Sequential vs' Parallel solvers for the systems of linear equations

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Sequential vs. Parallel solvers for the systems
of linear equations

• PhD student Taras Grytsenko
• Supervisor Dr. Andres Peratta
• Wessex Institute of Technology, March 2006

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Outline

• Introduction
• Parallel solver structure
• Krylov subspace methods
• Packages
• Matrix partitioning
• Data formats
• Parallel platforms
• Software structure
• Test results
• Conclusions and future work

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Applications

• Solving very large systems of linear equations
  is central to many numerical simulations, and is
  the most time-consuming part of the computation
• Discretisation (and linearisation) of partial
  differential equations of elliptic and parabolic
  type
• Design and computer analysis of
• circuits
• power system networks
• chemical engineering processes
• macroeconomics models
• queuing systems

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Clip

• Example of the results

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Harwell-Boeing, CSR, Coordinate format, MSR, VBR
Modified Mondriaan
Internal mechanism
Extended Aztec library
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About the solvers

• There are very simple solvers for the system of
  linear equations
• (SOLE, SLES) such as Cramers method. But these
  methods can
• produce only exact solution.
• What if we cant find exact solution?
• Iterative methods. In this case we can solve a
  n-dimensional problem
• Ax b up to a residual of Ax-b lt epsb
• where
• A - matrix
• x,b - vectors
• eps - is a stopping criterion.

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Krylov subspace methods

• CQ Conjugate gradient (not yet implemented)
• GMRES restarted generalised minimal residual
• CGS conjugate gradient squared
• TFQMR transpose-free quasi-minimal residual
• BICGSTAB Bi-conjugate gradient with
  stabilisation
• These methods have sequential implementation as
  well
• as parallel.

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Different solvers

• Why do we have so many solvers instead of having
  only one
• universal?
• CQ Conjugate gradient is only applicable to
  symmetric positive definite matrices
• BICGSTAB Bi-conjugate gradient with
  stabilisation solves not only the original system
  Ax b but also a dual linear system ATx b
• LU is a general direct solver for the
  non-singular sparse system
• In principal matrix could be
• Symmetric, Unsymmetric, Hermitian, Skew
  symmetric, Rectangular

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List of the libraries

• Aztec
• http//www.cs.sandia.gov/CRF/aztec1.html
• BlockSolve95
• http//www.mcs.anl.gov/blocksolve95/.
• BPKIT2
• http//www.cs.umn.edu/Echow/bpkit.html/
• IML
• http//math.nist.gov/iml/
• Itpack 2C / ItpackV 2D
• ftp//ftp.ma.utexas.edu/pub/CNA/ITPACK
• Laspack
• http//www.math.tu-dresden.de/skalicky/laspack/i
  ndex.html
• ParPre
• http//www.cs.utk.edu/eijkhout/
• and about 20-30 more

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Comparison chart of features
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About partitioning

• For large-scale computing the calculation of Axb
  is very time
• comsuming when computed without paying particular
  attention to
• data structure and computer architecture.
• Partitioning the original problem into blocks
• Distributing them to different processors and
  performing the computation in parallel.
• Criteria
• Should partition the vectors x and b and the
  nonzero
• entries of A
• Each block should contain almost the same number
  of entries and be as independent as possible from
  other blocks
• When the blocks are distributed, communication
  between them should be low.

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Matrix partitioning

• Block distribution of 59x59 matrix with 312
• nonzeros, for p 4
• nonzeros per processor 76, 76, 80, 80

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Popular partitioners

• PaToH
• hMeTiS
• These partitioners use the hypergraph model to
  partition initial matrix.
• Mondriaan
• It partitions both the matrix A and the vectors
  x and b. It partitions both the rows and the
  columns of the matrix. Mondriaan implements a
  recursive bipartitioning algorithm.
• Monet
• Monet is used to partition and reorder an
  unsymmetric matrix into singly bordered blocked
  diagonal (SBBD) form and is commonly used with
  direct methods such as LU method. It implements a
  multilevel recursive bisection algorithm and the
  Kernighan-Lin refinement method.

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About the data formats

• If we store sparse matrix without modification we
  lose memory for
• zero entries which are not important at all. It
  is why several
• advanced data formats were been proposed to avoid
  storing of
• zero entries.
• Coordinate format
• CSR
• VBR
• MSR

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Parallel platforms

• MPI (C, Fortran)
• HPF (new language)
• OpenMP (directives)
• DVM (C-DVM, Fortran-DVM)
• Test NPB 2.3 (BT, CG, FT, LU, MG, SP) class ?
• Speedup Ts/Tp
• Efficiency Speedup/Np

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MPI time / HPF time on Origin 2000
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MPI time / OpenMPMPI time on IBM SP
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MPI time / DVM time on MVS-1000m
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MPI speedup on Origin 2000
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MPI Parallel Efficiency on Origin 2000
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Test results
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Conclusions

• Developed test platform based on MPI and Aztec
  library. Since new ideas can be tested and
  compared to existing methods
• Developed converter for different matrix storing
  formats (Harwell-Boeing, CSR, Coordinate format,
  MSR, VBR). It gives possibility to use matrices
  from different sources
• Investigated existing partitioning schemes and
  algorithms. Adopted one of them (Mondriaan) for
  our test platform
• Compared popular Krylov subspace methods for
  solving SLES.

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Future work

• We are working with new solver based on
  hierarchical matrices and it promises to be very
  efficient solver of linear complexity for the
  kind of matrices which we are dealing with, i.e.
  for BEM and FEM methods
• To make necessary corrections to existing
  software and test it on 4-processor cluster
• To develop an approach for analysing of the
  initial information about matrix structure and
  hardware platform and make decision about
  appropriate partitioner, preconditioner and
  solver to apply in every case