Concepts¶

Ways to use simsopt¶

Simsopt is a collection of classes and functions that can be used in several ways. One application is to solve optimization problems involving stellarators, similar to STELLOPT. You could also define an objective function using simsopt, but use an optimization library outside simsopt to solve it. Or, you could use the simsopt optimization infrastructure to optimize your own objects, which may or may not have any connection to stellarators. Alternatively, you can use the stellarator-related objects in a script for some purpose other than optimization, such as plotting how some code output varies as an input parameter changes, or evaluating the finite-difference gradient of some code outputs. Or, you can manipulate the objects interactively, at the python command line or in a Jupyter notebook.

Input files¶

Simsopt does not use input data files to define optimization problems, in contrast to STELLOPT. Rather, problems are specified using a python driver script, in which objects are defined and configured. However, objects related to specific physics codes may use their own input files. In particular, a simsopt.mhd.vmec.Vmec object can be initialized using a standard VMEC input.* input file, and a simsopt.mhd.spec.Spec object can be initialized using a standard SPEC *.sp input file.

Optimization¶

To do optimization using simsopt, there are four basic steps:

Create some objects that will participate in the optimization.
Define the independent variables for the optimization, by choosing which degrees of freedom of the objects are free vs fixed.
Define an objective function.
Solve the optimization problem that has been defined.

This pattern is evident in the examples in this documentation and in the examples directory of the repository.

Some typical objects are a MHD equilibrium represented by the VMEC or SPEC code, or some electromagnetic coils. To define objective functions, a variety of additional objects can be defined that depend on the MHD equilibrium or coils, such as a simsopt.mhd.boozer.Boozer object for Boozer-coordinate transformation, a simsopt.mhd.spec.Residue object to represent Greene’s residue of a magnetic island, or a simsopt.geo.objectives.LpCurveCurvature penalty on coil curvature.

More details about setting degrees of freedom and defining objective functions can be found on the Defining optimization problems page.

For the solution step, two functions are provided presently, simsopt.solve.serial.least_squares_serial_solve() and simsopt.solve.mpi.least_squares_mpi_solve(). The first is simpler, while the second allows MPI-parallelized finite differences to be used in the optimization.

MPI Partitions and worker groups¶

To use MPI parallelization with simsopt, you must install the mpi4py package.

MPI simsopt programs can generally be launched using mpirun or mpiexec, e.g.

mpiexec -n 32 ./name_of_your_python_driver_script

where 32 can be replaced by the number of processors you want to use. There may be specific instructions to run MPI programs on your HPC system, so consult the documentation for your system.

Given a set of \(M\) MPI processes, an optimization problem can be parallelized in different ways. At one extreme, all \(M\) processes could work together on each evaluation of the objective function, with the optimization algorithm itself only initiating a single evaluation of the objective function \(f\) at a time. At the other extreme, the optimization algorithm could request \(M\) evaluations of \(f\) all at once, with each evaluation performed by a single processor. This type of parallelization can be useful for instance when finite differences are used to evaluate gradients. Or, both types of parallelization could be used at the same time. To allow all of these types of parallelization, simsopt uses the concepts of worker groups and MPI partitions.

A worker group is a set of MPI processes that works together to evaluate the objective function. An MPI partition is a division of the total set of \(M\) MPI processors into one or more worker groups. If there are \(W\) worker groups, each group has approximately \(M/W\) processors (approximate because one may need to round up or down.) Simsopt has a class simsopt.util.mpi.MpiPartition to manage this division of processors into worker groups. Each MPI-aware simsopt object, such as simsopt.mhd.vmec.Vmec, keeps an instance of MpiPartition which can be set from its constructor. The number of worker groups can be set by an argument to MpiPartition. For instance, to tell a Vmec object to use four worker groups, one could write

from simsopt.mhd import Vmec
from simsopt.util.mpi import MpiPartition

mpi = MpiPartition(4)
equil = Vmec("input.li383_low_res", mpi=mpi)

The same MpiPartition instance should be passed to the solver:

# ... code to define an optimization problem "prob" ...

from simsopt.solve.mpi import least_squares_mpi_solve

least_squares_mpi_solve(prob, mpi, grad=True)

Many optimization algorithms that do not use derivatives do not support concurrent evaluations of the objective. In this case, the number of worker groups should be \(W=1\). Any algorithm that uses derivatives, such as Levenberg-Marquardt, can take advantage of multiple worker groups to evaluate derivatives by finite differences. If the number of parameters (i.e. independent variables) is \(N\), you ideally want to set \(W=N+1\) when using 1-sided finite differences, and set \(W=2N+1\) when using centered differences. These ideal values are not required however - simsopt will evaluate finite difference derivatives for any value of \(W\), and results should be exactly independent of \(W\). Other derivative-free algorithms intrinsically support parallelization, such as HOPSPACK, though no such algorithm is available in simsopt yet.

An MPI partition is associated with three MPI communicators, “world”, “groups”, and “leaders”. The “world” communicator represents all \(M\) MPI processors available to the program. (Normally this communicator is the same as MPI_COMM_WORLD, but it could be a subset thereof if you wish.) The “groups” communicator also contains the same \(M\) processors, but grouped into colors, with a different color representing each worker group. Therefore operations such as MPI_Send and MPI_Bcast on this communicator exchange data only within one worker group. This “groups” communicator is therefore the one that must be used for evaluation of the objective function. Finally, the “leaders” communicator only includes the \(W\) processors that have rank 0 in the “groups” communicator. This communicator is used for communicating data within a parallel optimization algorithm (as opposed to within a parallelized objective function).

Given an instance of simsopt.util.mpi.MpiPartition named mpi, the number of worker groups is available as mpi.ngroups, and the index of a given processor’s group is mpi.group. The communicators are available as mpi.comm_world, mpi.comm_groups, and mpi.comm_leaders. The number of processors within the communicators can be determined from mpi.nprocs_world, mpi.nprocs_groups, and mpi.nprocs_leaders. The rank of the present processor within the communicators is available as mpi.rank_world, mpi.rank_groups, and mpi.rank_leaders. To determine if the present processor has rank 0 in a communicator, you can use the variables mpi.proc0_world or mpi.proc0_groups, which have type bool.

Geometric Objects¶

Simsopt contains implementations of two commonly used geometric objects: curves and surfaces.

Curves¶

A simsopt.geo.curve.Curve is modelled as a function \(\Gamma:[0, 1] \to \mathbb{R}^3\). A curve object stores a list of \(n_\phi\) quadrature points \(\{\phi_1, \ldots, \phi_{n_\phi}\} \subset [0, 1]\) and return all information about the curve, at these quadrature points.

Curve.gamma(): returns a (n_phi, 3) array containing \(\Gamma(\phi_i)\) for \(i\in\{1, \ldots, n_\phi\}\), i.e. returns a list of XYZ coordinates along the curve.
Curve.gammadash(): returns a (n_phi, 3) array containing \(\Gamma'(\phi_i)\) for \(i\in\{1, \ldots, n_\phi\}\), i.e. returns the tangent along the curve.
Curve.kappa(): returns a (n_phi, 1) array containing the curvature \(\kappa\) of the curve at the quadrature points.
Curve.torsion(): returns a (n_phi, 1) array containing the torsion \(\tau\) of the curve at the quadrature points.

The different curve classes, such as simsopt.geo.curverzfourier.CurveRZFourier and simsopt.geo.curvexyzfourier.CurveXYZFourier differ in the way curves are discretized. Each of these take an array of dofs (e.g. Fourier coefficients) and turn these into a function \(\Gamma:[0, 1] \to \mathbb{R}^3\). These dofs can be set via Curve.set_dofs(dofs) and dofs = Curve.get_dofs(), where n_dofs = dofs.size. Changing dofs will change the shape of the curve. Simsopt is able to compute derivatives of all relevant quantities with respect to the discretisation parameters. For example, to compute the derivative of the coordinates of the curve at quadrature points, one calls Curve.dgamma_by_dcoeff(). One obtains a numpy array of shape (n_phi, 3, n_dofs), containing the derivative of the position at every quadrature point with respect to every degree of freedom of the curve. In the same way one can compute the derivative of quantities such as curvature (via Curve.dkappa_by_dcoeff()) or torsion (via Curve.dtorsion_by_dcoeff()).

A number of quantities are implemented in simsopt.geo.curveobjectives and are computed on a simsopt.geo.curve.Curve:

CurveLength: computes the length of the Curve.
LpCurveCurvature: computes a penalty based on the \(L_p\) norm of the curvature on a curve.
LpCurveTorsion: computes a penalty based on the \(L_p\) norm of the torsion on a curve.
MinimumDistance: computes a penalty term on the minimum distance between a set of curves.

The value of the quantity and its derivative with respect to the curve dofs can be obtained by calling e.g., CurveLength.J() and CurveLength.dJ().

Surfaces¶

The second large class of geometric objects are surfaces. A simsopt.geo.surface.Surface is modelled as a function \(\Gamma:[0, 1] \times [0, 1] \to \mathbb{R}^3\) and is evaluated at quadrature points \(\{\phi_1, \ldots, \phi_{n_\phi}\}\times\{\theta_1, \ldots, \theta_{n_\theta}\}\).

The usage is similar to that of the Curve class:

Surface.gamma(): returns a (n_phi, n_theta, 3) array containing \(\Gamma(\phi_i, \theta_j)\) for \(i\in\{1, \ldots, n_\phi\}, j\in\{1, \ldots, n_\theta\}\), i.e. returns a list of XYZ coordinates on the surface.
Surface.gammadash1(): returns a (n_phi, n_theta, 3) array containing \(\partial_\phi \Gamma(\phi_i, \theta_j)\) for \(i\in\{1, \ldots, n_\phi\}, j\in\{1, \ldots, n_\theta\}\).
Surface.gammadash2(): returns a (n_phi, n_theta, 3) array containing \(\partial_\theta \Gamma(\phi_i, \theta_j)\) for \(i\in\{1, \ldots, n_\phi\}, j\in\{1, \ldots, n_\theta\}\).
Surface.normal(): returns a (n_phi, n_theta, 3) array containing \(\partial_\phi \Gamma(\phi_i, \theta_j)\times \partial_\theta \Gamma(\phi_i, \theta_j)\) for \(i\in\{1, \ldots, n_\phi\}, j\in\{1, \ldots, n_\theta\}\).
Surface.area(): returns the surface area.
Surface.volume(): returns the volume enclosed by the surface.

A number of quantities are implemented in simsopt.geo.surfaceobjectives and are computed on a simsopt.geo.surface.Surface:

ToroidalFlux: computes the flux through a toroidal cross section of a Surface.

The value of the quantity and its derivative with respect to the surface dofs can be obtained by calling e.g., ToroidalFlux.J() and ToroidalFlux.dJ_dsurfacecoefficients().

Cacheing¶

The quantities that Simsopt can compute for curves and surfaces often depend on each other. For example, the curvature or torsion of a curve both rely on Curve.gammadash(); to avoid repeated calculation geometric objects contain a cache that is automatically managed. If a quantity for the curve is requested, the cache is checked to see whether it was already computed. This cache can be cleared manually by calling Curve.invalidate_cache(). This function is called everytime Curve.set_dofs() is called (and the shape of the curve changes).