PDMP-control

This repository contains all code necessary to reproduce the simulation study presented in the manuscript ...

PDMP parameters

All PDMP parameters (flow, intensity, noise), MPD parameters (horizon, decisions) are encoded in file R/Parameters.R An interested user might modify these parameters and run grids constructions. If flows should not be exponentials, the simulator (encoded in file R/SimulationsFunctions) should be modified accordingly. So far codes are not implemented to easily allow such modifications.

Cost parameters

Cost functions and parameters are implemented in file R/Costs.R Three cost functions are proposed :

• the time dependent cost : c(x,d,x')=beta_l*r if l is not the appropriate treatment
• the marker dependent cost : c(x,d,x')=kappa|zeta'-zeta0|* r +beta_l*r if process is in mode 0
• the clinic-like cost : c(x,d,x')=kappa|zeta'-zeta0|* r if zeta' is above a given threshold +beta_l*r if process is in mode 0. This last cost is combined with a minimal treatment duration

Grids construction

To construct first discretisation grid, run R/1_XGrid.R Points will be added in the grids automatically to respect frontiers as described in the paper. This code will create the grid, estimate the transition matrix P and store output in an RData file.

To construct the second discretisation grid, run R/2_ThetaGrid-Init.R This code creates a minimal grid with probability measures charging a point of the first discretisation grid with probability 95% and dispatches the rest of the mass to each other point with a Dirichlet probability. Parameters of this initial grid can easily be modified within the code.

Dynamic programming

The dynamic programming iterations are encoded in file R/ProgDyn.R. This function can then be launched using R/3_RunDynProg.R using parameters -cost function -grid identification through its size -distance function

Iterative grid construction

The initial second discretisation grid can be enriched through simulation in the following manner:

• code R/6_EvalGrids.R evaluates through a given number of simulations (entry parameter) the use of each point of the current grid, and removes points whose density is below a given threshold
• code R/4_AddPoints.R simulates a given number of trajectories(entry parameter) with current optimal strategy and adds to the current grid each filters whose projection to the grid is above a given threshold (entry parameter).
• code R/5_ThetaGrid-Added.R then computes the transition matrix of the updated grid, and dynamic programming can be run.

Simulations in the paper

The grids were computed by running file Scripts-Paper/RunGridConstruction.sh All RData files were saved in directory RData. Then simulations and comparisons were run using Sweave codes Scripts-Paper/GridsEvaluation.Rnw to evaluate the iterative procedure, Scripts-Paper/StrategiesComparisons.Rnw to compare our approach with other strategies, and Scripts-Paper/Filter.Rnw to compare the use of different filter projections. Finally, the codes to produce the figures of the manuscript can be found in directory figures.