Sampling of random fields on manifolds

We are three researchers, Erik Jansson (PhD student), Annika Lang (professor) and Mike Pereira (postdoc), from the Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, who work together on the development of new efficient algorithms for the sampling of random fields on manifolds. Special emphasis is put on Gaussian random fields, also known as Gaussian processes, which for example can be used to model spatial data. For an introduction to the topic refer to AdlerAdler & TaylorCressieMarinucci & Peccati.

Random fields are random functions defined on a spatial domain. As an example we can consider the sphere as a model for our planet Earth and the temperature at each point on the surface as a function disturbed by some noise. This noise can be due to measurement errors, missing data, or inaccuracy of the used models. At the same time, the temperature is not totally random since we know that at nearby places it is obviously related. Another application is the generation of ice crystals which are all similar but different (see pictures). Having this picture in mind, we want to generate structured randomness described by so-called correlations between space points.

Simulation of a Gaussian random field on the surface of a sphere.

In more general terms, Gaussian random fields are widely used to model spatial data and for performing tasks ranging from prediction to filtering and uncertainty quantification. This is due to their relative flexibility and ease of implementation. So far, research has mainly focused on random fields on Euclidean domains but applications in material sciences, environmental sciences and cosmology all call for methods to model data that lie on curved surfaces (e.g. spheres). This highlights the need to study Gaussian random fields on more general domains as well as the need to develop accurate and efficient approximation algorithms. Random field samples are used as input in simulations, e.g. of models involving partial differential equations, where uncertainty needs to be taken into account.

Simulation of a Gaussian random field on a surface shaped like a cow.

Supported by the Swedish Research Council for the project “Efficient Approximation Methods for Random Fields on Manifolds”, we work with different characterizations of random fields on Riemannian manifolds. One example is a basis expansion known as the Karhunen–Loève expansion. Approximating fields by truncation of this series expansion leads to spectral methods which are in some sense generalizations of well-known Fourier methods. Special emphasis is put on expansions with respect to the eigenfunctions of the Laplace–Beltrami operator. In the Gaussian setting these expansions include the spherical Matérn–Whittle random fields, which are widely used in spatial statistics. These fields are connected to stochastic partial differential equationsinvolving the Laplace–Beltrami operator and white noise.

Simulation of an ice crystal obtained from the simulation of a “rough” Gaussian random field on a sphere.

One way to generate Gaussian random fields is by employing this connection to obtain a class of stochastic partial differential equations. We propose to work with numerical approximations of these equations based on the surface finite element method (SFEM). With this method we can generate random fields on surfaces that are polyhedral approximations of the real geometry, e.g. surface meshes, as shown in the pictures on the sphere or a cow. This step allows us to use the sparsity of the mass and stiffness matrices encountered in finite element methods to propose scalable and efficient algorithms for the sampling and predictions of Gaussian random fields.

Besides the development of new algorithms, we are especially interested in the analysis of the quality of the generated random fields. We derive error bounds on the accuracy and estimates on the computational complexity. Therefore, we provide the user not only with approximation algorithms but also with information on the quality and efficiency for a chosen mesh size.

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