Extremes of random fields over arbitrary domains with application to concrete rupture stresses

To find the exact probability distribution of the global maximum or minimum of a random field within a bounded domain is a pending problem even for Gaussian fields. Except for very special examples of fields, recourse must be taken to approximate reasoning or asymptotic considerations to be judged with respect to accuracy by simulations. In this paper, the problem is addressed through a functional equation that leads to the definition of a class of distribution functions that depend solely on process or field characteristics and domain quantities that can be calculated explicitly. This distribution function class is studied for Gaussian processes in earlier works by the author and it has been obtained explicitly for Gaussian fields on rectangular domains in the plane. Simulation studies show that rather good predictions are obtained for sufficiently smooth wide band Gaussian processes and fields. In this paper, the distribution function is obtained in general for Gaussian fields over arbitrary bounded domains with piecewise continuous and differentiable boundaries, and as in earlier works the distribution function is tested against empirical distribution functions obtained by simulation of sample functions of a smooth approximately Gaussian field, herein called a broken line Hino field. For completeness this particular field type is defined in appendix a and appendix b. The paper concludes with a statistical application on data for plain concrete tensile strength.