Sensitivity Analysis

Definition

Sensitivity analysis is intended to assess the influence of perturbing parameters on a functional (output parameter, system response, etc.) allowing one to determine the parameters which require more attention while analyzing a system. To quantify the influence, the sensitivities (coefficients) are used. The sensitivity $S(R,\alpha)$ of an arbitrary functional $R$ that depends on a parameter $\alpha$ is defined as (fractional/relative) derivative:

$$S(R,\alpha)\equiv\frac{\alpha}{R}\frac{\textnormal{d}R}{\textnormal{d}\alpha}$$

Using this value one can assess the change of the functional $R$ from the value of $R_0$ in a linear approximation. If $I$ parameters $\alpha_i$ are perturbed by a value of $\delta\alpha_i$, $i\in I$, it can be represented as first-order Taylor series.

$$R=R_0+\sum_{i=1}^I{S_i \delta\alpha_i}$$

where $S_i\equiv S(R_i,\alpha_i)$ is used for brevity.

Sensitivity assessment

Sensitivities are quantified via different approaches that can be divided into two groups relied on the direct and perturbation/variational methods. The first method is usually not considered due because it requires at least one additional simulation per parameter to perturb. The perturbation-based approach generally uses first-order perturbation theory. For the eigenvalue $k$ the equation has the following form:

$$S(k,\alpha)=\frac{\alpha}{k}\frac{\textnormal{d} k}{\textnormal{d} \alpha} = -\alpha \frac{\langle\psi^*,\left(\frac{\partial \hat{A}}{\partial \alpha} - \frac{1}{k}\frac{\partial \hat{F}}{\partial \alpha}\right)\rangle}{\langle \psi^*,\frac{1}{k}\hat{F}\psi \rangle} +\mathcal{O}(\delta\psi)$$

where $\langle f,g \rangle \equiv \int f(\vec{\xi})g(\vec{\xi}) \textnormal{d}\vec{\xi}$ is the scalar product of arbitrary functions $f(\vec{\xi})$ and $g(\vec{\xi})$; $\vec{\xi}$ is the phase-space vector; $\hat{A}$ is the transport operator; $k$ is the eigenvalue (multiplication factor); $\hat{F}$ is the fission operator; $\delta\psi$ is the perturbation in $\psi$ due to the perturbation in $\alpha$.

This formula clearly shows the advantage of the perturbation-based approach: it requires only the knowledge of the forward and adjoint fluxes, i.e. requires two neutron transport simulations to get the sensitivities to all the $I$ parameters while the direct approach requires at least $I$ perturbed simulations besides one unperturbed forward simulation, making this approach rather inefficient.

This is the classical approach for the eigenvalue $k$, and, to calculate the sensitivity of other functionals, one shall turn to Generalized perturbation theory also known as GPT introduced by Usachev. GPT has a similar advantage, but it might require two addition simulations if the functional is bi-linear (or their ratio) such as the effective delayed neutron fraction $\beta_{eff}$ or the other kinetic parameters. GPT has two limitations: 1) the functional $R$ has to satisfy the condition $\langle \psi, \frac{\partial R}{\partial \psi} \rangle= 0$ (e.g. $^{135}\textnormal{Xe}$ equilibrium concentration); 2) if the number of functionals is large enough (e.g. power distribution), the advantage of the perturbation-based approach diminishes.

SAUNA does not compute sensitivities: it relies on the tools, having these techniques implemented such as Serpent and KENO-VI of SCALE. Still, they may be created manually, but the approach is rather inefficient.

The formulations were developed long ago though they are still widely used, and their development is rather concerned with how a stochastic neutron transport code can compute sensitivities in a single simulation. These techniques are implemented in a number of well-known tools such as MCNP, KENO-VI, Serpent, TRIPOLI-4, etc.