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Particle Weighting

Particle Weighting. D Adey CM27 Analysis Session 9/7/10. Weighting. Option to weight a particle distribution according to its similarity to a desired case Using an approximate Voronoi diagram + covariance matrix defined from desired parameters ( ε,β,p). Weight each particle.

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Particle Weighting

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  1. Particle Weighting D Adey CM27 Analysis Session 9/7/10

  2. Weighting • Option to weight a particle distribution according to its similarity to a desired case • Using an approximate Voronoi diagram + covariance matrix defined from desired parameters (ε,β,p)

  3. Weight each particle Measure phase space distribution in 1st tracker Apply weights to particle in tracker 2 and plot new phase space co-ordinates Cooling Channel Tracker 1 Tracker 2 Calculate emittance of weighted set

  4. f(x) 1/dx . w = f(x) w = f(x).dx • Integrate function over single valued bin • 1D Voronoi diagram • Expand to N-D • Approximate for processing and integration w 1/dx dx

  5. Creating Approximate Diagram • Bisect region at midpoint. • Add another point • Find which regions contains point • Bisect with existing point in region in dimension with largest difference • Integrate probability density over each region Only store 2 points regardless of n-D

  6. Simple Test Results • Take 4D distribution of points, sampled from uni-variate Gaussians • Draw diagram and weight to different set of uni-variate Gaussians. Original distribution σx = 30 Weighted distribution σx = 5

  7. Original distribution σx = 30 σy = 10 Weighted distribution σx = 5 σy = 15

  8. Diagonalise covariance matrix Rotate points into diagonalised basis Draw diagram Integrate Multivariate Integration • Brute force • Numerically integrate 4D Gaussian over all regions • Improve search timing with KD tree • More refined initial region definition

  9. Original basis Rotated basis

  10. Summary/Plans • Even if not (4D)viable-needed-palatable, there is a case for selecting for RF • Integration to multivariate still not satisfactory – poor variance and covariance, large number of zero weights • Need to check rotations, verify with numerical integration, scaling of weights • Make it work.. (+tidy, +comment, +commit) • Full Voronoi + Numerical integration • Mismatched beam tests • Phase space co-ordinates from recent data taking

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