In many reliability problems, particularly in the area of offshore and marine safety analysis, the maxima of stochastic processes and random fields, describing random load responses and resistances, are critically important. In the case of stationary differentiable Gaussian processes, approximations for the distribution of the maxima using upcrossing rates are well known. In this paper, we consider nonstationary differentiable Gaussian random fields. A new technique first applied by Sun (1993) can be used to derive direct approximations for the tail probability. This is an alternative approach based on an expansion of the covariance operator and on more or less geometric concepts. The method does not rely on assumptions regarding the point process of upcrossings. It provides, directly, an asymptotic approximation for the distribution of the maximum of the random field.
Direct Approximation of the Extreme Value Distribution of Nonhomogeneous Gaussian Random Fields
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Maes, M. A., and Breitung, K. W. (November 1, 1997). "Direct Approximation of the Extreme Value Distribution of Nonhomogeneous Gaussian Random Fields." ASME. J. Offshore Mech. Arct. Eng. November 1997; 119(4): 252–256. https://doi.org/10.1115/1.2829104
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