ODE group integrator Function

1 Description

All the preivous functions solves for a single ODE system. Torsten also provides group modeling counterparts for ODE integrators. The functions solve for a group of ODE systems that share an ODE RHS but with different parameters. They have similar signatures to single-ODE integration functions.

2 Usage

matrix pmx_integrate_ode_group_[adams || rk45 || bdf](ODE_system, real[ , ] y0, real t0, int[] len, real[] ts, real[ , ] theta, real[ , ] x_r, int[ , ] x_i, [ real rtol, real atol, int max_step ] );

Here [adams || rk45 || bdf] indicates the function name can be of any of the three suffixes. See Section ODE integrator function.

• ODE_rhs Function that specifies the right-hand-side \(f\). See Section ODE integrator function.
• y0 Initial condition \(y_0\) for each subsystem in the group. The first dimension equals to the size of the group.
• t0 Initial time \(t_0\).
• len A vector that contains the number of output time points for each subsystem. The lenght of the vector equals to the size of the group.
• ts Output time when solution is seeked, consisting of ts of each subsystem concatenated.
• theta 2d-array parameters to be passed to ODE_rhs function. Each row corresponds to one subsystem.
• x_r 2d-array real data to be passed to ODE_rhs function. Each row corresponds to one subsystem.
• x_i 2d-array integer data to be passed to ODE_rhs function. Each row corresponds to one subsystem.
• rtol Relative tolerance.
• atol Absolute tolerance.
• max_step Maximum number of steps allowed between neighboring time in ts.

3 Return value

An n-by-nd matrix, where n is the size of ts and nd the dimension of the system.

4 Note

• With optional arguments indicated by square bracket, the following calls are allowed:

pmx_integrate_group_[adams || rk45 || bdf](..., x_i);
pmx_integrate_group_[adams || rk45 || bdf](..., x_i, rel_tol, abs_tol, max_step);

• The group integrators support paralleisation through Message Passing Interface(MPI). One can access this feature through cmdstan or cmdstanr interface.

# cmdstan interface user need to add "TORSTEN_MPI=1" and
# "TBB_CXX_TYPE=gcc" in "cmdstan/make/local" file. In linux & macos
# this can be done as
echo "TORSTEN_MPI=1" > cmdstan/make/local
echo "TBB_CXX_TYPE=gcc" > cmdstan/make/local # "gcc" should be replaced by user's C compiler
make path-to-model/model_name
mpiexec -n number_of_processes model_name sample... # additional cmdstan options

library("cmdstanr")
cmdstan_make_local(cpp_options = list("TORSTEN_MPI" = "1", "TBB_CXX_TYPE"="gcc"))  # "gcc" should be replaced by user's C compiler
rebuild_cmdstan()
mod <- cmdstan_model(path-to-model-file, quiet=FALSE, force_recompile=TRUE)
f <- mod$sample_mpi(data = ..., chains = 1, mpi_args = list("n" = number_of_processes), refresh = 200)

Here \(n\) denotes number of MPI processes, so that \(N\) ODE systems are distributed to and solved by \(n\) processes evenly. Note that to access this feature user must have MPI installed, and some MPI installation may require set additional compiler arguments, such as CXXLFAGS and LDFLAGS.