Monte Carlo Estimates of Domain-Deformation Sensitivities

It is shown that, when a Monte Carlo algorithm is used for estimation of any physical quantity $A$, a simple and fast additional procedure can be implemented that simultaneously estimates the sensitivity of $A$ to any problem parameter. The proposed approach is general and systematic in the sense that: (i) it includes domain-deformation sensitivities, i.e., cases where a change in the parameter modifies the domain over which the sampled random variables are defined and (ii) a simple generic procedure is presented to address all remaining free choices in terms of variance minimization.

Figure 1

The domain $\mathcal{D}$ in ${\mathbb{R}}^n$ can always be defined as $\{\mathbf{x}\in {\mathbb{R}}^n|f(\mathbf{x},\lambda )<0\}$, where $f$: ${\mathbb{R}}^n\times \mathbb{R}\to \mathbb{R}$ is $C_1$ in $\lambda$. ${\mathbf{V}}_{\lambda }$ is constrained by $\frac{d}{du}f(\mathbf{x}+u{\mathbf{V}}_{\lambda },\lambda +u){|}_{u=0}=0$ which only fixes the normal component at the boundary $\partial \mathcal{D}$ [8]. If $\mathbf{W}$ is any vector field tangential to $\partial \mathcal{D}$, ${\mathbf{V}}_{\lambda }+\mathbf{W}$ is also solution, which means that there is a large freedom of choice for the transformation velocity field.