22 Feb 2005AGATA Week1 David Radford ORNL Signal Decomposition Algorithm for GRETINA

22 Feb 2005AGATA Week2 Introduction to the problem Candidate algorithms Recent progress –Adaptive Grid Search –Singular Value Decomposition Plans and conclusions Outline

22 Feb 2005AGATA Week3 Signal Decomposition Digital pulse processing of segment data Uses data from both hit segments and image charges Extracts multiple interaction positions & energies Must allow for at least two interactions per hit segment Uses a set of calculated basis pulse shapes Done on a per-crystal basis Ideally suited to parallel processing Requires about 90% of total CPU cycles - The major processing bottleneck - Risk: baseline design allows ~ 4 ms/crystal/CPU for decomposition

22 Feb 2005AGATA Week4 Event Processing Crystal Event Builder Segment events Crystal events Signal Decomposition Interaction points Global Event Builder Tracking 36 segments per detector 1-30 crystals Global Events Data from Auxiliary Detectors Event Building Data Flow: Analysis & Archiving

22 Feb 2005AGATA Week5 Signal Decomposition Candidate algorithms: Adaptive Grid Search Singular Value Decomposition Constrained Least-Squares / Sequential Quadratic Programming When work begun in 2003, “best” algorithm was taking ~ 7 s / segment / CPU - would require ~ 10 5 CPUs

22 Feb 2005AGATA Week6 Signal Decomposition Most hit crystals have one or two hit segments Most hit segments have one or two interactions CPU time goes as AGS : ~ 100 n SVD : ~ n SQP : ~ n for n interactions

22 Feb 2005AGATA Week7 Signal Decomposition: AGS Adaptive Grid Search algorithm: Start on a course grid, to roughly localize the interactions, then refine the grid close by.

22 Feb 2005AGATA Week8 Signal Decomposition: SVD Singular Value Decomposition algorithm: Very roughly: Full “Least Squares” matrix is underdetermined (singular). But it can be decomposed into the product of three matrices, one of which contains the correlations (eigenvalues). By neglecting the small eigenvalues, the product can be inverted. Then an approximate fit can be obtained with very little computational effort, using a precalculated SVD inverse. The more eigenvalues kept, the higher the quality of the fit.

22 Feb 2005AGATA Week9 Signal Decomposition: SVD Singular Value Decomposition algorithm (early work, LBNL): Results of SVD in three dimensions for two interactions indicated by red squares. The upper row shows obtained probability distributions taking all possible positions into account. The bottom row shows the results after limiting the radius to the regions between the green lines. The final result agrees well with the input positions.

22 Feb 2005AGATA Week10 AGS: First efforts completed in 2003: Adaptive Grid Search, followed by constrained least-squares Grid search in position only; energy fractions are fitted (see following slides) interactions per hit segment Excellent results for single-segment events : – Converges for 100% of events – Reproduces positions of simulated events to < ½ mm – Very fast; ~ 7 ms/event/CPU (2 GHz P4)

22 Feb 2005AGATA Week11 AGS + SQP Example events Blue: measured Red: fitted

22 Feb 2005AGATA Week12 Some Math

22 Feb 2005AGATA Week13 Math Cont’d

22 Feb 2005AGATA Week14 AGS Some numbers for adaptive grid search: ~35000 grid points in 1/6 crystal (one column, 1x1x1 mm) 2x2x2mm (slices 1-3) or 3x3x3 mm (slices 4-6) coarse grid gives N  600 course grid points per segment. For two interactions in one segment, have N(N-1)/2  1.8 x 10 5 pairs of points for grid search. This takes ~ 3 ms/cpu to run through. But (N(N-1)/2) 2 ~ 3.2 x combinations for two interactions in each of 2 segments; totally unfeasible! Limit N to only 64 points; then (N(N-1)/2) 2 ~ 4 x 10 6 But (N(N-1)/2) 3 ~ 8 x 10 9 combinations for two interactions in each of 3 segments; still impossible.

22 Feb 2005AGATA Week15 AGS Cont’d Adaptive grid search fitting: Energies e i and e j are constrained, such that 0.1(e i +e j )  e i  0.9(e i +e j ) Once the best pair of positions (lowest  2 ) is found, then all neighbor pairs are examined on the finer (1x1x1 mm) grid. This is 9x9 = 81 pairs. If any of them are better, the procedure is repeated. For this later procedure, the summed signal-products cannot be precalculated. Finally, nonlinear least-squares (SQP) can be used to interpolate off the grid. This improves the fit ~ 50% of the time.

22 Feb 2005AGATA Week16 AGS: Recent progress Code has been completely rewritten in C - Translated from FORTRAN - Cleaner, easier to develop and maintain - Results verified to be identical, and slightly faster Extended to handle two-segment events - Up to four interactions total - Starts with one interaction per hit segment, then adds interactions - AGS again followed by constrained least-squares - Converges for 100% of events - Excellent speed; ~ 3-8 ms/event/CPU for 1 seg; ~ ms/event/CPU for 2 seg. (2GHz P4)

22 Feb 2005AGATA Week17 SVD: DOE SBIR (Phase I) with Tech-X Corp. - Funded to investigate alternative algorithms - Started with Singular Value Decomposition - Used signal basis developed for AGS Two-step SVD: - 2 mm grid (50 eigenvalues) to localize interaction region, followed by 1 mm grid (200 e.v.) over reduced space - Works perfectly for single interaction - Currently tested for up to 3x2 interactions - Results certainly good enough for input to SQP - CPU time linear in number of interactions: ~ 6 ms/segment/CPU (2GHz G5)

22 Feb 2005AGATA Week18 2D projections of SVD amplitudes Interaction sites at (13,9,11) and (8,11,11) New SVD Results x y z y

22 Feb 2005AGATA Week19 Current Concept 1 segment: AGS + SQP 2 segments: AGS + SQP or SVD + SQP ≥ 3 segments: SVD + SQP Need to include fitting of variable event time Cylindrical coordinate system (rather than Cartesian coordinates used presently) should save time and improve accuracy - constraints programmed into SQP Need to develop good metrics for performance

22 Feb 2005AGATA Week20 To-Do List Try to include three-segment events in AGS? Continue SVD development with Tech-X Corp SVD  least-squares SVD  grid search  least-squares Replace Cartesian coordinate system with cylindrical or quasi-cylindrical coord’s (begun) Allow for variation in event start time Allow for occasional three interactions/segment? Compare reliability of AGS and SVD results Examine failure modes in detail, develop metrics Deal with irregular-hexagonal detectors

22 Feb 2005AGATA Week21 Conclusion Excellent progress made over past 12 months - AGS for two-segment events - SVD development with Tech-X - looks very promising for 2 or more segments Final algorithm speed should be sufficient - Moore’s Law over 3-4 years