# Computational Harmonic Analysis and Approximation

### Abstract

Many problems in science and engineering require the reconstruction of an object - an image, signal or high-dimensional function, for example - from a collection of measurements. Due to time, cost or other constraints, one is often severely limited by the amount of data that can be collected, which significantly affects ones ability to recover the unknown object accurately. This research project addresses the development and analysis of new methods for this problem using the techniques of computational harmonic analysis and approximation theory, with an emphasis on the development of new compressed sensing-based methods. Examples of relevant applications include medical imaging, microscopy, uncertainty quantification in physical systems and the numerical solution of PDEs. Its primary goals are (i) to develop and study compressed sensing-based methods for high-dimensional function approximation, (i) to develop new structured compressed sensing algorithms for imaging problems that leverage the connected tree structure of standard multiresolution representations (e.g. wavelets), and (iii) the investigation of the limits of stability and accuracy for sampling-based algorithms in scientific computing, and the design of new computational methods based on conformal mappings and redundant approximations (i.e. frames) that attain such limits.