Pseudo-Likelihood Estimation for Parameters of Stochastic Time-Fractional Diffusion Equations

Although stochastic fractional partial differential equations have received increasing attention in the last decade, the parameter estimation of these equations has been seldom reported in literature. In this paper, we propose a pseudo-likelihood approach to estimating the parameters of stochastic time-fractional diffusion equations, whose forward solver has been investigated very recently by Gunzburger, Li, and Wang (2019). Our approach can accurately recover the fractional order, diffusion coefficient, as well as noise magnitude given the discrete observation data corresponding to only one realization of driving noise. When only partial data is available, our approach can also attain acceptable results for intermediate sparsity of observation.

Differences and similarities between precipitation patterns of different climates

In recent years water-related issues are increasing globally, some researchers even argue that the global hydrological cycle is accelerating, while the number of meteorological extremities is growing. With the help of large number of available measured data, these changes can be examined with advanced mathematical methods. In the outlined research we were able to collect long precipitation datasets from two different climatical regions, one sample area being Ecuador, the other one being Kenya. Using the methodology of spectral analysis based on the discrete Fourier-transformation, several deterministic components were calculated locally in the otherwise stochastic time series, while by the comparison of the results, also with previous calculations from Hungary, several global precipitation cycles were defined in the time interval between 1980 and 2019. The results of these calculations, the described local, regional, and global precipitation cycles can be a helpful tool for groundwater management, as precipitation is the major resource of groundwater recharge, as well as with the help of these deterministic cycles, precipitation forecasts can be delivered for the areas.

Error estimates of a finite element method for stochastic time-fractional evolution equations with fractional Brownian motion

The aim of this paper is to consider the convergence of the numerical methods for stochastic time-fractional evolution equations driven by fractional Brownian motion. The spatial and temporal regularity of the mild solution is given. The numerical scheme approximates the problem in space by the Galerkin finite element method and in time by the backward Euler convolution quadrature formula, and the noise by the [Formula: see text]-projection. The strong convergence error estimates for both semi-discrete and fully discrete schemes are established. A numerical example is presented to verify our theoretical analysis.