Multi processing least squares collocation applications to gravity field analysis
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Multi-Processing Least Squares Collocation: Applications to Gravity Field Analysis. Kaas . E., B. Sørensen , C. C. Tscherning, M. Veicherts. Introduction. LSC is used for gravity field modeling. This includes the determination of parameters and the estimation of errors.

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Multi processing least squares collocation applications to gravity field analysis

Multi-Processing Least Squares Collocation: Applications to Gravity Field Analysis.

Kaas. E., B. Sørensen,

C. C. Tscherning, M. Veicherts


Introduction

Introduction

  • LSC is used for gravity field modeling.

  • This includes the determination of parameters and the estimation of errors.

  • The quantity to be modeled is the so-called anomalous potential T.


Remove restore

Remove-Restore

  • An EGM is removed and laterrestored:

  • The change of summationorder enables the use of multiprocessing of a sum for eachorder.


Timing results

Timing Results

Summation times for EGM2008 for 26 points

(at different latitude) including read overhead

of coefficients.


Covariance computation c c p

Covariancecomputation, C, CP

Computation of N*(N+1)/2 covariances. Time in seconds as a function of number of processors and of block-size k*k.

N 37971 22464 22464 22464

Processors 22 22 4 4

Blocksize, k.

s sss

05 8486 32044 31341

10 3737 1101 3703 4784

15 4623 1268 3159 4430

20 3547 895 2847 3851

25 3600 1047 2974 3694

30 3621 1031 3101 3804


Solution of equations upper triangular part divided in blocks collected in chunks c ij

Solution of EquationsUpper triangular part divided in blocks, collected in ”Chunks” Cij


Cholesky reduction row wise

CholeskyreductionRow-wise

Inner sum over block k, Outer sum over all blocksm/b in a column of blocks of size b.


Timing of cholesky reduction omp

Timing of Choleskyreduction: OMP

N 37971 22464 22464 22464

Processors 22 22 4 4

Blocksize, k.

05 369 5157 1698

10 440 136 421 639

15 966 208 419 612

20 1013 238 391 591

25 1307 346 538 755

30 1542 411 668 1060


Time depends on block size and number of working processors

Time depends on block-size and number of working processors


Mpi timing

MPI timing

???


Conclusion

Conclusion

Standard software for Choleskyreductioncan not beused in the general setting, wherealso parameters areunknowns: Summation in the reduction must bechanged from positive accumulation to negative accumulation.

Use of OMP and MPI makes LSC feasibleeven for very large number of data. Bothcovariancecomputation and Choleskyreductionbecomesmuch faster.


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