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<p>For readers working in applications, this paper serves as a basic primer for the numerical calculation of (pseudo-) random functions. Keeping probabilistic arguments at a minimum in favour of computation, random functions are studied by randomizing coefficients of expansions. This reveals a natural connection to Reproducing Kernel Hilbert spaces. Randomizing the coefficients of orthonormal Newton bases of Reproducing Kernel Hilbert spaces allows a comprehensive, constructive, and computational presentation, providing regularity results and error bounds in terms of variances. The approach is unexpectedly general, because paths of most random fields are shown to arise this way. A natural extension allows to use expansions in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_2\)</EquationSource> </InlineEquation> spaces, in particular wavelets and finite elements. Smoothing operators are introduced for the transition from white noise to random functions in higher-order Sobolev spaces, leading to algorithms without solving stochastic partial differential equations. Numerical examples serve for illustration.</p>

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A computational introduction to random functions

  • 期刊论文

  • 发表日期：

    2026年3月19日

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  • Emilio Porcu,
  • Robert Schaback

摘要

For readers working in applications, this paper serves as a basic primer for the numerical calculation of (pseudo-) random functions. Keeping probabilistic arguments at a minimum in favour of computation, random functions are studied by randomizing coefficients of expansions. This reveals a natural connection to Reproducing Kernel Hilbert spaces. Randomizing the coefficients of orthonormal Newton bases of Reproducing Kernel Hilbert spaces allows a comprehensive, constructive, and computational presentation, providing regularity results and error bounds in terms of variances. The approach is unexpectedly general, because paths of most random fields are shown to arise this way. A natural extension allows to use expansions in \(L_2\) spaces, in particular wavelets and finite elements. Smoothing operators are introduced for the transition from white noise to random functions in higher-order Sobolev spaces, leading to algorithms without solving stochastic partial differential equations. Numerical examples serve for illustration.