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

Φ ND c j = Φ FD ij i ND off-axis spectrum Oscillated FD target flux 1e 8 1e 15 33 4.0 3.5 3.5 3.0 3.0 2.5 2.5 D OA (m) (arb.) 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0 0 5 10 0 2 4 6 8 10 E (GeV) E (GeV) How to find � c ? D. Douglas Michigan State University 2018-10-25 Linear Algebra 2 / 7

If Φ ND is invertible, simply do � − 1 � Φ ND Φ FD c j = ij i Coefficients Oscillated FD flux fits 1e 8 1e 15 4.0 Target 6 Matrix inversion 3.5 Minuit fit 4 3.0 2.5 2 (arb.) 2.0 0 1.5 2 1.0 4 0.5 6 0.0 0 5 10 15 20 25 30 0 2 4 6 8 10 D OA (m) E (GeV) D. Douglas 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 χ 2 : i ( c i − c i +1 ) 2 χ 2 reg = e − λ � For the linear algebra fit, make a “difference matrix” A :  1 − 1 0 ... 0  0 1 − 1 ... 0     A = 0 0 1 ... 0     ... ... ... ... ...   0 0 0 ... 1 And define a tunable “penalty matrix” Γ = λ A : Φ FD | 2 + | Γ � c − � And minimize | Φ ND � c | 2 (“residual norm” + “solution norm”) D. Douglas Michigan State University 2018-10-25 Linear Algebra 4 / 7

This is Tikhonov regularization. The solution is � − 1 � �� Φ ND � T Φ ND � T Φ ND + Γ T Γ Φ FD � c = i Using λ = 10 − 9 (quick guess): Coefficients Oscillated FD flux fits 1e 8 1e 15 4.0 Target 4 Regularized matrix inversion 3.5 Minuit fit 3.0 2 2.5 (arb.) 2.0 0 1.5 1.0 2 0.5 0.0 4 0 5 10 15 20 25 30 0 2 4 6 8 10 D OA (m) E (GeV) D. Douglas Michigan State University 2018-10-25 Linear Algebra 5 / 7

To find the optimal value of λ , we construct an “L-Curve”: L-curve 10 13 10 14 Solution norm | Ac | 10 15 10 16 10 17 10 18 10 19 10 34 10 33 10 32 10 31 10 30 10 29 10 28 Residual norm | ND c FD | D. Douglas 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”: Curvature 1.5 ξ = log | Γ � c | 1.0 c − � ρ = log | Φ ND � Φ FD | ρ ′ ξ ′′ − ρ ′′ ξ ′ 0.5 curv = 2 (( ρ ′ ) 2 + ( ξ ′ ) 2 ) 3 / 2 0.0 0.5 10 10 10 9 10 8 10 7 10 6 10 5 Here, λ opt = 5 . 86 × 10 − 9 D. Douglas Michigan State University 2018-10-25 Linear Algebra 7 / 7

With λ opt = 5 . 86 × 10 − 9 : Coefficients Oscillated FD flux fits 1e 8 1e 15 4.0 Target Regularized matrix inversion 4 3.5 Minuit fit 3.0 2 2.5 (arb.) 2.0 0 1.5 1.0 2 0.5 4 0.0 0 5 10 15 20 25 30 0 2 4 6 8 10 D OA (m) E (GeV) D. Douglas Michigan State University 2018-10-25 Linear Algebra 8 / 7

With λ opt = 5 . 86 × 10 − 9 : Oscillated FD flux fits Flux fit residuals 1e 15 1e 16 4.0 Target Regularized matrix inversion 4 Regularized matrix inversion Minuit fit 3.5 Minuit fit 3.0 2 2.5 Residual (arb.) (arb.) 2.0 0 1.5 2 1.0 0.5 4 0.0 0 2 4 6 8 10 0 2 4 6 8 10 E (GeV) E (GeV) D. Douglas Michigan State University 2018-10-25 Linear Algebra 9 / 7

Disclaimers The above fits have slightly different targets (gaussian tail for low-E in Minuit fit) Runs much faster than Minuit fit! ( 1s) Finding λ opt takes a long time, depending on sparsity of scan I think it’s feasible to run this on the fly (with a pre-determined λ opt Implementation done (mostly) in Eigen (very portable LA package) D. Douglas Michigan State University 2018-10-25 Linear Algebra 10 / 7

TO DO Integrate into DUNEPrismTools look at λ opt 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? D. Douglas Michigan State University 2018-10-25 Linear Algebra 11 / 7