# Meeting 2015 10 16¶

## Objectives Proposed during last meeting¶

1. (DONE) Finish all calculations internally in Geant4. * RHS calculations. See last commits by Lucio. * Error calculations. See my last commit.
The errors are calculated 100% internally and in the numerically correct way (in the r_error case) also are not varying with epsilon. Actually, they vary very slightly and they become larger when we decrease epsilon.
2. (DONE partially) Run again all experiments (in Geant4 with the new internal calculations, and in QSS instead of PD)
3. (DONE) Add boundaries to the QSS solver.
4. (DONE partially: missing error calculation) In parallel to solving the epsilon mistery try running an experiment with a stand-alone implementation of RungeKutta 4/5. Proposal: do this using the QSS Solver!
5. Keep in mind the relation between epsilon and deltaInt in each of the experiments (more of a reminder than a task).

## Generic settings¶

/N02/det/trackMaxSteps 1000000000
/N02/det/trackMaxLen 10 km
/gun/energy 10.9181415106 MeV
/N02/prec/setDeltaOne 1.0e-2 mm
/N02/prec/setDeltaInt 1.0e-5 mm
/N02/prec/setEpsilonMin 1.0e-3
/N02/prec/setEpsilonMax 1.0e-3
/N02/prec/setDeltaChord 0.25 mm
/N02/det/stepMax 20. mm

• All measured times correspond to executions (both of G4 or PD) which don't produce output at all (or produce a minimum logging).
• MSE is the Mean Squared Error of the position
• MSE = 1/N * sum_{i=1}^{N} (G4x_i - ANx_i)^2 + (G4y_i - ANy_i)^2
• N = Number of output points
• r_RMSE = sqrt(MSE)
• max(X_Error) is the maximum of all the absolute errors in x, i.e. max(abs(G4x_i - ANx_i))
• circle_RMSE is the RMSE of the accumulated distances from the simulated position to the analytical circle.
• circle_RMSE = sqrt(1/N * sum_{i=1}^{N} circleError_i^2)
• circleError_i = |sqrt(G4x_i^2 + (G4y_i + 38.0854)^2) - 38.0854|

## Experiment 1.1 - Vary epsilon in G4 & Calculate errors inside G4¶

epsilonMax = epsilonMin = epsilon
The errors still show an unusual behavior. They don't vary, or even grow, when making epsilon smaller. This is counterintuitive. Also, the circle_RMSE doesn't vary much with distance whereas the other errors do, showing the movement is stable around the circle, but the error comes from a "time phase" which seems to increase over time.

epsilon Simulation Time[s] RHS evaluation steps Output Steps round(Steps Ratio / 11) r_RMSE [mm] Max(X_Error) [mm] circle_RMSE [mm] r_RMSE [mm] Max(X_Error) [mm] circle_RMSE [mm] r_RMSE [mm] Max(X_Error) [mm] circle_RMSE [mm] r_RMSE [mm] Max(X_Error) [mm] circle_RMSE [mm] r_RMSE [mm] Max(X_Error) [mm] circle_RMSE [mm] 100 mm 1 m 10 m 100 m 1 km 1E-3 1.58 1.65E6 50001 3 3.3E-5 1.85E-5 1.64E-4 2.5E-4 1.75E-4 1.81E-4 2.89E-3 1.73E-3 1.87E-4 2.95E-2 1.73E-2 1.87E-4 2.99E-1 1.73E-1 1.87E-4 1E-4 1.52 1.65E6 50001 3 3.3E-5 1.85E-5 1.64E-4 2.5E-4 1.75E-4 1.81E-4 2.89E-3 1.73E-3 1.87E-4 2.95E-2 1.73E-2 1.87E-4 2.99E-1 1.73E-1 1.87E-4 1E-5 1.54 1.65E6 50001 3 3.3E-5 1.85E-5 1.64E-4 2.5E-4 1.75E-4 1.81E-4 2.89E-3 1.73E-3 1.87E-4 2.95E-2 1.73E-2 1.87E-4 2.99E-1 1.73E-1 1.87E-4 1E-6 2.41 3.85E6 50001 7 3.3E-5 1.87E-5 1.67E-4 2.52E-4 1.76E-4 1.82E-4 2.91E-3 1.74E-3 1.88E-4 2.97E-4 1.74E-2 1.88E-4 3.01E-1 1.74E-1 1.88E-4 1E-7 3.19 6.05E6 50001 11 3.3E-5 1.87E-5 1.67E-4 2.52E-4 1.76E-4 1.82E-4 2.91E-3 1.74E-3 1.88E-4 2.97E-4 1.74E-2 1.88E-4 3.01E-1 1.74E-1 1.88E-4 1E-8 4.20 1.05E7 50001 19 3.3E-5 1.87E-5 1.67E-4 2.52E-4 1.76E-4 1.82E-4 2.91E-3 1.74E-3 1.88E-4 2.97E-4 1.74E-2 1.88E-4 3.01E-1 1.74E-1 1.88E-4 1E-9 6.13 1.59E7 50001 29 3.3E-5 1.87E-5 1.67E-4 2.52E-4 1.76E-4 1.82E-4 2.91E-3 1.74E-3 1.88E-4 2.97E-4 1.74E-2 1.88E-4 3.01E-1 1.74E-1 1.88E-4 1E-10 9.30 2.70E7 50001 49 3.3E-5 1.87E-5 1.67E-4 2.52E-4 1.76E-4 1.82E-4 2.91E-3 1.74E-3 1.88E-4 2.97E-4 1.74E-2 1.88E-4 3.01E-1 1.74E-1 1.88E-4 1E-11 15.06 4.57E7 50001 83 3.3E-5 1.87E-5 1.67E-4 2.52E-4 1.76E-4 1.82E-4 2.91E-3 1.74E-3 1.88E-4 2.97E-4 1.74E-2 1.88E-4 3.01E-1 1.74E-1 1.88E-4 1E-12 24.73 7.87E7 50001 143 3.3E-5 1.87E-5 1.67E-4 2.52E-4 1.76E-4 1.82E-4 2.91E-3 1.74E-3 1.88E-4 2.97E-4 1.74E-2 1.88E-4 3.01E-1 1.74E-1 1.88E-4

## Plots using the Solver¶

No planes, simulation during 3339 ns (equivalente to 1km distance)
Fig 1: {x,y}[mm] vs time [ns], QSS3, Tolerance = 1E-3 Fig 2: {x,y}[mm] vs time [ns], QSS3, Tolerance = 1E-5 Fig 3: {x,y}[mm] vs time [ns], DOPRI5, Tolerance = 1E-3 Fig 4: {x,y}[mm] vs time [ns], DOPRI5, Tolerance = 1E-5 ## Simulation Time vs # Planes for each method¶

Fig 5: Simulation Time [ms] vs #Planes (linear scale in y axis) Fig 6: Simulation Time [ms] vs #Planes (log scale in y axis) 