Transcript MICE input beam and weighting Dr Chris Rogers Analysis PC 05/09/2007 Overview  Recap - MICE input beam alignment & matching       X, y, px, py misalignment.

MICE input beam and weighting
Dr Chris Rogers
Analysis PC
05/09/2007
Overview
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Recap - MICE input beam alignment & matching
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X, y, px, py misalignment effect on cooling
Energy “misalignment” effect on cooling
Beta, alpha mismatch effect on cooling
Dispersion effect on cooling cooling
Third (& higher) moment effect on cooling
A possible reweighting algorithm to realign beam “offline”
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Apply continuous polynomial weighting
Discuss only 1D case but extensible to 6D etc
Choose desired output moments
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=> output emittance, alignment, amplitude moment corr etc
Alignment Sensitivity
Energy Alignment sensitivity
Linear Mismatch Sensitivity
Dispersion Sensitivity
Amplitude-Momentum Corr
Energy Dependent Beta
Reweighting
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What if we don’t get the desired beam
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Reweighting in 6D is difficult
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No real way to measure particle density in a region
Binning algorithms break down as phase space density is too
sparse in high-dimensional spaces
FT/Voronoi type algorithms seem to become analytically
challenging in > 3 dimensions
If I can’t measure density I can’t calculate weight needed to get a
particular pdf
Propose a reweighting algorithm based around beam moments
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May need to reweight input beam
This is true for bunch emittance and particle amplitude analyses
Beam optics can be expressed purely in terms of moments of the
beam
No need to discuss actual pdfs at all
Weight using a polynomial series and assess the quality of the
weighting by looking at the moments before and after
Reweighting Principle
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Say we have some (1D) input distribution f(x) with known raw
moments like <x>f, <x2>f etc
Say we have some desired output distribution g(x) with known
raw moments like <x>g, <x2>g etc
Apply some weighting w(x) to each event
w( x)  (1 a1x  a2 x2  a3 x3  ...)
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so that
g ( x)  (1 a1x  a2 x2  a3 x3  ...) f ( x)
Then the ai can be found in terms of input and output moments
analytically
Say we calculate coefficients up to aN
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Then N is the largest moment that we can choose in the target
distribution
Then we need to invert an NxN matrix
And we need to calculate a 2Nth moment from input distribution
Some maths details which I don’t reproduce here
Reweighting effects
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input
For 10,000 events, N=12
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(Line) Parent pdf
(Hist) Unweighted events
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output
Input gaussian with:
Variance 1
Mean 0.1
Output gaussian with moments:
Moment
Target
Actual
1
0
0
2
0.9
0.9
3
0
0
2.43
2.43
10.935
10.935
8
68.891
68.8905
10
558.01
558.01
11
5524.3
5524.3
12
1041.97
948.22
(Line) Expected analytical4
Pdf
6
(Hist) Weighted events
Technique goes awry for large N
Output
N=8
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Output
N=16
Largest coefficient calculated is aN
As I ramp up N the technique breaks down
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Numerical errors creeping in
Can compare output calculated moment with target moment to find
when the technique breaks down
Failure vs N
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Consider output moment/target moment
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What is the cause of the failure?
Calculation of moments?
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I am using CLHEP for linear algebra
Better linear algebra libraries exist
This is still a feasible algorithm
In principle this technique can be extended to 6D phase space
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Matrix becomes larger
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May be a better way
Inversion of matrix?
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“Relative error”
See a clear transition at N=12
6x6 for 1st moments
~24x24 for 2nd moments
~200x200 for 3rd moments
But inverting a matrix is easy?
Conclusions
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Some study of alignment and matching sensitivity of MICE
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Perhaps conflicts with earlier studies
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Needs to be resolved
Perhaps needs another look with higher statistics
A proposal for a reweighting algorithm
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Looks encouraging in 1D
Some computational error for reweighting the tails of the
distribution
See how it extends up to higher dimensional phase spaces