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Yihua Bai Department of Mathematics and Computer Science Indiana State University

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High-Performance Eigensolver for Real Symmetric Matrices:Parallel Implementations and Applications in Electronic Structure Calculation

Yihua Bai

Department of Mathematics and Computer Science

Indiana State University

- Current status of real symmetric eigensolvers
- Motivation
- BD&C algorithm – a high performance approximate eigensolver
- Parallel implementations of BD&C algorithm
- Applications in electronic structure calculation and numerical results
- Summary and Future Work

Current Status of Dense Symmetric Eigensolvers

PDSYEVD

PDSYEVX

PDSYEVR

- Reduction to symmetric tridiagonal form
A=HTHT

- Eigen-decomposition of the tridiagonal matrix
T=VΛVT

- Cuppen’s divide-and-conquer
- Bisection and inverse iteration
- Multiple Relatively Robust Representations (MRRR)

- Back-transformation of the eigenvectors
X=HV

- Reduction time is the bottleneck

PDSYEVR

PDSYEVD

Robert C. Ward and Yihua Bai, Performance of Parallel Eigensolvers on Electronic Structure Calculations II, Technical Report UT-CS-06-572, University of Tennessee August 2006

- Compute eigen-solution to full accuracy, while lower accuracy frequently sufficient in electronic structure calculation

Questions:

Trade accuracy for efficiency?

How?

Motivation

A high performance approximate eigensolver for electronic structure calculation

- Solve Schrödinger’s Equation efficiently
- Different approximation methods
- Hartree-Fock approximation
- density functional theory
- configuration interaction
- …, etc.

- Self-Consistent Field method
- Solve generalized non-linear real symmetric eigenvalue problem iteratively
- A standard linear eigenvalue problem solved in each iteration.
- Typically the most time consuming part of electronic structure calculation
- Low accuracy suffices in earlier iterations
- Matrices from application problems may have locality properties

Given a real symmetric matrix A and accuracy

tolerance , want to compute

where and contain the approximate eigenvectors

and eigenvalues, respectively, and satisfy

1)Block-tridiagonal divide-and-conquer (BD&C)

– The centerpiece

2) Block tridiagonalization (BT)

– Block tridiagonalization of sparse and

“effectively” sparse matrices

3) Orthogonal reduction of full matrix to block-

tridiagonal form (OBR)

– Orthogonal transformations to produce

block-tridiagonal matrix

Decompose:

where

numerically orthogonal eigenvector matrix

diagonal matrix of eigenvalues

block tridiagonal matrix

accuracy tolerance

number of blocks

* W. N. Gansterer, R. C. Ward, R. P. Muller and W. A. Goddard III, Computing Approximate Eigenpairs of Symmetric Block Tridiagonal Matrices, SIAM J. Sci. Comput., 25 (2003), pp. 65 – 85.

1. Subdivision

with

2. Solve Sub-problem

decompose:

where:

,

,

3. Synthesis – the most time consuming step

decompose

, then multiply Vi and Z

Complexity:

a function of deflation, rank, and size

- An approximation to the original full matrix
- May require eigenvectors from previous iteration

Complexity:

* Y. Bai, W. N. Gansterer and R. C. Ward, Block-Tridiagonalization of “Effectively” Sparse Symmetric Matrices, ACM Trans. Math. Softw., 30 (2004), pp. 326 – 352.

• A full matrix that cannot be sparsified

•A sequence of Householder transformations

Complexity:

message passing latency

time to transfer one floating point number

time for one floating point operation ranks for off-diagonal blocks

- Parallel block divide-and-conquer (PBD&C) *
- Preprocessing
- Parallel block tridiagonalization (PBT)
- Parallel orthogonal block-tridiagonal reduction (POBR) **

* Yihua Bai and Robert C. Ward, A Parallel Symmetric Block-Tridiagonal Divide-and-Conquer Algorithm, Technical Report UT-CS-06-571, University of Tennessee, December 2005. Submitted to ACM TOMS

** Yihua Bai and Robert C. Ward, Parallel Block Tridiagonalization of Real Symmetric Matrices, Technical Report UT-CS-06-578, University of Tennessee, June 2006. Submitted to ACM TOMS

Implementations of PBD&C

Mixed data/task parallel implementation

versus

complete data parallel implementation

- Mixed parallelism – data/task

- Data distribution and redistribution

- Merging sequence and workload balance

- Deflation

- Divide processors into groups of sub-grids
- Assign each sub-grid to a sub-problem

Block-tridiagonal matrix

with q diagonal blocks

2D block cyclic distribution

on each sub-grid

Each diagonal block

assigned a sub-grid

Redistribute data from one sub-grid to another one (subdivision step)

Distribute from a 22 grid to a 3 3 grid

Redistribute data for each merging operation from

two sub-grids to one super-grid (synthesis step)

Distribute from a 22 and a 24 grids to a 34 grid

Level 4

Level 3

Level 2

Level 1

Level 0

Idle time

hright

hlett

Final merging operation

Final merging operation counts for up to 75% of total computational cost. Consider low computational complexity and workload balance at the same time for the final merge.

- Subgrid construction
- Example: subgrid 1: 2X2
subgrid 2: 5X5

supergrid: 1X29?

- Example: subgrid 1: 2X2
- Many communicator handles
- Can use up to 2k handles, where k=max(number of diagonal blocks, number of total processors)

- Portability on different MPI implementations
- Example: need minor modification of code when use mpimx (myrinet mpi)

- Assign all processors to each block in block-tridiagonal matrix

Assume a 2X2 processor grid,

Assigned to B1, B2, …, Bq,

and C1, C2, …, Cq-1.

Block-tridiagonal matrix

with q diagonal blocks

- Advantages
- One communicator
- One processor grid
- Portability to different MPI platform

- Disadvantages
- Not all processors involved in some steps
- SVD of off-diagonal blocks
- Decomposition of diagonal blocks
- Merge smaller sub-problems

- Still need data redistribution for each merging operation

- Not all processors involved in some steps

- Mixed data/task parallel BD&C subroutine PDSBTDC vs. ScaLAPACK PDSYEVD
- Matrices with different eigenvalue distributions and different sizes
- Banded application matrix

- Complete data parallel BD&C subroutine PDSBTDCD vs. Mixed data/task parallel BD&C subroutine PDSBTDC

Arithmetically distributed

eigenvalues

Geometrically distributed

eigenvalues

=10-6, b = 20

Residual:

Departure from orthogonality:

Polyalanine matrix,

n = 5027, b = 79

PDSBTDC with different

tolerances

Similar performance and

scales a little better

Block-tridiagonal matrix with arithmetically distributed eigenvalues,

Matrix size = 12000, block size = 20, tolerance = 10-6.

Data parallel implementation scales down in SVD of off-diagonal blocks and solving sub-problems.

- Trans-Polyacetylene

- Simple chemical structure
- Semiconducting conjugated polymer
- Light emitting devices, flexible
- Fast nonlinear optical response
- Strong nonlinear susceptibility

Yihua Bai, Robert C. Ward, and Guoping Zhang, Parallel Divide-and-Conquer Algorithm for

Computing Full Spectrum of Polyacetylene, Poster at the Division of Atomic, Molecular and

Optical Physics (DAMOP) 2006 meeting, Knoxville, Tennessee.

- Construct block-tridiagonal matrix from the original dense matrix H
- M = H + E, where M is block tridiagonal
- Algorithm: PBT

- Compute eigensolutions to reduced accuracy
- User defined accuracy, typically 10-6
- Algorithm: PBD&C

Trans-(CH)16000.

n=16000, =10-6.

With lower accuracy (i.e., 10-6),

the savings in execution time is

order of magnitude.

With fixed per-processor problem size,

The relative execution time for an O(n3)

algorithm should be

as the reference line shows. The curve

for our new parallel algorithm shows a computational complexity between

O(n2) and O(n3)

Conclusion and Future Work

- PBD&C: very efficient on block tridiagonal matrices with
- Low ranks for off-diagonal blocks
- High ratio of deflation

- Comparison of PDSBTDC and PDSBTDCD
- PDSBTDCD performs better with smaller number of processors in use
- PDSBTDC scales better as the number of processors in use increases

- PBD&C combined with PBT
- Efficient on application matrices with specific locality property

Incorporate PBD&C and PBT into SCF for trans-PA

Fine tuning of PDSBTDCD

Alternative method for computation of eigenvectors

Approximation in sparse eigensolver

A Parallel Adaptive Eigensolver

End of Presentation

Thank you!

Dr. R. P. Muller Sandia National Laboratories

Dr. G. Zhang Indiana State University

Major Efficiency improvements from

• Reduced accuracy in early iterations of SCF

• Reducing the reduction bottleneck

• Eigenvectors may be required if efforts made to improve efficiency

message passing latency

time to transfer one floating point number

time for one floating point operation

nbblock size for parallel 2D matrix distribution