The linear algebra of compressed sensing, with applications to PDEs

• Antragstellerin: Professorin Dr. Olga Holtz
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2008 bis 2013
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 78969239

1. Exploring sufficient conditions for optimal reconstruction of functions that are 'sparse' in terms of dictionaries consisting of orthogonal polynomials. We suggest a detailed investigation of sufficient conditions in these cases, e.g., Spark, coherence, RIP, the null space property and others.2. Deterministic construction of compressed sensing matrices. We continue to study matrices withvarious structures: Kronecker product, Toeplitz, cyclic, generalized Vandermonde, etc. Methods from graph theory are suggested to analyse matrices with underlying graph structure, and specific structured matrix techniques are to be applied to matrices with low displacement rank and other known structures. This study should lead to a deterministic construction of optimally performing compressive sensing matrices with a given structure.3. Applications of frame theory to compressive sensing and vice versa. We investigate potentialapplications of compressive sensing techniques to the Feichtinger and the Paving conjectures. A related task is to understand the existence and compressive properties of equiangular tight frames.4. Applications of compressive sensing techniques to PDEs. We solve discretized boundary valueproblems for elliptic PDEs, using compressive sensing ideas.

• DFG-Verfahren: Schwerpunktprogramme
• Teilprojekt zu: SPP 1324: Mathematische Methoden zur Extraktion quantifizierbarer Information aus komplexen Systemen