ABSTRACT

Bodmann
DMS-1109545

The concept of frame mechanics addresses the need for constructing an abundance of optimal redundant, stable expansions with frames, which have become central to applications of mathematics in remote sensing or wireless transmissions, in analog-digital conversion such as audio and video encoding, in packet-based network communications, noise-insensitive quantum computing and recently also in compressive sensing. Despite its popularity, the search for near-optimal frames has been successful mostly in small dimensions, or it had to rely on specific group-representation properties, or the use of randomization principles. In frame mechanics, the investigator is studying an alternative to the conventional, structured or random design methods by letting frames evolve under flows which drive them towards optimality, instead of constructing them directly. The general objectives are to find (1) appropriate frame dynamics, (2) suitable initializations, and to obtain (3) deterministic control of the approximation error. The envisioned outcome of the project includes leveraging recently established numerical results on the construction of equiangular tight frames for the verification of Zauner's conjecture (the existence of maximal Gabor frames in all finite-dimensional Hilbert spaces), constructing controlled approximations of Grassmannian frames and fusion frames for loss-insensitive transmissions in wireless or packet-based network communications, and the design of matrices for compressive sensing based on quantum chaotic dynamics which improve the restricted isometry properties of sensing matrices.

The mathematics of redundant signal representations is called frame theory. For practical purposes, a frame is a tool which incorporates or removes repetitive information when data is stored, transmitted or received. Frames have become essential in many data-intensive areas of modern technology, because the repetitive information helps compensate errors of transmission devices and sensors. However, over the last decades, progress in the optimal design of frames has been outpaced by the rapid growth of data generated by our hardware. In frame mechanics, the investigator and his students explore a fundamentally new strategy to overcome this problem: The burden of constructing such optimal frames is put on the computer, which lets frames evolve in a way that drives them towards optimality. The goal of this project is to demonstrate that this dynamic design strategy is mathematically guaranteed to find many optimal frames where previous attempts failed. Frame mechanics allows us to maximize performance in remote sensing, seismic and medical imaging, wireless and fiber-optic communications, and to make internet transmissions robust to network outages.

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