Representation of Gaussian random fields on spheres

Abstract: Motivated by biological application, such as cell-biology, partial differential equations on curved (moving) domains have become a flourishing mathematical field. Moreover, including uncertainty into these models is natural due to the lack of precise initial data or randomness of the processes itself. One of the basic questions in these models is how to represent random field on a curved domain?

In this presentation we will first give a brief insight into different possibilities of representing isotropic Gaussian random fields defined on a flat domain and their importance. In particular, we will recall the standard Karhunen-Loeve expansions. Next, we will consider Gaussian random fields on a sphere. The main goal of the talk will be to present the construction of a multilevel expansions of isotropic Gaussian random fields on a sphere with independent Gaussian coefficients and localized basis functions (modified spherical needlets). In the last part we show numerical illustrations and an application to random elliptic

PDEs on a sphere. This is a joint work with Markus Bachmayr.

differential geometryprobability

Audience: advanced learners

Young Researchers between Geometry and Stochastic Analysis 2021