Use of interpolation and FSAL property for integration in field Dmitry Sorokin, MIPT John Apostolakis, CERN

Content Introduction Part 1: New design of the propagation classes Part 2: The Bulirsch-Stoer method Part 3: Using FSAL in integration Results Future work: Using interpolation in integration Sample tracks from Geant4 b2a example

Introduction Large number of track segments (steps) in each simulation Robust integration of a variety of equations (with a user provided field) High cost of field evaluation Need to find the intersection point between a curved trajectory and the surfaces of the experiment

Part 1: New design of the propagation classes

Part 2: The Bulirsch-Stoer method Idea: Use midpoint method to estimate integral Vary number of intermediate points Approximate the integral using rational functions & extrapolate to n = ∞ Advantages: Step size and order control Very good for smooth problems and large steps Can provide dense output n = 2i+2, for i = 0,1,2…. n = 4i+2, for i = 0,1,2….

Part 3: Using FSAL in integration Derivative at endpoint (for successful step) The decrease in the number of field evaluations when its FSAL property is used (the estimate of the final derivative is reused if valid), in comparison with its use when it’s FSAL property is not used.

Test: propagation in uniform magnetic field 1 MeV proton in the uniform 1 tesla magnetic field. Radius of the circle in xz plane is 102.20 mm Momentum is 43.33 MeV/c. delta1step = 1e-4 mm, deltaIntersection = 1e-5

Test: NTST Main integral test to check the overall performance of the simulation which is used is test NTST. It simulates the inner detector of the Babar experiment, with the realistic interpolated Babar field. Sample events are sample Babar events using the Pythia 6 event generator. File range cut looper cut min Ecat run2a.mac 1 mm 200 MeV 1 MeV run2b.mac Not applied run2c.mac

Next step: Using interpolation in integration Old strategy Make series of steps without error control to predict the step size (satisfy sagitta distance d < δchord) Make a step with error control to improve the accuracy (ΔB < δ1step) If the chord intersects: Make a series of substeps with error control to locate the intersection point New strategy Make one step with error control Use dense output to divide the step to substeps (satisfying d < deltachord) For each substep: If the chord intersects: Use dense output to locate the intersection point Pros: a lot fewer field evaluations required for large steps Cons: dense output is less accurate than the solution

Other new integration methods

New methods for step size control (Runge Kutta) How to calculate the size of the next step after a successful step ? New method – more stable (vs oscillations) Old method Standard driver Gustafsson driver Decrease, % run2a.mac 178’346 178’340 .003 run2b.mac 635’086 627’125 1.25 run2c.mac 2’068’425 2’016’590 2.50 Number of calls to field per event in the NTST test. Stepper is g4dormandprince745

Summary Introduced FSAL method to reuse endpoint derivative Alternative Estimation of Next Step size for RK steppers – small benefit New methods (Bulirsch-Stoer) for Integration Provided Prepared for use of Interpolation – a revolution in integration

Thank you

RK method adapted for oscillatory problems Proof of principle: use method adapted for oscillatory solutions based on classical RK4 method G4ClassicalRK4 ARK4 Decrease, % run2a.mac 259’111 237’066 8.5% run2b.mac 1’251’644 1’047’696 16.3% run2c.mac 4’283’088 3’565’924 16.7% Test NTST, number of field evaluations per event for G4ClassicalRK4 stepper and ARK4 stepper (adapted for oscillatory solutions classical RK4 stepper)