Linear Algegra Flux Fitting Dan Douglas Michigan State University - - PowerPoint PPT Presentation
Linear Algegra Flux Fitting Dan Douglas Michigan State University October 25, 2018 D. Douglas Michigan State University 2018-10-25 Linear Algebra 1 / 7 ND c j = FD ij i ND off-axis spectrum Oscillated FD target flux 1e 8 1e 15
Dan Douglas Michigan State University October 25, 2018
Michigan State University 2018-10-25 Linear Algebra 1 / 7
ΦND
ij
cj = ΦFD
i
5 10 E (GeV) 33 DOA (m)
ND off-axis spectrum
0.5 1.0 1.5 2.0 2.5 3.0 3.5 1e 8 2 4 6 8 10 E (GeV) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (arb.) 1e 15
Oscillated FD target flux
How to find c?
Michigan State University 2018-10-25 Linear Algebra 2 / 7
If ΦND is invertible, simply do cj =
ij
−1 ΦFD
i
5 10 15 20 25 30 DOA (m) 6 4 2 2 4 6 1e 8
Coefficients
2 4 6 8 10 E (GeV) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (arb.) 1e 15
Oscillated FD flux fits
Target Matrix inversion Minuit fit
Michigan State University 2018-10-25 Linear Algebra 3 / 7
How to include regularization? For the Minuit fit, just add a penalty term to the χ2: χ2
reg = e−λ
i(ci−ci+1)2
For the linear algebra fit, make a “difference matrix” A: A = 1 −1 ... 1 −1 ... 1 ... ... ... ... ... ... ... 1 And define a tunable “penalty matrix” Γ = λA: And minimize |ΦND c − ΦFD|2 + |Γ c|2 (“residual norm” + “solution norm”)
Michigan State University 2018-10-25 Linear Algebra 4 / 7
This is Tikhonov regularization. The solution is
ΦND + ΓTΓ −1 ΦNDT ΦFD
i
Using λ = 10−9 (quick guess):
5 10 15 20 25 30 DOA (m) 4 2 2 4 1e 8
Coefficients
2 4 6 8 10 E (GeV) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (arb.) 1e 15
Oscillated FD flux fits
Target Regularized matrix inversion Minuit fit
Michigan State University 2018-10-25 Linear Algebra 5 / 7
To find the optimal value of λ, we construct an “L-Curve”:
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Residual norm |
NDc FD|
10
19
10
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10
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10
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10
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10
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10
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Solution norm |Ac|
L-curve
Michigan State University 2018-10-25 Linear Algebra 6 / 7
The optimal value is at the “bend” in the L-curve. Look at the “curvature”: ξ = log |Γ c| ρ = log |ΦND c − ΦFD| curv = 2 ρ′ξ′′ − ρ′′ξ′ ((ρ′)2 + (ξ′)2) 3/2
10
10
10
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10
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10
5
0.5 0.0 0.5 1.0 1.5
Curvature
Here, λopt = 5.86 × 10−9
Michigan State University 2018-10-25 Linear Algebra 7 / 7
With λopt = 5.86 × 10−9:
5 10 15 20 25 30 DOA (m) 4 2 2 4 1e 8
Coefficients
2 4 6 8 10 E (GeV) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (arb.) 1e 15
Oscillated FD flux fits
Target Regularized matrix inversion Minuit fit
Michigan State University 2018-10-25 Linear Algebra 8 / 7
With λopt = 5.86 × 10−9:
2 4 6 8 10 E (GeV) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (arb.) 1e 15
Oscillated FD flux fits
Target Regularized matrix inversion Minuit fit 2 4 6 8 10 E (GeV) 4 2 2 4 Residual (arb.) 1e 16
Flux fit residuals
Regularized matrix inversion Minuit fit
Michigan State University 2018-10-25 Linear Algebra 9 / 7
The above fits have slightly different targets (gaussian tail for low-E in Minuit fit) Runs much faster than Minuit fit! ( 1s) Finding λopt takes a long time, depending on sparsity of scan I think it’s feasible to run this on the fly (with a pre-determined λopt Implementation done (mostly) in Eigen (very portable LA package)
Michigan State University 2018-10-25 Linear Algebra 10 / 7
Integrate into DUNEPrismTools look at λopt as a function of oscillation parameters. Is it safe to use one value for all? Look at variation of c over oscillation parameters. Other things?
Michigan State University 2018-10-25 Linear Algebra 11 / 7