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S. Lungten

Promotor: W. H. A. Schilders

Supervisor: J. M. L. Maubach

SOLUTION OF INDEFINITE LINEAR

SYSTEMS

Center for Analysis, Scientific Computing and

Applications (CASA)

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Outline

Introduction

Factorization of indefinite matrices

Numerical experiments

Conclusion and future work

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• Previous methods

• Current method

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Introduction

known as saddle point problems:

• is an symmetric and positive (semi)

definite.

To solve symmetric indefinite linear systems of the form

• is an symmetric and positive definite matrix,

• is an matrix of rank with ,

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• mixed finite element methods

• regularized, weighted least squares

• discretization of PDEs

• constraint optimization

• Stokes

• electric circuits and networks

• economic models

Introduction

Applications leading to saddle point problems:

The saddle point matrix appear in two forms:

Case 1.

Case 2.

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Introduction

• Direct solvers

• Iterative solvers

• Preconditioning techniques

Solution methods:

• existence

• sparsity permutation

• stability

Each method has issues with one of the following:

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Introduction

For example (in terms of sparsity), consider a matrix

,

and its permuted form:

.

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Introduction

The decomposition: gives:

while gives:

.

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PART I

Saddle point matrices

with the case

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where is an upper triangular and is an

matrix.

Solution method

I. Transform into as follows:

.

• is a permutation or an orthogonal matrix

• is congruent to

• is equivalent to such that

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Solution method

III. Factorize .

II. Split into a block 3 by 3 structure [S., 2009]:

.

IV. Solve by using the factors of where

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Existence of factorization (previous result)

Schilders [S., 2009] showed the factorization of the form :

exists, i.e.,

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because of the existence of :

This existence proof requires to be an

upper triangular form.

Existence of factorization (previous result)

.

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We do the following transformation:

such that where is an

lower triangular matrix.

Current approach : Transformation

Aim: sparse block factors

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Method relies on:

• Crout decomposition:

Current approach

• Micro-block factorization [Maubach and S., 2012]

(applied for the case upper triangular form of ).

-has computational efficiency equivalent to that of

Cholesky for a symmetric positive definite matrix .

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Micro-block partitioning

Algorithm for the factorization

Micro-block factorization

Back permutation

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Block partitioning: Example ( )

is macro-block partitioned:

Define a permutation matrix as in [S., 2009]:

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which leads to the micro-block partitioned matrix,

with micro-blocks of order 1 and 2.

Block partitioning: Example ( )

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leads to:

Micro-block factorization: eg ( )

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Back permuting and give:

Induced macro-block factors: eg ( )

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Macro-block factorization: Existence

For general and ,

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Micro-block partitioning

Algorithm for the factorization

Micro-block factorization

Back permutation

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Define,

Comparison with Schilders’

Eliminating and from and gives another

form, which resembles Schilders’ form.

Then,

where .

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Comparison with Schilders’

Differences between the non-trivial blocks:

Similarity between the non-trivial blocks:

Both and are determined from

, where is the basis of null space of .

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current through

resistors

injected current

through nodes

Resistor network modeling [Rommes and Schilders, 2010]:

Numerical experiments

voltage

across nodes

diagonal matrix

(resistance values

of resistors)

incidence matrix

• full rank

• (entries:

at most 2 nonzero in

each column )

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Two visual representations of graphs related to resistor

networks:

Numerical experiments

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Saddle point matrix representation of the graphs of

and :

Numerical experiments

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Transformation: of

Schilders’ , nz = 22371 Current , nz = 22371

Numerical experiments

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Macro-block factors of of :

Schilders’ , Current ,

Numerical experiments

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Schilders’ , nz = 23526 Current , nz = 23526

Numerical experiments

Transformation: of

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Schilders’ , Current ,

Numerical experiments

Macro-block factors of of :

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PART II

Saddle point matrices

with the case

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Regularized saddle point systems

Consider a regularized saddle point matrix:

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Consider for the case is a diagonal matrix.

• is an symmetric and positive (semi)

definite.

• is an symmetric and positive definite matrix,

• is an matrix of rank with ,

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Transformation of to

Due to :

• is congruent to

• is congruent to

• is equivalent to such that

, where is an

upper triangular matrix.

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Macro-block factorization of

The macro-block factorization is of the form:

This factorization exists and has its blocks with these

shapes, which is shown in [J.M.L. Maubach and S., “Micro- and

macro-block factorizations for regularized saddle point problems”,

submitted]

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In particular,

where

Macro-block factorization of

• The presented factorizations are all exact, but

can be used as a basis for preconditioning.

• Especially the method with lower triangular is

attractive for preconditiong (topic of further

research).

• More numerical experiments from various

applications.

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Concluding remarks and future work

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References

[1] W. H. A. Schilders, Solution of indefinite linear systems using an LQ

decomposition for the linear constraints, Linear Algebra and Applications,

pp. 381-395, 431 (2009).

[2] J. M. L. Maubach, W. H. A. Schilders, Micro- and macro-block factorizations for

regularized saddle point systems, Technical report, Center for Analysis, Scientific

computing and Applications, Eindhoven University of Technology, April 2012.

[3] J. Rommes and W. H. A. Schilders, Efficient Methods for Large Resistor

Networks, IEEE Transactions on Computer-Aided Design of Integrated Circuits

and Systems, 29 (2010), 28-39.

[4] S. Lungten, J. M. L. Maubach, W. H. A. Schilders, Sparse inverse incidence

matrices for Schilders’ factorization applied to resistor network modeling,

submitted to NACO, Dec. 2013.

THANK YOU

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