Event-by-Event Fission Modeling
297
4 Parameter Optimization for Spontaneous Fission
The five FREYA parameters, e 0 , c S , x, c, and dTKE, are all physics-based and
affect some observables directly without having any significant effect on others. For
example, c S , the parameter related to spin fluctuations, controls the photon energy
and multiplicity but has a negligible effect on the neutron observables. (The only
effect of c S on neutron emission comes from the fact that it controls the division
of the total excitation energy into rotational and statistical excitation. Giving more
energy to rotation by a large c S reduces the energy available for neutron emission.)
The parameter x controlling the excitation energy advantage given to the light
fragment has a direct effect on the neutron multiplicity as a function of fragment
mass, ν(A), while, e.g. having no effect on the neutron multiplicity distribution
P (ν). The parameter x is also the only parameter to have a strong effect on the
neutron–neutron angular correlations, as will be discussed later. The parameter
controlling the width of the thermal fluctuations, c, conversely, has a strong effect on
P (ν) and its moments but no effect on ν(A). All the parameters, however, have some
effect on the prompt fission neutron spectrum because all affect the energy available
for neutron emission, either directly, as through c S and dTKE, or indirectly, through
the excitation energy sharing via x or the fluctuations controlled by c. Indeed, the
only observable that is affected by e 0 is the neutron spectrum. See Refs. [2, 4] for
more discussion on how the parameter choices affect observables.
A first attempt to make a global fit of the five FREYA parameters was made in
Ref. [5] using a grid search method. More recently an optimization using simulated
annealing was able to generally reproduce these results as well as provide variances
and covariance on the parameter values [6]. Similar studies were carried out for all
spontaneous fission isotopes in FREYA.
Techniques such as a brute force grid search are computationally intensive and,
in order to avoid local minima, which are not the global minimum and thus not
physically relevant, it is necessary to move away from simpler optimization schemes
like gradient descent. Optimized parameters were determined for FREYA using the
simulated annealing method [7] in Ref. [6] which injects a certain randomness into
the process to allow for the procedure to occasionally jump in a seemingly “worse”
direction in order to move out of a potential local minimum and eventually find
the global solution. The general flow of such an algorithm is to first generate a
random solution, calculate its cost using some objective function, generate a random
neighboring solution, calculate the cost of this new solution with the same objective
function, and then compare these costs using an acceptance probability function.
This acceptance probability is calculated by comparing the difference of the two
costs with the so-called temperature. This is a parameter which is initially equal
to unity and is decreased to a new value, T , after each iteration of the algorithm
employing a scale factor α, generally between 0.8 and 1, allowing the algorithm
to become less stochastic as the number of iterations is increased. This procedure
helps prevent the algorithm from sinking into a local minimum. In the FREYA
297
4 Parameter Optimization for Spontaneous Fission
The five FREYA parameters, e 0 , c S , x, c, and dTKE, are all physics-based and
affect some observables directly without having any significant effect on others. For
example, c S , the parameter related to spin fluctuations, controls the photon energy
and multiplicity but has a negligible effect on the neutron observables. (The only
effect of c S on neutron emission comes from the fact that it controls the division
of the total excitation energy into rotational and statistical excitation. Giving more
energy to rotation by a large c S reduces the energy available for neutron emission.)
The parameter x controlling the excitation energy advantage given to the light
fragment has a direct effect on the neutron multiplicity as a function of fragment
mass, ν(A), while, e.g. having no effect on the neutron multiplicity distribution
P (ν). The parameter x is also the only parameter to have a strong effect on the
neutron–neutron angular correlations, as will be discussed later. The parameter
controlling the width of the thermal fluctuations, c, conversely, has a strong effect on
P (ν) and its moments but no effect on ν(A). All the parameters, however, have some
effect on the prompt fission neutron spectrum because all affect the energy available
for neutron emission, either directly, as through c S and dTKE, or indirectly, through
the excitation energy sharing via x or the fluctuations controlled by c. Indeed, the
only observable that is affected by e 0 is the neutron spectrum. See Refs. [2, 4] for
more discussion on how the parameter choices affect observables.
A first attempt to make a global fit of the five FREYA parameters was made in
Ref. [5] using a grid search method. More recently an optimization using simulated
annealing was able to generally reproduce these results as well as provide variances
and covariance on the parameter values [6]. Similar studies were carried out for all
spontaneous fission isotopes in FREYA.
Techniques such as a brute force grid search are computationally intensive and,
in order to avoid local minima, which are not the global minimum and thus not
physically relevant, it is necessary to move away from simpler optimization schemes
like gradient descent. Optimized parameters were determined for FREYA using the
simulated annealing method [7] in Ref. [6] which injects a certain randomness into
the process to allow for the procedure to occasionally jump in a seemingly “worse”
direction in order to move out of a potential local minimum and eventually find
the global solution. The general flow of such an algorithm is to first generate a
random solution, calculate its cost using some objective function, generate a random
neighboring solution, calculate the cost of this new solution with the same objective
function, and then compare these costs using an acceptance probability function.
This acceptance probability is calculated by comparing the difference of the two
costs with the so-called temperature. This is a parameter which is initially equal
to unity and is decreased to a new value, T , after each iteration of the algorithm
employing a scale factor α, generally between 0.8 and 1, allowing the algorithm
to become less stochastic as the number of iterations is increased. This procedure
helps prevent the algorithm from sinking into a local minimum. In the FREYA