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New experiment: error measures in one cycle, varying epsilon

Motivation

As we already discussed, past experiments have shown some counterintuitive facts that are yet to be explained. On the one hand, the errors computed did not seem to change as epsilon was decreased [1]. Also, we recently observed that Geant4 calls AccurateAdvance depending at least on the values of stepMax and epsilon [2]. After further inspection, we have seen that this function is used even though QuickAdvance seems to provide acceptable results (fourth item on Nico's last email as of 2015-11-30).

In order to explore this, we designed a new experiment. The goal was not only to have a controlled trajectory with a fixed and well-known number of steps, but also to change the accuracy parameters together and keep them synchronized, since we were usually focusing on epsilonMin and epsilonMax alone. As we show below, these changes follow from the relationship between deltaOneStep, stepMax, epsilonMin and epsilonMax [3].

Outline of the experiment

  • The simulation consists of one full cycle of the particle. This was achieved by setting trackMaxLen = p, where p is the circle perimeter (see below).
  • The idea was to keep the number of steps small, and so we defined stepMax = p / SLICES, being SLICES a constant with the value 10.
  • As we are focusing on how epsilon and deltaOneStep interact with the step taken so as to decide whether or not AccurateAdvance is called, we set deltaChord = p just to keep it high enough and make sure that it does not interfere with the experiment.
  • deltaOneStep, epsilonMin and epsilonMax are defined in terms of stepMax and an additional value f, which is the factor by which stepMax is divided:
    • deltaOneStep = stepMax / f
    • epsilonMin = epsilonMax = 1 / f
    • This way, epsilon = deltaOneStep / stepMax = stepMax / (f * stepMax) = 1 / f, which explains why epsilonMin and epsilonMax take this value.
  • Having this framework, the expriment iterates increasing values of f and takes measures of the maximum r_error obtained and the substep ratio for each run.
  • Finally, for this experiment we added a new output line inside G4ChordFinder::AdvanceChordLimited so as to expose the values of dyErr, epsStep and stepPossible after calling FindNextChord, for each step of each run.

Settings and definitions

  • p: circle perimeter. Since its radius is ~38.0854 mm, p ~= 239.2976256980574 mm.
  • SLICES: a constant value used to control the number of steps in each run. Set to 10 by default and generic settings.
  • f: variable that defines how the accuracy parameters are obtained from stepMax. Each run uses a different value for f. The range we swept for the results below is [100, 10^5) with leaps of 10^4.
  • r_error(t) = sqrt((G4x(t) - ANx(t))^2 + (G4y(t) - ANy(t))^2)
  • StepRatio = RHS steps / (output steps * 11)

Results

  • The following table summarizes the results we found:
f stepMax [mm] epsilonMin epsilonMax deltaOneStep [mm] max(r_error) [mm] StepRatio
100 23.9297625698 0.01 0.01 0.2392976257 0.000680661443008 1
10100 23.9297625698 0.0000990099 0.0000990099 0.0023692834 0.000689586107749 3
20100 23.9297625698 0.0000497512 0.0000497512 0.0011905355 0.000706697247766 4
30100 23.9297625698 0.0000332226 0.0000332226 0.0007950087 0.000719450484745 4
40100 23.9297625698 0.0000249377 0.0000249377 0.0005967522 0.000721110255093 4
50100 23.9297625698 0.0000199601 0.0000199601 0.0004776400 0.000722221572649 4
60100 23.9297625698 0.0000166389 0.0000166389 0.0003981658 0.000723336712742 4
70100 23.9297625698 0.0000142653 0.0000142653 0.0003413661 0.000723336712742 4
80100 23.9297625698 0.0000124844 0.0000124844 0.0002987486 0.000723895710721 4
90100 23.9297625698 0.0000110988 0.0000110988 0.0002655911 0.000723895710721 4

Discussion

  • We could not see significant changes in the errors. In fact, the variance of the errors is very small (~2.27e-10).
  • Apparently dyErr is not changing, which is also rather surprising. The debug line we added always shows that dyErr = 1.338101e-02. In consequence, Geant4 should call AccurateAdvance if epsStep <= 5.59178e-4 = dyErr / stepMax (as stepMax is fixed). This last inequality holds for every f but 100.
  • The StepRatio grows with f, which is reasonable. Lower values of f make epsilon larger, and thus less work is needed to fulfill this precision requirement.

References