SFB 1481:  Sparsity and Singular Structures

• Subject Area: Mathematics
• Term: since 2022
• Website: Homepage
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 442047500

Despite vast gains in computational power in the past decades, the deluge of data and complexity of models in current applications pose fundamental challenges that cannot be surmounted by computational resources alone. Two critical areas are (1) machine learning and signal processing with high dimensional data and (2) partial differential equations (PDEs) with singularities. Significantly expanding the frontier in these areas requires new insight into the underlying mathematical structure of the problems at hand. While the two mentioned challenges may appear to have little in common, we are convinced that their analysis will benefit from closely related ideas and algorithms, in particular, from those based on sparsity: The crucial challenge is to control low complexity structures in high, potentially infinite dimensions. We will explore ways in which a predictor in machine learning, a signal, or the solution of a (singular) PDE can be described and efficiently computed based on a small number of parameters. Concrete examples from the proposal include sparsity in the sense of a few non-zero coefficients in a suitable basis representation, low rank matrices and tensors, neural networks representing complicated functions with relatively few parameters, and finite element methods utilizing specially chosen, singular ansatz functions. The proposed CRC aims at a coordinated research effort on the mathematical foundations and algorithms related to sparsity and partial differential equations with singularities. The main research tasks can be summarized as follows.• Development of cutting edge algorithms and novel theory related to sparsity and low rank concepts in mathematical signal processing (compressive sensing) and deep learning.• Exploiting sparsity, low rank matrix and tensor as well as neural network concepts systematically for highly efficient, numerical solution algorithms for partial differential equations. Particular emphasis will be placed on parametric equations, kinetic modelsand geometric equations.• Development and analysis of numerical methods for challenging partial differential equations with singularities, especially methods that exploit sparsity concepts.The exchange of ideas and mathematical tools among the different involved fields of analysis, numerics, probability, optimization and algebra will prove fruitful and foster significant progress. Rooted in the expertise of the consortium and driven by the selected exampleproblems, we expect to impact both underlying mathematical theory and corresponding computational methods. With these developments, we will lay foundations that will contribute in the future to advancing methodology and technology in a broad spectrum of applications, including artificial intelligence; data processing tasks in industry, society and science; simulation techniques in engineering; materials science and more.

• DFG Programme: Collaborative Research Centres
• International Connection: United Kingdom

Current projects

• A01 - Gradient descent for deep neural network learning (Project Heads Rauhut, Holger ;  Westdickenberg, Michael )
• A02 - Scattering transforms of sparse signals (Project Head Führ, Hartmut )
• A03 - Group actions and t-designs in sparse and low rank matrix recovery (Project Heads Führ, Hartmut ;  Nebe, Gabriele ;  Rauhut, Holger )
• A06 - Theta tensor norms and low rank recovery (Project Heads Fourier, Ghislain ;  Rauhut, Holger )
• A07 - Signal processing on graphs and complexes (Project Head Schaub, Ph.D., Michael )
• A08 - Sparse exit wave reconstruction via deep unfolding (Project Head Berkels, Benjamin )
• A09 - Regularizing neural network classification using random perturbations (Project Heads Krumscheid, Sebastian ;  Rauhut, Holger ;  Tempone, Ph.D., Raul )
• B01 - Nonlinear reduced modeling for state and parameter estimation (Project Heads Bachmayr, Markus ;  Dahmen, Wolfgang )
• B02 - Robust sparse low rank approximation of multi-parametric partial differential equations (Project Heads Bachmayr, Markus ;  Grasedyck, Lars )
• B03 - Robust data-driven coarse-graining for surrogate modeling (Project Head Krumscheid, Sebastian )
• B04 - Sparsity promoting patterns in kinetic hierarchies (Project Heads Herty, Michael ;  Torrilhon, Manuel )
• B05 - Sparsification of time-dependent network flow problems by discrete optimization (Project Heads Büsing, Christina ;  Herty, Michael ;  Koster, Arie )
• B06 - Kinetic theory meets algebraic systems theory (Project Heads Herty, Michael ;  Zerz, Eva )
• C01 - Singularity formation in dissipative harmonic flows (Project Heads Melcher, Christof Erich ;  Reusken, Arnold )
• C02 - Intrinsic convexity in the Mullins-Sekerka evolution (Project Heads Westdickenberg, Michael ;  Westdickenberg, Maria G. )
• C04 - Mathematical analysis of domain decomposition methods for the efficient solution of continuum solvation models (Project Heads Reusken, Arnold ;  Stamm, Benjamin )
• C05 - Numerical approximation of the Gross-Pitaevskii equation via vortex tracking (Project Heads Melcher, Christof Erich ;  Stamm, Benjamin )
• C06 - Elimination theory for deformed differential calculus (Project Head Robertz, Daniel )
• MGK - Integrated Graduate Program (Project Heads Krumscheid, Sebastian ;  Melcher, Christof Erich )
• Z - Central Tasks of the Collaborative Research Centre (Project Heads Rauhut, Holger ;  Westdickenberg, Michael )

• Applicant Institution: Rheinisch-Westfälische Technische Hochschule Aachen
• Spokespersons: Professor Dr. Holger Rauhut, until 7/2023; Professor Dr. Benjamin Stamm, from 8/2023 until 9/2023; Professor Dr. Michael Westdickenberg, since 9/2023