Robust sparse recovery and deep learning algorithms in computational mathematics

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Sparse recovery is the art of computing sparse representations of objects like signals or functions with respect to suitable bases. It lies at the core of key applications and techniques such as signal compression (JPEG, MPEG, MP3), regression and variable selection in statistical data analysis, and inverse problems in imaging (e.g., denoising, deblurring, super-resolution). One of the greatest success stories in this field is compressed sensing, which allows to acquire sparse signals via linear measurements and efficient sparse recovery techniques. Sparse recovery has also led to breakthroughs in computational mathematics, such as wavelet methods for Partial Differential Equations (PDEs). Moreover, over the last decade compressed sensing had a significant impact in computational mathematics, with applications to uncertainty quantification and numerical methods for PDEs. More recently, deep learning emerged as a new essential paradigm in data science. It is revolutionizing technology and scientific research by quickly becoming a critical element in virtually every data-driven application. Deep neural networks outperformed state-of-the-art approaches in extremely challenging tasks such as image classification, speech recognition, or playing strategic games. Moreover, they are becoming increasingly popular in inverse problems and in computational mathematics. In sparse recovery and deep learning, a crucial issue is the development of robust algorithms. This includes robustness to various sources of errors such as physical noise, model error, numerical error, and adversarial attacks. For this reason, the overarching goal of this research program is twofold: (i) to develop a new generation of robust algorithms for sparse recovery and deep learning and (ii) to apply them to solve challenging problems in computational mathematics. In particular, my research program has three long-term goals: 1. to develop a comprehensive algorithmic framework for robust sparse recovery and optimal compressed sensing strategies; 2. to devise robust algorithms for high-dimensional problems in uncertainty quantification and numerical methods for PDEs based on sparse recovery; 3. to design a new generation of robust neural network architectures and demonstrate their potential in uncertainty quantification. This work includes applications to compressed sensing and its use in Magnetic Resonance Imaging (MRI). It also involves the development of new approaches for uncertainty quantification and numerical approximation of PDEs, essential tools in computational science and engineering, and has highly inter-disciplinary elements. It harnesses new mathematical breakthroughs to propose efficient algorithms for applications in aerospace engineering and computational chemistry through collaboration with experts in these fields.