1 Response Matrix Analytical Implementation of TwissResponse - - PowerPoint PPT Presentation

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Response Matrix

Analytical Implementation of TwissResponse

Joschua Dilly VIA at CERN

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Analytical Formulas Implementation Testing Setup Response Comparison MAD-X Comparison Conclusions

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Analytical Formulas Implementation Testing Setup Response Comparison MAD-X Comparison Conclusions

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Phase Advances

ΔΦz,wj ΔΦz,wj τz,wj Πwj = ( − ) Φz,j Φz,w = ( − ) + 2π Φz,j Φz,w Qz = Δ − π Φz,wj Qz = < sw sj , if > Φz,j Φz,w , if <= Φz,j Φz,w

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Advances

Hints:

• : plane or
• : elements of interest
• : magnets contibuting to K
• : in our case

∝ δK1

δQz δβz,j δΦz,wj = = = ± δ ∑

m

K1,m βz,m 4π ∓ δ × βz,j ∑

m

K1,m βz,m 2 cos(2 ) τz,mj sin(2π ) Qz ± δ × ∑

m

K1,m βz,m 4 {2 [ − + ] + } Πmj Πmw Πjw sin(2 ) − sin(2 ) τz,mj τz,mw sin(2π ) Qz z x y j m δ w ≡ j − 1

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Dispersion Advances

Hints:

• : plane or
• : elements of interest
• : magnets contibuting to K

δDx,j δDy,j = = + δ × βx,j ‾ ‾ ‾ √ ∑

m

K0,m βx,m ‾ ‾ ‾‾ √ 2 cos( ) τx,mj sin(π ) Qx + δ × βx,j ‾ ‾ ‾ √ ∑

m

K1S,mDy,m βx,m ‾ ‾ ‾‾ √ 2 cos( ) τx,mj sin(π ) Qx − δ × βy,j ‾ ‾ ‾ √ ∑

m

K0S,m βy,m ‾ ‾ ‾‾ √ 2 cos( ) τy,mj sin(π ) Qy + δ × βy,j ‾ ‾ ‾ √ ∑

m

K1S,mDx,m βy,m ‾ ‾ ‾‾ √ 2 cos( ) τy,mj sin(π ) Qy z x y j m δ

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Analytical Formulas Implementation Testing Setup Response Comparison MAD-X Comparison Conclusions

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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TwissResponse Class

twiss = TwissResponse(path_sequence, path_model, path_variables, exclude_categories)

where:

• path_sequence: madx-sequence file, e.g. saved from madx by

save, sequence=lhcb1, file=lhcb1_full.seq;

• path_model: tfs model file, e.g. saved from madx by

twiss, sequence=lhcb1, file=twiss_lhcb1.dat;

• path_variables: .json file, containing the different madx-variables that will be used. The

variables are assumed to be in categories (or at least one category). An easy way to create such a file can be found in twiss_optics/sequence_parser.py:

sequence_parser.varpmap_variables_to_json(path_sequence)

• exclude_categories: Categories of the json file, that will not be used.
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Response Matrix - Analytical Implementation of TwissResponse

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TwissResponse Class

• Generates response matrices for beta, dispersion, phase and tune.
• Via analytical formulas from previous section.
• A mapping is a applied: From the MadX-variables to the actual magnets, hence the

return is not a response of but .

• Response is returned via getter-functions of class-object:

twiss.get_beta()

• Return of the K response via:

twiss.get_beta(mapped=False)

Returns dictionaries with X and Y entries, containing the response matrices as

• dataframes. (In case of dispersion there are: X_K0L, X_K1SL, Y_K0SL, Y_K1SL

entries)

• Imitating behavoiur of response_pandas.py also possible:

twiss.get_fullresponse()

Returns a dictionary with BETX, BETY, MUX, MUY, DX, DY and Q dataframes.

δK δVar δ

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Hints

• Generating the response matrices takes a few seconds, but is about 4 times faster

than MadX (on my setup 30s analytical vs 130s MadX).

• TwissResponse is not parallelized (as is MadX), so maybe some tuning is possible (e.g.

parallel calculation for the different beam parameters.) Yet: Half of the time is dedicated to phase advance response calculation.

• To perform matrix multiplication

for all paramters automatically, use from twiss_optics/response_class.py:

fullresponse = twiss.get_fullresponse() response_class.get_delta(fullresponse, delta_k)

⋅ δK Mresp

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Sequence parsing assumptions:

• Magnets defined as "mb" are characterized by their bending radius and are hence

immutable!

• Sequence is saved in a way, that magnets definitions contain 'knl:={}', 'ksl:={}' or

'K#:='.

• There is only one value in each knl/ksl array.
• Zero-offset, i.e. no fixed number summation (e.g. no 'magnet := 2 + kq.xxx').*
• linearity, i.e. variables do not multiply with each other (will result in zeros).*
• the variable-name is final (i.e. it is not reassigned by ':=' somewhere else).
• the variable-name does not start with "l." (reserved for length multiplier).
• the length multiplier is given by l.<magnettype>.
• apart from the length, there are no other named multipliers (numbers are fine).
• If a magnet is redefined, last definition is used.

* this is due to the way the mapping is found. All variables apart from one are set to zero, and then the magnet values are checked. It is of course possible to implement it in a different, way e.g. with multiple values for the variables.

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Analytical Formulas Implementation Testing Setup Response Comparison MAD-X Comparison Conclusions

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Error Function

The physical parameter-functions can be arbitrarily close to zero, e.g. dispersion:

5000 10,000 15,000 20,000 25,000 −3 −2 −1 1 2 3 twiss results madx results

Position [m] Dispersion X [m]

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Error Function

Hence, the relative error between two different calculation methods, can be arbitrarily large: A solution can be to use a "normalized" error instead:

5000 10,000 15,000 20,000 25,000 −12 −10 −8 −6 −4 −2 2 4 error relative error mean-normalized

Position [m] Error

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Processing math: 100%

Gathering Statistics

• 25x: Pick 1(="single") or 10(="multi") variables at random from list (e.g. "K1L").
• 5x: Change variable strength(s) by

"gaussian w/ cutoff" distributed around , i.e. for .

• 12x: Increase stepwise:
• At each step: Calculate the

via twiss response, madx response and madx directly.

• Calculate RMS-Error of the statistics.
• Average over the different variable sets. Min/Max define error-bars.
• Plot over

±ϵ 10λ ϵ ∈ [.4 ⋅ , 1.6 ⋅ ] 10λ 10λ λ λ ∈ [−6 : .5 : 0) δ(D/β/ΔΦ/Q) ⇒ λ

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Analytical Formulas Implementation Testing Setup Response Comparison MAD-X Comparison Conclusions

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Comparison Analytical- and Mad-X Response Matrices

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Response Matrix - Analytical Implementation of TwissResponse

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Single Variable Change

• BET-Y shows the most differences between results (max ~5% RMS)
• Still all results in agreement with each other.

δK1L

−6 −5 −4 −3 −2 −1

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

1 10 10

2

BETX BETY DX DY MUX MUY QX QY

Approximate log(10) of δK RMS Error

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Multi Variable Change

• Tunes show wider spread, disagreement up to 40% RMS
• Other parameters in agreement

δK1L

−6 −5 −4 −3 −2 −1

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

1 10 10

2

BETX BETY DX DY MUX MUY QX QY

Approximate log(10) of δK RMS Error

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Single Variable Change

• DX values agree with each other

δK0L

−6 −5 −4 −3 −2 −1

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

1 10 10

2

BETX BETY DX DY MUX MUY QX QY

Approximate log(10) of δK RMS Error

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Multi Variable Change

• Wider spread compared to single-

test, but still good agreement of DX

δK0L

δK

−6 −5 −4 −3 −2 −1

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

1 10 10

2

BETX BETY DX DY MUX MUY QX QY

Approximate log(10) of δK RMS Error

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Single Variable Change

• DY results are in full agreement

δK1SL

−6 −5 −4 −3 −2 −1

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

1 10 10

2

BETX BETY DX DY MUX MUY QX QY

Approximate log(10) of δK RMS Error

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Multi Variable Change

• Mean of DY raised a bit compared to single-

test, still in perfect agreement

δK1SL

δK

−6 −5 −4 −3 −2 −1

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

1 10 10

2

BETX BETY DX DY MUX MUY QX QY

Approximate log(10) of δK RMS Error

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Analytical Formulas Implementation Testing Setup Response Comparison MAD-X Comparison Conclusions

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Comparison to Mad-X results

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Single Variable Change

• Compared to MAD-X calculations the results are actually pretty bad
• But "at least" equally bad for both response matrices
• Best for

~ [10 , 10 ]

δK1L

δK

• 4.5
• 3.5

10

−6

10

−4

10

−2

1 10

2

BETX BETY DX DY MUX MUY QX QY

Analytical RMS Error

−6 −5 −4 −3 −2 −1

10

−6

10

−4

10

−2

1 10

2

BETX BETY DX DY MUX MUY QX QY

Approximate log(10) of δK Mad-X Resp. RMS Error

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Multi Variable Change

• Compared to MAD-X calculations the results are actually pretty bad
• But "at least" equally bad for both response matrices
• Best for

~ [10 , 10 ]

δK1L

δK

• 4.5
• 3.5

10

−6

10

−4

10

−2

1 10

2

BETX BETY DX DY MUX MUY QX QY

Analytical RMS Error

−6 −5 −4 −3 −2 −1

10

−6

10

−4

10

−2

1 10

2

BETX BETY DX DY MUX MUY QX QY

Approximate log(10) of δK Mad-X Resp. RMS Error

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Single Variable Change

δK0L

10

−6

10

−4

10

−2

1 10

2

BETX BETY DX DY MUX MUY QX QY

Analytical RMS Error

−6 −5 −4 −3 −2 −1

10

−6

10

−4

10

−2

1 10

2

BETX BETY DX DY MUX MUY QX QY

Approximate log(10) of δK Mad-X Resp. RMS Error

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Multi Variable Change

δK0L

10

−6

10

−4

10

−2

1 10

2

BETX BETY DX DY MUX MUY QX QY

Analytical RMS Error

−6 −5 −4 −3 −2 −1

10

−6

10

−4

10

−2

1 10

2

BETX BETY DX DY MUX MUY QX QY

Approximate log(10) of δK Mad-X Resp. RMS Error

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Single Variable Change

δK1SL

10

−6

10

−4

10

−2

1 10

2

BETX BETY DX DY MUX MUY QX QY

Analytical RMS Error

−6 −5 −4 −3 −2 −1

10

−6

10

−4

10

−2

1 10

2

BETX BETY DX DY MUX MUY QX QY

Approximate log(10) of δK Mad-X Resp. RMS Error

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Multi Variable Change

δK1SL

10

−6

10

−4

10

−2

1 10

2

BETX BETY DX DY MUX MUY QX QY

Analytical RMS Error

−6 −5 −4 −3 −2 −1

10

−6

10

−4

10

−2

1 10

2

BETX BETY DX DY MUX MUY QX QY

Approximate log(10) of δK Mad-X Resp. RMS Error

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Response Matrix - Analytical Implementation of TwissResponse

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Analytical Formulas Implementation Testing Setup Response Comparison MAD-X Comparison Conclusions

• J. Dilly

Response Matrix - Analytical Implementation of TwissResponse

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Conclusions & Summary

• Analytical response matrices fully implemented and tested
• Accuracy wise MAD-X response and analytical response show similar results, for

"expected" parameters

• Influences on other parameters not more accurate in MAD-X response

Switch to analytical response possible

• Using response matrices always incorporates an error
• Should be coupled to an appropriate optimization algorithm

⇒

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Questions ?

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