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Published by Fusion Energy Division, Oak Ridge National Laboratory
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Editor: James A. Rome Issue 123 December 2009
E-Mail: jar@ornl.gov Phone (865) 482-5643
On the Web at http://www.ornl.gov/sci/fed/stelnews
Optimization of stellarators
using metaheuristics
and grid computing
Stellarator optimization requires many different executions
of an equilibrium code such as VMEC and the applications
that estimate the target functions. Often, it is very
difficult to get the best configuration because of the huge
parameter space. Trying to explore this vast solution space
by using brute force algorithms or by specification of all
of the different configurations becomes an impossible
task. In the case of optimization problems, the use of
metaheuristics such as genetic or evolutionary algorithms
has been traditionally considered one of the best solutions.
Nevertheless, stellarator optimization using these kind of
tools requires huge computational resources; grid computing
technologies could be a good solution for this task.
The different computations that must be performed in
order to evaluate the selected individuals, i.e., the selected
stellarator configurations, can be executed in parallel in
the computational resources of the infrastructure, and the
management of the algorithm can be centralized in a single
machine in a master-slave process.
In our case and as a starting point, we focused on optimization
of the neoclassical transport of particles by estimating
the average drift and searching for
configuration equilibria that minimize this quantity (see
Fig. 1). We are using VMEC to estimate the magnetic configuration
and different metaheuristics to perform this
optimization. In all our cases, an individual is a magnetic
configuration (an input for VMEC, that is, an execution of
the code in a remote resource of the grid); a chromosome
is a single configuration parameter of the input (the Fourier
harmonics of the magnetic field), and an individual is
based on several chromosomes. The population consists of
all individuals that are being evaluated at a giving
moment. We have developed three different types of algorithms.
◗ Genetic Algorithms: These algorithms imitate evolution
strategies such as crossover or mutation of chromosomes
to create new individuals in a population.
They are iterative models that require a large population
(a few thousand individuals) to obtain good
results. The wall clock time for an optimization process
using this technique can be as much as one
month, representing more than one year of CPU time.
◗ Scatter Search: Scatter search (SS) is a metaheuristic
process using formulations that date back to the 1960s
B × ∇B
Fig. 1. Example of a two-period stellarator optimized with
the condition of minimum average drift.
In this issue . . .
Optimization of stellarators using metaheuristics
and grid computing
Stellarator equilibrium optimization requires numerous
lengthy calculations of equilibria and optimization criteria
for each configuration. But they can all be performed
in parallel using grid computing in a masterslave
configuration. Three different methods are being
used to guide the calculations: genetic algorithms,
scatter search, and an artificial bee colony.............. 1
Stellarator News -2- December 2009
for combining decision rules and problem constraints.
SS works over a set of solutions, combining them to
get new ones that improve the original set. In contrast
to other evolutionary methods, such as genetic algorithms,
SS is based not on large random populations
but on strategic selections among small populations.
SS also follows an iterative model that uses a small
population and several improvement processes over a
set of reference solutions. It is based on the idea of
keeping a good balance between convergence and dispersion
to find approximate solutions.
◗ Artificial Bee Colony Algorithm: This optimization
process is modeled on behavior observed in bee colonies:
“rover” bees that find a desirable food source
return to the colony and communicate the location of
the food source by “dancing.” Once the information is
exchanged, more bees are allocated to similar “flowers”
(see Fig. 2). In other words, some processes
search for approximate solutions for a given problem;
and, when an optimized solution is found, more processes
are allocated to look for approximate solutions
using the new optimized one as a base element to
explore. The exchange of information is, in fact, a
master process managing the overall execution. It is
not based on the idea of iteration and, although it is
more difficult to develop and deploy, it makes better
use of grid resources and offers the fastest results of
the different metaheuristics.
All these algorithms have been developed using a generic
schema, so any other optimization process suitable for
execution on the grid can be solved by means of these
techniques without substantial effort. Also, the scientist
doesn't need to know many things about the technology
being used, since these algorithms deal with the challenges
that a distributed infrastructure can present.
All three types of algorithm are suitable to be run on the
grid. Grid computing offers a large number of computational
resources and a distributed paradigm that creates a
perfect environment for large simulations where the communication
requirements between the different elements
involved in the simulation are low. These computational
resources, usually devoted to scientific calculations, offer
a high quality of service when in production state, with
large storage capability, and are easily accessible. They
create an attractive venue where scientists can carry out
their simulations. Grid computing is, therefore, especially
appropriate to run serial codes that can be executed in parallel,
with different data, to get different results. This is the
case of the stellarator optimization problem.
The next step is to introduce more target functions to be
optimized, beyond just neoclassical transport. Mercier and
ballooning stability criteria are the next optimization functions
that will be considered in our algorithm. The Mercier
criterion is estimated by the latest version of VMEC,
which is already running in the grid, while calculating ballooning
stability criterion requires porting the COBRA
code to the grid. This task is now under way.
In this way we are able to optimize any given configuration
by taking into account these three criteria for the
moment. The inclusion of further optimization criteria can
be done easily in this kind of algorithm.
Antonio Gómez and Francisco Castejón
Laboratorio Nacional de Fusión
Asociación Euratom/Ciemat para Fusión
E-mail antonio.gomez@ciemat.es
Fig. 2. Distribution in the parameter space of several types
of bees and view of their explored areas.