State-of-the-art stellarator optimization code
The MANGO library provides STELLOPT with many optimization algorithms from the packages PETSc, GSL, NLOpt, and HOPSPACK. Many of these algorithms are serial by nature, and MANGO provides them with parallelization through parallelized finite difference gradient calculations. Since MANGO is also available from the stellarator optimization system ROSE, STELLOPT and ROSE can be compared while using exactly the same implementation of the same optimization algorithm.
An example input file for running STELLOPT with a MANGO algorithm
can be found here,
also in BENCHMARKS/STELLOPT_TEST/MANGO.
To use the MANGO algorithms in STELLOPT, you must first build the MANGO library, following the directions here.
Next, before compiling STELLOPT, use a text editor to
open the SHARE/make_*.inc file for your system. Look for the section on MANGO,
set LMANGO = T, and set MANGO_DIR appropriately.
Now build STELLOPT following the standard instructions here.
To use an optimization algorithm from the MANGO collection,
just set OPT_TYPE to any of the available algorithms
described here.
Based on preliminary experience, the most effective algorithms
for STELLOPT problems are the ones designed for least-squares problems
rather than for more general single-objective optimization problems.
Good choices are gsl_lm, gsl_dogleg, gsl_ddogleg, gsl_subspace2d,
petsc_brgn (which requires PETSc v3.12 or later), petsc_pounders,
and mango_levenberg_marquardt.
If your choice of L*_OPT and SIGMA_* in the OPTIMUM namelist means
there are more independent variables than terms in the objective function,
certain algorithms will not work.
In particular, STELLOPT’s native Levenberg-Marquardt and GSL’s lm, dogleg, ddogleg,
and subspace2d algorithms will return an error in this case.
However other algorithms such as mango_levenberg_marquardt will still work.
Besides OPT_TYPE, several other variables in STELLOPT’s OPTIMUM namelist affect MANGO.
For MANGO algorithms that use finite difference gradients,
the value of EPSFCN from the OPTIMUM namelist is used for the finite difference step size.
Any bound constraints (a.k.a. box constraints) that are set by the *_MIN and *_MAX variables
in the OPTIMUM namelist are passed to MANGO. These values can be honored by some algorithms but not by others.
Some algorithms require bound constraints to be set. For the full list of which algorithms
allow or require bound constraints,
see the table here.
LCENTERED_DIFFERENCES can be set to T or F to use either centered or 1-sided
finite differences. This variable only has an effect when OPT_TYPE is set to a MANGO algorithm,
i.e. it has no effect on STELLOPT’s native Levenberg-Marquardt method, which only uses 1-sided differences.
Note that the setting for LCENTERED_DIFFERENCES affects the values of NOPTIMIZERS you should choose,
as discussed in more detail below.
The variable AXIS_INIT_OPTION affects how the magnetic axis in VMEC is initialized.
Available settings are "previous", "mean", "midpoint", "input", and "vmec".
The default value is "previous". For "previous", the initial guess for the next VMEC calculation will be the
final axis from the previous VMEC calculation on this processor. This is often an excellent
guess, but it has the downside that it makes the objective function slightly
dependent on the history of previous evaluations of the objective function.
This can be a problem when evaluating the finite-difference Jacobian, where
small differences in the objective function matter a lot. This history-dependence
can also introduce dependence on the number of MPI processes, which may be
undesirable. It is STRONGLY RECOMMENDED that you set AXIS_INIT_OPTION to something
other than "previous" to avoid these problems. A recommended setting is "vmec",
which invokes VMEC’s internal mechanism for initializing the axis shape as a deterministic
function of the boundary shape. See STELLOPTV2/Sources/General/stellopt_fcn.f90 for details
of the other options.
The best value for the NOPTIMIZERS variable in STELLOPT’s OPTIMUM namelist depends
on the number of MPI processes for you job, on your choice of optimization algorithm,
and on your choice of LCENTERED_DIFFERENCES.
For algorithms like petsc_pounders that do not support concurrent evaluations of the objective function,
you should set NOPTIMIZERS to 1. For algorithms like gsl_dogleg, petsc_brgn, mango_levenberg_marquardt etc
that use gradient information, the ideal value of NOPTIMIZERS is N + 1 for 1-sided differences,
or 2*N+1 for centered differences, where N is the number of parameters.
Such a value enables the entire finite-difference Jacobian and associated objective function to
be evaluated in the same wallclock time as a single function evaluation.
You may wish to round NOPTIMIZERS up to a value that is a factor of the number of processors you will have available, if it is not already a factor.