Robust sparse recovery and deep learning algorithms in computational mathematics

稀疏恢复是计算对象（如信号或函数）相对于合适的基的稀疏表示的技术。它是关键应用和技术的核心，如信号压缩（JPEG，MPEG，MP3），统计数据分析中的回归和变量选择，以及成像中的逆问题（例如，去噪、去模糊、超分辨率）。压缩感知是该领域最成功的案例之一，它允许通过线性测量和有效的稀疏恢复技术来获取稀疏信号。稀疏恢复也导致了计算数学的突破，例如偏微分方程（PDE）的小波方法。此外，在过去的十年中，压缩感知在计算数学中产生了重大影响，应用于不确定性量化和偏微分方程的数值方法。最近，深度学习成为数据科学中一种新的基本范式。它正在通过迅速成为几乎所有数据驱动应用程序中的关键元素来彻底改变技术和科学研究。深度神经网络在图像分类、语音识别或玩战略游戏等极具挑战性的任务中表现出色。此外，它们在反问题和计算数学中越来越受欢迎。在稀疏恢复和深度学习中，一个关键问题是开发鲁棒的算法。这包括对各种错误来源的鲁棒性，例如物理噪声，模型误差，数值误差和对抗性攻击。因此，该研究计划的总体目标是双重的：（i）开发新一代用于稀疏恢复和深度学习的鲁棒算法，以及（ii）将它们应用于解决计算数学中具有挑战性的问题。 具体来说，我的研究计划有三个长期目标：1。开发一个全面的算法框架，用于鲁棒稀疏恢复和最佳压缩感知策略; 2.针对高维不确定性量化问题和基于稀疏恢复的偏微分方程数值方法设计鲁棒算法; 3.设计新一代强大的神经网络架构，并展示其在不确定性量化方面的潜力。这项工作包括压缩传感及其在磁共振成像（MRI）中的应用。它还涉及到不确定性量化和偏微分方程数值近似的新方法的发展，在计算科学和工程的基本工具，并具有高度的跨学科元素。它利用新的数学突破，通过与这些领域的专家合作，为航空航天工程和计算化学的应用提出有效的算法。

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.