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Final Presentation

Parallel Covariance Matrix Creation. Final Presentation. Supervisor: Oded Green. Table of Contents - Overview. Introduction Building the covariance matrix The naïve algorithm Our algorithm Terminology The Algorithm Optimizations Results MVM on Plurality The MVM algorithm

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Final Presentation

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  1. Parallel Covariance Matrix Creation Final Presentation Supervisor: Oded Green

  2. Table of Contents - Overview • Introduction • Building the covariance matrix • The naïve algorithm • Our algorithm • Terminology • The Algorithm • Optimizations • Results • MVM on Plurality • The MVM algorithm • Plurality Platform • Results • Future Projects • Conclusions Parallel Covariance Matrix Creation - Final Presentation

  3. Table of Contents • Introduction • Building the covariance matrix • The naïve algorithm • Our algorithm • Terminology • The Algorithm • Optimizations • Results • MVM on Plurality • The MVM algorithm • Plurality Platform • Results • Future Projects • Conclusions Parallel Covariance Matrix Creation - Final Presentation

  4. Project’s Goals • Developing a parallel algorithm for the creation of a covariance matrix • Compatibility with Plurality’s HAL platform • Maximized parallelization and core utilization • Integrating the algorithm into Elta’s MVM (Minimum Variance Method) algorithm implementation Parallel Covariance Matrix Creation - Final Presentation

  5. MVM Algorithm MVM is a modern 2-D spectral estimation algorithm used by Elta’s Synthetic Aperture Radar (SAR). The MVM algorithm: • Improves resolution • Removes side lobe artifacts (noise) • Reduces speckle compared to what is possible with conventional Fourier transform SAR imaging techniques • One of MVM’s main building blocks is the creation of a covariance matrix Parallel Covariance Matrix Creation - Final Presentation

  6. Plurality Platform Plurality’s HyperCore Architecture Line (HAL) family ofmassively parallel manycore processors features: • Unique task-oriented programming model • Near-serial programmability • High performance at low cost per watt per square millimeter • Unique shared memory architecture - 2 MB cache size Parallel Covariance Matrix Creation - Final Presentation

  7. Table of Contents • Introduction • Building the covariance matrix • The naïve algorithm • Our algorithm • Terminology • The Algorithm • Optimizations • Results • MVM on Plurality • The MVM algorithm • Plurality Platform • Results • Future Projects • Conclusions Parallel Covariance Matrix Creation - Final Presentation

  8. Implementing the Naïve Algorithm Implementing the naïve algorithm will give us a greater understanding of the parallelization problem. • Motivation: Parallel Covariance Matrix Creation - Final Presentation

  9. The Naïve Algorithm Chip [NxM] Parallel Covariance Matrix Creation - Final Presentation

  10. The Naïve Algorithm Sub aperture [N1xM1] Parallel Covariance Matrix Creation - Final Presentation

  11. The Naïve Algorithm Parallel Covariance Matrix Creation - Final Presentation

  12. The NaïveAlgorithm Parallel Covariance Matrix Creation - Final Presentation

  13. The Naïve Algorithm • Every Sub-aperture holds its covariance • matrix Cov Parallel Covariance Matrix Creation - Final Presentation

  14. The NaïveAlgorithm • The covariance matrix Cov is the sum of all • Sub-aperturesCov matrixes Parallel Covariance Matrix Creation - Final Presentation

  15. The NaïveAlgorithm • Shortcomings • Each multiplication is executed many times • For a 32x32 chip, the total number of multiplies is 11.4M when the optimal number of multiplications is 208K (x28!) • The naïve algorithm is difficult to parallelize. Two main difficulties: • Simultaneous writing to the same Rcells – requires mutexes • Memory cost of holding a Cov matrix for every permutation (each is 250 KB) is too expensive Parallel Covariance Matrix Creation - Final Presentation

  16. The NaïveAlgorithm • Disadvantages • Mutexes - adds complexity • Memory space - cache size is only 2 MB • Plurality Platform • The problem requires different solution! Parallel Covariance Matrix Creation - Final Presentation

  17. Our Algorithm A Whole different Ball Game! Parallel Covariance Matrix Creation - Final Presentation

  18. But first … Before presenting the algorithm there is a need to create a common language for the terms we have created. Parallel Covariance Matrix Creation - Final Presentation

  19. Table of Contents • Introduction • Building the covariance matrix • The naïve algorithm • Our algorithm • Terminology • The Algorithm • Optimizations • Results • MVM on Plurality • The MVM algorithm • Plurality Platform • Results • Future Projects • Conclusions Parallel Covariance Matrix Creation - Final Presentation

  20. Terminology Examples • Permutation [1,0] • Permutation [1,1] M1 M2 Permutation Parallel Covariance Matrix Creation - Final Presentation

  21. Terminology Examples • Permutation [1,0] • Permutation [1,1] M1 M2 Permutation Parallel Covariance Matrix Creation - Final Presentation

  22. Terminology Examples • Permutation [1,0] • Permutation [1,1] M1 M2 Permutation Parallel Covariance Matrix Creation - Final Presentation

  23. Terminology Block M1 M2 Block Parallel Covariance Matrix Creation - Final Presentation

  24. Terminology Block Parallel Covariance Matrix Creation - Final Presentation

  25. BNW Terminology BNW Parallel Covariance Matrix Creation - Final Presentation

  26. Terminology Shifting • Shift onlyupwards • and leftwards Block M1 • The block is always inside • the shifted window M2 Parallel Covariance Matrix Creation - Final Presentation

  27. Terminology Shifting • Shift onlyupwards • and leftwards Block M1 • The block is always inside • the shifted window M2 Parallel Covariance Matrix Creation - Final Presentation

  28. Terminology Shifting • Shift onlyupwards • and leftwards Block M1 • The block is always inside • the shifted window M2 Parallel Covariance Matrix Creation - Final Presentation

  29. Terminology Shifting • Shift onlyupwards • and leftwards Block M1 • The block is always inside • the shifted window M2 • Shift of (0,0) is named • Zero iteration Parallel Covariance Matrix Creation - Final Presentation

  30. Terminology Cov- The covariance matrix[M∙N, M∙N] Parallel Covariance Matrix Creation - Final Presentation

  31. Terminology Rcell Parallel Covariance Matrix Creation - Final Presentation

  32. Table of Contents • Introduction • Building the covariance matrix • The naïve algorithm • Our algorithm • Terminology • The Algorithm • Optimizations • Results • MVM on Plurality • The MVM algorithm • Plurality Platform • Results • Future Projects • Conclusions Parallel Covariance Matrix Creation - Final Presentation

  33. Our Algorithm – Key Features • Parallel • Each multiplication is executed once (208k for 32x32 chip) • Memory efficient • Generic Concept: Each Rcell in Cov is calculated by one specific permutation. This enables different permutations to work simultaneously. Parallel Covariance Matrix Creation - Final Presentation

  34. Our Algorithm(simplified) 1. For each permutation (1:313) 1.1 For each legal BNW 1.1.1. Multiply the two multipliers 1.1.2. For each legal shift (including the zero iteration) 1.1.2.1. Add the multiplication product to the matching Rcell in Cov Parallel Covariance Matrix Creation - Final Presentation

  35. Our algorithm(simplified) Finding all unique permutations • Iterative algorithm 1. Initialize Delta (x,y) set and Permutation(x,y) set 2. For each pair of cells (M1,M2) in a N1xM1 matrix 2.1. If |M1-M2| is not in D 2.1.1. Add |M1-M2| to D 2.1.2. Add (M1,M2) to P • Unique permutation count is 313 ( for Sub-aperture [13x13]) • Executed off-line Parallel Covariance Matrix Creation - Final Presentation

  36. Our algorithm (simplified) Cov- The covariance matrix[M∙N, M∙N] Chip [NxM] Parallel Covariance Matrix Creation - Final Presentation

  37. Our algorithm (simplified) • For a given Permutation [1,1] M1 M2 Parallel Covariance Matrix Creation - Final Presentation

  38. Our algorithm (simplified) • There’s a Block Block M1 M2 Parallel Covariance Matrix Creation - Final Presentation

  39. Our algorithm (simplified) • Leagal BNWs for this Block BNW Block M1 M2 Parallel Covariance Matrix Creation - Final Presentation

  40. Our algorithm (simplified) • For a given BNW Block M1 M2 Parallel Covariance Matrix Creation - Final Presentation

  41. Our algorithm (simplified) RES • RES=M1∙M2* Block M1 M2 Parallel Covariance Matrix Creation - Final Presentation

  42. Our algorithm (simplified) • The multipliers Numbering Block Parallel Covariance Matrix Creation - Final Presentation

  43. Our algorithm (simplified) RES • The Zero Iteration Rcell (1,5) RES 1 Block Diag(5-1) 5 Main Diag Parallel Covariance Matrix Creation - Final Presentation

  44. Our algorithm (simplified) • Shifting Block Parallel Covariance Matrix Creation - Final Presentation

  45. Our algorithm (simplified) RES • Shifting Rcell (2,6) RES RES 2 Diag(5-1) Block 6 Main Diag Parallel Covariance Matrix Creation - Final Presentation

  46. Our algorithm (simplified) • Shifting 2 Block 6 Parallel Covariance Matrix Creation - Final Presentation

  47. Our algorithm (simplified) RES • Shifting RES RES 5 RES Diag(5-1) Block 9 Main Diag Parallel Covariance Matrix Creation - Final Presentation

  48. Our algorithm (simplified) We came across a regularity in the offset of the Rcell coordinates when shifting: Leftwards (+Sub-ap size, +Sub-ap size) Upwards (+1,+1) Parallel Covariance Matrix Creation - Final Presentation

  49. Our algorithm (simplified) • Each color represents adifferent permutation Parallel Covariance Matrix Creation - Final Presentation

  50. Our Algorithm(simplified) Summary For a given permutation: • RES is always written into the same group of Rcells • All on the same diagonal • Not necessarily all diagonal cells • There is no overlapping between Rcells of different permutations. • The basis for parallelism! • Each shift writes to one unique Rcell. • Theoretically enables parallelism of Rcell granularity(an instance per Rcell) Parallel Covariance Matrix Creation - Final Presentation

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