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Many physical and mathematical models involve random fields in their inputdata. Examples are ordinary differential equations, partial differentialequations and integro--differential equations with uncertainties in thecoefficient functions described by random fields. They also play a dominantrole in problems in machine learning. In this article, we do not assume to haveknowledge of the moments or expansion terms of the random fields but we insteadhave only given discretized samples for them. We thus model some measurementprocess for this discrete information and then approximate the covarianceoperator of the original random field. Of course, the true covariance operatoris of infinite rank and hence we can not assume to get an accurateapproximation from a finite number of spatially discretized observations. Onthe other hand, smoothness of the true (unknown) covariance function results ineffective low rank approximations to the true covariance operator. We deriveexplicit error estimates that involve the finite rank approximation error ofthe covariance operator, the Monte-Carlo-type errors for sampling in thestochastic domain and the numerical discretization error in the physicaldomain. This permits to give sufficient conditions on the three discretizationparameters to guarantee that an error below a prescribed accuracy $\varepsilon$is achieved.

On the Numerical Approximation of the Karhunen-Lo\`{e}ve Expansion for Random Fields with Random Discrete Data