Algorithms of greedy-type are a popular tool for sparse approximation. Sparse approximations of functions are beneficial for several reasons. Therefore, we will develop greedy algorithms for two classes of problems, probability density estimation and inverse problems.
The development of a greedy algorithm for density estimation was motivated by the desire to implement a simulation algorithm for so-called nonwovens, a particular type of technical textiles, which are widely used in industrial applications. We will propose such a simulation algorithm, which needs an estimation of the probability density of the fiber directions inside a nonwoven. Fortunately, these directions can be obtained from real nonwovens by a CT scan, which yields millions of data points. The incorporation of a probability density that is generated by the newly developed greedy algorithm reduces the computation time of the simulation algorithm from 80 days to 150 minutes by a factor of 750 in comparison to the use of a standard method for density estimation, namely kernel density estimators.
For inverse problems, we introduce two generalizations of the Regularized Functional Matching Pursuit (RFMP) algorithm, which is a greedy algorithm for linear inverse problems. For the first generalization, called RWFMP, an improved theoretical analysis is possible. Furthermore, using the RWFMP, it is possible to reduce the computation time of the RFMP by a factor of 10 without losing much of the accuracy. The second generalization is an RFMP for nonlinear inverse problems. We apply the algorithm to the nonlinear inverse gravimetric problem, which is concerned with the reconstruction of information about the interior of a planetary body from gravitational data. We obtain very good numerical results concerning the accuracy, the sparsity, and the interpretability of the results.