Gradient Estimation

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EP: This page should gear to the Gradient Estimation and not to Sensitivity Analysis. Furthermore the reference does not seem appropriate.


Local sensitivity analysis aims at exploring the changes of the quantity of interest due to some variations of the input parameters around a reference point \mathbf{x}_0 (e.g. mean value of the vector of input parameters).


\begin{matrix}
s_i=\frac{\partial y(\mathbf{x})}{\partial x_i}.
\end{matrix}


The quantity s_i can be approximated using finite differences (see e.g.[1])).

If using finite differences for the calculation of the sensitivities, s_i is approximated by the slope of a secant line through the reference point (\mathbf{x}_0,y(\mathbf{x}_0)) and the point where the variable x_i is affected by a small perturbation, which yields


\begin{matrix}
s_i=\frac{\partial y(\mathbf{x})}{\partial x_i} \approx \frac{y[x_{0,1}, \ldots, x_{0,i-1},x_{0,i}+h,x_{0,i+1}, \ldots, x_{0,n}]-y(\mathbf{x}_0)}{h}.
\end{matrix}


An important consideration for the numerical differentiation is the selection of the value of h . If it is selected too small, rounding errors can lead to erroneous results, if h is too large, the slope of the tangent may be considerably different from the derivative at point \mathbf{x}_0 .

In case of a large number of input parameters n , the computational efforts associated with the determination of all s_i,\, i=1,\ldots, n may lead to the limits of practical applicability. Therefore, efficient Monte Carlo Gradient Estimation procedures are available for the computation of gradients.


References

  1. M. Kleiber, H. Antunez, T.D. Hien, P. Kowalczyk. Parameter sensitivity in nonlinear mechanics: theory and finite element computations, Wiley, New York, 1997.


See Also