Asymptotic approximations and importance sampling methods are presented for evaluating a class of\ud probability integrals with multiple design points that may arise in the calculation of the reliability of\ud uncertain dynamical systems. An approximation based on asymptotics is used as a first step to provide a\ud computationally efficient estimate of the probability integral. The importance sampling method utilizes\ud information of the integrand at the design points to substantially accelerate the convergence of available\ud importance sampling methods that use information from one design point only. Implementation issues\ud related to the choice of importance sampling density and sample generation for reducing the variance of\ud the estimate are addressed. The computational efficiency and improved accuracy of the proposed methods\ud is demonstrated by investigating the reliability of structures equipped with a tuned mass damper for which\ud multiple design points are shown to contribute significantly to the value of the reliability integral
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