SOLUTION OF INDEFINITE LINEAR SYSTEMS · S. Lungten PAGE 36 References [1] W. H. A. Schilders,...
Transcript of SOLUTION OF INDEFINITE LINEAR SYSTEMS · S. Lungten PAGE 36 References [1] W. H. A. Schilders,...
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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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