions

A version of an in omplete LU de omposition has been presented that performs dropping

with respe t to the growth of the inverses of the triangular fa tors. We have illustrated that

the resulting pre onditioner is very robust. Often one an avoid adapting the parameters

to a spe i� matrix and still get a pre onditioner that is omputed in a sensible time with

moderate �ll{in. For many examples this has turned out to be a good ompromise between

sparse dire t solvers and standard in omplete LU de ompositions. Sin e this pre onditioner

shares several properties with sparse dire t solvers, an implementation based on modi�ed

dire t solvers seems to be reasonable. Currently odes from dire t solvers like the Harwell{

Subroutine{Library are under investigation to build this kind of pre onditioner. Real{time

results for bigger problems will be presented in a forth oming paper.

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Laboratory, S ienti� Computing Group, 1999.

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16

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Institue, University of Minnesota, 2000.

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[16℄ S. Khar henko, L. Kolotilina, A. Nikishin, and A. Yeremin. A reliable AINV{type pre ondi-

tioning method for onstru ting sparse approximate inverse pre onditioners in fa tored form.

Te hni al report, Russian A ademy of S ien es, Mos ow, 1999.

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17

Other titles in the SFB393 series:

99-01 P. Kunkel, V. Mehrmann, W. Rath. Analysis and numeri al solution of ontrol problems

in des riptor form. January 1999.

99-02 A. Meyer. Hierar hi al pre onditioners for higher order elements and appli ations in om-

putational me hani s. January 1999.

99-03 T. Apel. Anisotropi �nite elements: lo al estimates and appli ations (Habilitationss hrift).

January 1999.

99-04 C. Villagonzalo, R. A. R�omer, M. S hreiber. Thermoele tri transport properties in disor-

dered systems near the Anderson transition. February 1999.

99-05 D. Mi hael. Notizen zu einer geometris h motivierten Plastizit�atstheorie. Februar 1999.

99-06 T. Apel, U. Rei hel. SPC-PM Po 3D V 3.3, User's Manual. February 1999.

99-07 F. Tr�oltzs h, A. Unger. Fast solution of optimal ontrol problems in the sele tive ooling

of steel. Mar h 1999.

99-08 W. Rehm, T. Ungerer (Eds.). Ausgew�ahlte Beitr�age zum 2. Workshop Cluster-Computing

25./26. M�arz 1999, Universit�at Karlsruhe. M�arz 1999.

99-09 M. Arav, D. Hershkowitz, V. Mehrmann, H. S hneider. The re ursive inverse eigenvalue

problem. Mar h 1999.

99-10 T. Apel, S. Ni aise, J. S h�oberl. Crouzeix-Raviart type �nite elements on anisotropi

meshes. May 1999.

99-11 M. Jung. Einige Klassen iterativer Aufl�osungsverfahren (Habilitationss hrift). Mai 1999.

99-12 V. Mehrmann, H. Xu. Numeri al methods in ontrol, from pole assignment via linear

quadrati to H

1

ontrol. June 1999.

99-13 K. Bernert, A. Eppler. Two-stage testing of advan ed dynami subgrid-s ale models for

Large-Eddy Simulation on parallel omputers. June 1999.

99-14 R. A. R�omer, M. E. Raikh. The Aharonov-Bohm e�e t for an ex iton. June 1999.

99-15 P. Benner, R. Byers, V. Mehrmann, H. Xu. Numeri al omputation of de ating subspa es

of embedded Hamiltonian pen ils. June 1999.

99-16 S. V. Nepomnyas hikh. Domain de omposition for isotropi and anisotropi ellipti prob-

lems. July 1999.

99-17 T. Stykel. On a riterion for asymptoti stability of di�erential-algebrai equations. August

1999.

99-18 U. Grimm, R. A. R�omer, M. S hreiber, J. X. Zhong. Universal level-spa ing statisti s in

quasiperiodi tight-binding models. August 1999.

99-19 R. A. R�omer, M. Leadbeater, M. S hreiber. Numeri al results for two intera ting parti les

in a random environment. August 1999.

99-20 C. Villagonzalo, R. A. R�omer, M. S hreiber. Transport Properties near the Anderson Tran-

sition. August 1999.

99-21 P. Cain, R. A. R�omer, M. S hreiber. Phase diagram of the three-dimensional Anderson

model of lo alization with random hopping. August 1999.

99-22 M. Bollh�ofer, V. Mehrmann. A new approa h to algebrai multilevel methods based on

sparse approximate inverses. August 1999.

99-23 D. S. Watkins. In�nite eigenvalues and the QZ algorithm. September 1999.

99-24 V. Uski, R. A. R�omer, B. Mehlig, M. S hreiber. In ipient lo alization in the Anderson

model. August 1999.

99-25 A. Meyer. Proje ted PCGM for handling hanging in adaptive �nite element pro edures.

September 1999.

99-26 F. Milde, R. A. R�omer, M. S hreiber. Energy-level statisti s at the metal-insulator transi-

tion in anisotropi system. September 1999.

99-27 F. Milde, R. A. R�omer, M. S hreiber, V. Uski. Criti al properties of the metal-insulator

transition in anisotropi systems. O tober 1999.

99-28 M. The�. Parallel multilevel pre onditioners for thin shell problems. November 1999.

99-29 P. Biswas, P. Cain, R. A. R�omer, M. S hreiber. O�-diagonal disorder in the Anderson

model of lo alization. November 1999.

99-30 C. Mehl. Anti-triangular and anti-m-Hessenberg forms for Hermitian matri es and pen ils.

November 1999.

99-31 A. Barinka, T. Bars h, S. Dahlke, M. Konik. Some remarks for quadrature formulas for

re�nable fun tions and wavelets. November 1999.

99-32 H. Harbre ht, C. Perez, R. S hneider. Biorthogonal wavelet approximation for the oupling

of FEM-BEM. November 1999.

99-33 C. Perez, R. S hneider. Wavelet Galerkin methods for boundary integral equations and the

oupling with �nite element methods. November 1999.

99-34 W. Dahmen, A. Kunoth, R. S hneider. Wavelet least squares methods for boundary value

problems. November 1999.

99-35 S. I. Solovev. Convergen e of the modi�ed subspa e iteration method for nonlinear eigen-

value problems. November 1999.

99-36 B. Heinri h, B. Nkemzi. The Fourier-�nite-element method for the Lam�e equations in ax-

isymmetri domains. De ember 1999.

99-37 T. Apel, F. Milde, U. Rei hel. SPC-PM Po 3D v 4.0 - Programmers Manual II. De ember

1999.

99-38 B. Nkemzi. Singularities in elasti ity and their treatment with Fourier series. De ember

1999.

99-39 T. Penzl. Eigenvalue de ay bounds for solutions of Lyapunov equations: The symmetri

ase. De ember 1999.

99-40 T. Penzl. Algorithms for model redu tion of large dynami al systems. De ember 1999.

00-01 G. Kunert. Anisotropi mesh onstru tion and error estimation in the �nite element

method. January 2000.

00-02 V. Mehrmann, D. Watkins. Stru ture-preserving methods for omputing eigenpairs of large

sparse skew-Hamiltonian/Hamiltonian pen ils. January 2000.

00-03 X. W. Guan, U. Grimm, R. A. R�omer, M. S hreiber. Integrable impurities for an open

fermion hain. January 2000.

00-04 R. A. R�omer, M. S hreiber, T. Vojta. Disorder and two-parti le intera tion in low-

dimensional quantum systems. January 2000.

00-05 P. Benner, R. Byers, V. Mehrmann, H. Xu. A uni�ed de ating subspa e approa h for lasses

of polynomial and rational matrix equations. January 2000.

00-06 M. Jung, S. Ni aise, J. Tabka. Some multilevel methods on graded meshes. February 2000.

00-07 H. Harbre ht, F. Paiva, C. Perez, R. S hneider. Multis ale Pre onditioning for the Coupling

of FEM-BEM. February 2000.

00-08 P. Kunkel, V. Mehrmann. Analysis of over- and underdetermined nonlinear differential-

algebrai systems with appli ation to nonlinear ontrol problems. February 2000.

00-09 U.-J. G�orke, A. Bu her, R. Krei�ig, D. Mi hael. Ein Beitrag zur L�osung von

Anfangs-Randwert-Problemen eins hlie�li h der Materialmodellierung bei �niten elastis h-

plastis hen Verzerrungen mit Hilfe der FEM. M�arz 2000.

00-10 M. J. Martins, X.-W. Guan. Integrability of the D

2

n

vertex models with open boundary.

Mar h 2000.

00-11 T. Apel, S. Ni aise, J. S h�oberl. A non- onforming �nite element method with anisotropi

mesh grading for the Stokes problem in domains with edges. Mar h 2000.

00-12 B. Lins, P. Meade, C. Mehl, L. Rodman. Normal Matri es and Polar De ompositions in

Inde�nite Inner Produ ts. Mar h 2000.

00-13 C. Bourgeois. Two boundary element methods for the lamped plate. Mar h 2000.

00-14 C. Bourgeois, R. S hneider. Biorthogonal wavelets for the dire t integral formulation of the

heat equation. Mar h 2000.

00-15 A. Rathsfeld, R. S hneider. On a quadrature algorithm for the pie ewise linear ollo ation

applied to boundary integral equations. Mar h 2000.

00-16 S. Meinel. Untersu hungen zu Dru kiterationsverfahren f�ur di htever�anderli he Str�omungen

mit niedriger Ma hzahl. M�arz 2000.

00-17 M. Konstantinov, V. Mehrmann, P. Petkov. On Fra tional Exponents in Perturbed Matrix

Spe tra of Defe tive Matri es. April 2000.

00-18 J. Xue. On the blo kwise perturbation of nearly un oupled Markov hains. April 2000.

00-19 N. Arada, J.-P. Raymond, F. Tr�oltzs h. On an Augmented Lagrangian SQP Method for a

Class of Optimal Control Problems in Bana h Spa es. April 2000.

00-20 H. Harbre ht, R. S hneider. Wavelet Galerkin S hemes for 2D-BEM. April 2000.

00-21 V. Uski, B. Mehlig, R. A. R�omer, M. S hreiber. An exa t-diagonalization study of rare

events in disordered ondu tors. April 2000.

00-22 V. Uski, B. Mehlig, R. A. R�omer, M. S hreiber. Numeri al study of eigenve tor statisti s

for random banded matri es. May 2000.

00-23 R. A. R�omer, M. Raikh. Aharonov-Bohm os illations in the ex iton lumines en e from a

semi ondu tor nanoring. May 2000.

00-24 R. A. R�omer, P. Zies he. Hellmann-Feynman theorem and u tuation- orrelation analysis

of i the Calogero-Sutherland model. May 2000.

00-25 S. Beu hler. A pre onditioner for solving the inner problem of the p-version of the FEM.

May 2000.

00-26 C. Villagonzalo, R.A. R�omer, M. S hreiber, A. Ma Kinnon. Behavior of the thermopower

in amorphous materials at the metal-insulator transition. June 2000.

00-27 C. Mehl, V. Mehrmann, H. Xu. Canoni al forms for doubly stru tured matri es and pen ils.

June 2000. S. I. Solov'ev. Pre onditioned gradient iterative methods for nonlinear eigenvalue

problems. June 2000.

00-29 A. Eilmes, R. A. R�omer, M. S hreiber. Exponents of the lo alization lengths in the bipartite

Anderson model with o�-diagonal disorder. June 2000.

00-30 T. Grund, A. R�os h. Optimal ontrol of a linear ellipti equation with a supremum-norm

fun tional. July 2000.

The omplete list of urrent and former preprints is available via

http://www.tu- hemnitz.de/sfb393/preprints.html.