1. Multi-Processing Least Squares Collocation: Applications to Gravity Field Analysis. Kaas. E., B. Sørensen, C. C. Tscherning, M. Veicherts

2. 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. (A2)

3. Remove-Restore An EGM is removed and later restored: The change of summation order enables the use of multiprocessing of a sum for each order. (1)

4. Timing Results N – max. degree Number of processors Time (s) 40010.8 40.53 240.39 1000154.0 424.6 24 5.8 21901293.0 4116.8 24 26.1 Summation times for EGM2008 for 26 points (at different latitude) including read overhead of coefficients.

5. Covariance computation, C, C P 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 s s s 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

6. Solution of Equations Upper triangular part divided in blocks, collected in ”Chunks” C ij

7. Cholesky reduction Row-wise Inner sum over block k, Outer sum over all blocks m/b in a column of blocks of size b.

8. Timing of Cholesky reduction: 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

9. Time depends on block-size and number of working processors

10. MPI timing ???

11. Conclusion Standard software for Cholesky reduction can not be used in the general setting, where also parameters are unknowns: Summation in the reduction must be changed from positive accumulation to negative accumulation. Use of OMP and MPI makes LSC feasible even for very large number of data. Both covariance computation and Cholesky reduction becomes much faster.