MICE input beam and weighting

Dr Chris RogersAnalysis PC05/09/2007

Overview

Recap - MICE input beam alignment & matching 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” Apply continuous polynomial weighting Discuss only 1D case but extensible to 6D etc Choose desired output moments

=> output emittance, alignment, amplitude moment corr etc

Alignment Sensitivity

Energy Alignment sensitivity

Linear Mismatch Sensitivity

Dispersion Sensitivity

Amplitude-Momentum Corr

Energy Dependent Beta

Reweighting What if we don’t get the desired beam

May need to reweight input beam This is true for bunch emittance and particle amplitude analyses

Reweighting in 6D is difficult 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 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 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

so that

Then the ai can be found in terms of input and output moments analytically Say we calculate coefficients up to aN

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

)(...)1()( 33

221 xfxaxaxaxg

...)1()( 33

221 xaxaxaxw

Reweighting effects For 10,000 events, N=12

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

4 2.43 2.43

6 10.935 10.935

8 68.891 68.8905

10 558.01 558.01

11 5524.3 5524.3

12 1041.97 948.22

output

input (Line) Parent pdf(Hist) Unweighted events

(Line) Expected analytical Pdf(Hist) Weighted events

Technique goes awry for large N

Largest coefficient calculated is aN

As I ramp up N the technique breaks down Numerical errors creeping in Can compare output calculated moment with target

moment to find when the technique breaks down

Output N=8

Output N=16

Failure vs N Consider output moment/target moment

“Relative error” See a clear transition at N=12

What is the cause of the failure? Calculation of moments?

May be a better way Inversion of matrix?

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

Matrix becomes larger 6x6 for 1st moments ~24x24 for 2nd moments ~200x200 for 3rd moments

But inverting a matrix is easy?

Conclusions

Some study of alignment and matching sensitivity of MICE Perhaps conflicts with earlier studies

Needs to be resolved Perhaps needs another look with higher statistics

A proposal for a reweighting algorithm Looks encouraging in 1D Some computational error for reweighting the tails of the distribution See how it extends up to higher dimensional phase spaces