SNSF Starting grant: Multiresolution methods for unstructured data
In our daily lives, unstructured data is ubiquitous, while the amount of data is immense and rapidly increasing. The processing of social network data, text data, audio files, photos and videos, but also scientific data, like measurements and simulation data has become vital to our modern society. Multiresolution methods in general and wavelets in particular are well established tools for nonlinear approximation, image analysis, signal processing and machine learning. They have successfully been employed to process the data sources mentioned above. With a few notable exceptions however, these methods rely on an embedding of discrete data into a continuous, functional setting, for example by means of regression or interpolation. The project aims at the development and the numerical analysis of novel, fully discrete and data centered multiresolution methods for unstructured data. We shall develop the corresponding analytical framework, the arithmetic, high-dimensional variants and adaptive strategies for such methods. As a highly relevant application, we consider physical models with random input data, such as diffusion problems with uncertain permeability or mechanical problems with uncertain material parameters.
Fast multiresolution covariance analysis (FMCA)
FMCA is a header only library for the multiresolution analysis of scattered data and kernel matrices. It is developed at the Università della Svizzera italiana in the research group of Michael Multerer. Currently, the library features the construction of samplet bases and different versions of the pivoted Cholesky decomposition. The fast samplet covariance compression introduced in Samplets: Construction and scattered data compression will be added soon.
Bembel is a Boundary Element Method Based Engineering Library written in C and C++ to solve boundary value problems governed by the Laplace, Helmholtz or electric wave equation. It was written as part of a cooperation between the TU Darmstadt and the University of Basel, coordinated by H. Harbrecht, S. Kurz and S. Schöps. It is based on the Laplace BEM of J. Dölz, H. Harbrecht and M. Multerer as well as the spline and geometry framework of F. Wolf.
A Package which implements the anisotropic sparse grid quadrature for arbitrary downward closed index sets. To provide maximum flexibility, the underlying univariate quadrature rules as well as the criterion for the sparse index set can be defined by the user. Optimised versions of the total degree index set and the hyperbolic cross index set are provided by default. Currently, the package comes with a simple Matlab interface.