Project B10 (finished)

Project B10 - Sparse Frequency Estimation: Stability and Algorithms

Principal Investigator: Armin Iske

Background and Motivation

Parameter estimation of exponential sum has a long and fruitful history. Exponential sums can be used as a model for the superposition of waves with different frequencies, which is one reason for their common occurrence. In the second half of the last century, the problem was studied intensely, as it occurs in direction-of-arrival estimation, where one uses a sensor array to determine the direction of an incoming superposition of wavefronts. In recent years, the estimation problem again attracted a lot of attention, as it was proposed as a mathematical model for super-resolution, i.e., as a technique to overcome natural resolution limits. Here the multivariate case is of particular importance.

Aims and Objectives

The aim of this project is twofold. First, it strives to clarify basic stability properties of the parameter estimation of exponential sums. While in full generality it cannot be expected that the parameter estimation is well-posed, we study under which additional model assumptions we can establish well-posedness. A fine quantitative analysis then allows us to establish a posteriori error estimates. The second part is focused on the development of novel algorithms in the higher dimensional case. We hope to break the curse of dimensionality and to reduce the number of samples as well as the computational complexity significantly.

PhD Student: Benedikt Diederichs

Benedikt Diederichs successfully defended his PhD on 17 August 2018

Publications

Benedikt Diederichs and Armin Iske: Improved Estimates for Condition Numbers of Radial Basis Function Interpolation Matrices.
Journal of Approximation Theory. DOI: 10.1016/j.jat.2017.10.004, Published electronically, 16 October 2017.

B. Diederichs and A. Iske: Projection-based multivariate frequency estimation.
IEEE International Conference Sampling Theory and Applications (SampTA2017), 2017, 360-363, DOI: 10.1109/SAMPTA.2017.8024446.

B. Diederichs and A. Iske: Parameter Estimation for Bivariate Exponential Sums.
IEEE International Conference Sampling Theory and Applications (SampTA2015), 2015, 493-497, DOI: 10.1109/SAMPTA.2015.7148940.