3 research outputs found
Approximation of probability density functions for PDEs with random parameters using truncated series expansions
The probability density function (PDF) of a random variable associated with
the solution of a partial differential equation (PDE) with random parameters is
approximated using a truncated series expansion. The random PDE is solved using
two stochastic finite element methods, Monte Carlo sampling and the stochastic
Galerkin method with global polynomials. The random variable is a functional of
the solution of the random PDE, such as the average over the physical domain.
The truncated series are obtained considering a finite number of terms in the
Gram-Charlier or Edgeworth series expansions. These expansions approximate the
PDF of a random variable in terms of another PDF, and involve coefficients that
are functions of the known cumulants of the random variable. To the best of our
knowledge, their use in the framework of PDEs with random parameters has not
yet been explored
Inferring ecosystem states and quantifying their resilience : linking theories to ecological data
The core of my thesis concerns addressing the ecosystem resilience in a data-driven manner. In this direction, I have tried to make a bridge between advanced mathematical models and existing ecological data. I could come up with some quantitative measures of resilience and applied them to some ecological field and experimental data. These measures are more exact compared with the classical measures mentioned by Holling. I show that Holling measures are just two extremes of the measure I introduced and they do not necessarily capture the notion of resilience in its real sense of the word. Furthermore, I could also address the resilience of low-resolution tropical satellite data across the tropics (South America, Africa, south east Asia and, Australia). Besides, my thesis also sheds more light on the concept of ‘alternative stable states’ which is an important concept in ecology. I argue that advanced ‘system reconstruction’ approaches should be applied first, from where one can better justify weather or not an ecosystem has alternative stable states. </p