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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 77
PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON CIVIL AND STRUCTURAL ENGINEERING COMPUTING Edited by: B.H.V. Topping
Paper 120
Numerical Estimation of Sensitivities for Complex Probabilistically-Described Systems R.E. Melchers and M. Ahammed
Centre for Infrastructure Performance and Reliability, The University of Newcastle, Australia Full Bibliographic Reference for this paper
R.E. Melchers, M. Ahammed, "Numerical Estimation of Sensitivities for Complex Probabilistically-Described Systems", in B.H.V. Topping, (Editor), "Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 120, 2003. doi:10.4203/ccp.77.120
Keywords: Monte Carlo, simulation, gradients, parameters, sensitivities, system performance, constraints.
Summary
For many practical systems, the overall performance will be described by one or
more performance indicators. However, some systems are so complex that they can
be described only in probabilistic terms. Monte Carlo simulation is then the only
feasible approach for evaluating system performance. The system 'performance
functions' then involve random variables and stochastic process(es) X, each
described by statistical moments, such as the mean (or expected value), the standard
deviation, etc. collected in the vector v.
The performance is the expectation over a (preferably large) number of sample performances: where ![]() ![]() ![]() ![]() ![]()
For system optimization or design refinement, it is helpful to have available the
sensitivity of
The approach proposed herein to estimate the gradients and sensitivities at the
point of maximum likelihood x*
and to construct a 'response surface'
that is, the 'expected' system performance ![]() ![]() ![]() ![]() where ![]() ![]() ![]()
The gradients (e.g.
When there are constraint functions
G One example is given to illustrate that the approximate technique produces results for gradient and for sensitivities very close to the theoretical results. The second example shows how the gradients and sensitivities change with changes in the constraint equation. The change in behaviour is smooth, indicating a gradual change in sensitivity as the boundary moves. However, when the boundary reaches the point of maximum likelihood for the unconstrained performance function the gradients are then uninfluenced by a change in location of the constraint. There are corresponding jump changes in parameter sensitivities. The reason for this is discussed. The examples illustrate that the proposed technique produces satisfactory results even for highly nonlinear performance functions, implying very considerable approximations in the use of a tangent hyperplane. Importantly, the results for gradients and sensitivities were obtained without repeated Monte Carlo simulation runs. The proposed technique requires little computational effort beyond that required for the original Monte Carlo simulation. It was found that the proposed technique provided a satisfactory approach for estimating gradients and parameter sensitivities for a number of highly non-linear performance functions, including some with constraints on admissible regions. References
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