Probabilistic estimates of the maximum norm of random Neumann Fourier series

61. Dirk Blömker, Philipp Wacker, Thomas Wanner:
    Probabilistic estimates of the maximum norm of random Neumann Fourier series
    Communications in Nonlinear Science and Numerical Simulation 47, pp. 348-369, 2017.

Abstract

We study the maximum norm behavior of $L^2$-normalized random Fourier cosine series with a prescribed large wave number. Precise bounds of this type are an important technical tool in estimates for spinodal decomposition, the celebrated phase separation phenomenon in metal alloys. We derive rigorous asymptotic results as the wave number converges to infinity, and shed light on the behavior of the maximum norm for medium range wave numbers through numerical simulations. Finally, we develop a simplified model for describing the magnitude of extremal values of random Neumann Fourier series. The model describes key features of the development of maxima and can be used to predict them. This is achieved by decoupling magnitude and sign distribution, where the latter plays an important role for the study of the size of the maximum norm. Since we are considering series with Neumann boundary conditions, particular care has to be placed on understanding the behavior of the random sums at the boundary.

Links

The preprint version of the paper can be downloaded from https://arxiv.org/abs/1603.04300, while the published version of the paper can be found at https://doi.org/10.1016/j.cnsns.2016.11.023.

Bibtex

@article{bloemker:etal:17a,
   author = {Dirk Bl\"omker and Philipp Wacker
             and Thomas Wanner},
   title = {Probabilistic estimates of the maximum norm of random
            {N}eumann {F}ourier series},
   journal = {Communications in Nonlinear Science and Numerical Simulation},
   year = 2017,
   volume = {47},
   pages = {348--369},
   doi = {10.1016/j.cnsns.2016.11.023}
   }