Stochastic Gradient Descent in Banach Spaces

逆问题是指从给定的观测数据中重建物理现象的原因。它们在医学成像、信号处理和机器学习等科学和工程领域有着广泛的应用。迭代方法是一种特别强大的范例，用于解决各种各样的反问题。它们通常是通过定义一个目标函数来提出的，该目标函数包含有关数据保真度的信息和有关所寻求的数量的假设，然后通过迭代过程将其最小化。数学在分析反问题和相应的算法中起着至关重要的作用。数据采集和精度的最新进展导致了大量问题的数据集越来越大，包括计算机和正电子发射断层扫描。这种数据量的增加对传统重建方法提出了巨大的计算挑战，传统重建方法通常需要在每次迭代中使用所有观测数据。随机迭代方法通过在每次迭代中仅使用一小部分观测值来解决这个计算瓶颈。由此产生的方法是高度可扩展的，并已成功地部署在广泛的问题。然而，迄今为止，随机方法的使用仅限于一组限制性的几何假设，需要希尔伯特或欧几里得空间。拟议的奖学金旨在解决这些问题，通过发展随机梯度方法解决反问题提出的Banach空间。非希尔伯特空间的使用在逆问题和机器学习社区中越来越受到关注。Banach空间提供了更丰富的几何结构，是偏微分方程和医学层析成像中许多问题的自然问题域。此外，Banach空间范数有利于保持重要性质，例如稀疏性。该奖学金将把现代优化方法引入经典Banach空间理论，其成功完成将为逆问题和机器学习创造新的研究机会。

Inverse problems are concerned with the reconstruction of the causes of a physical phenomena from given observational data. They have wide applications in many problems in science and engineering such as medical imaging, signal processing, and machine learning. Iterative methods are a particularly powerful paradigm for solving a wide variety of inverse problems. They are often posed by defining an objective function that contains information about data fidelity and assumptions about the sought quantity, which is then minimised through an iterative process. Mathematics has played a critical role in analysing inverse problems and corresponding algorithms.Recent advances in data acquisition and precision have resulted in datasets of increasing size for a vast number of problems, including computed and positron emission tomography. This increase in data size poses significant computational challenges for traditional reconstruction methods, which typically require the use of all the observational data in each iteration. Stochastic iterative methods address this computational bottleneck by using only a small subset of observation in each iteration. The resulting methods are highly scalable, and have been successfully deployed in a wide range of problems. However, the use of stochastic methods has thus far been limited to a restrictive set of geometric assumptions, requiring Hilbert or Euclidean spaces. The proposed fellowship aims to address these issues by developing stochastic gradient methods for solving inverse problems posed in Banach spaces. The use of non-Hilbert spaces is gaining increased attention within inverse problems and machine learning communities. Banach spaces offer much richer geometric structures, and are a natural problem domain for many problems in partial differential equation and medical tomography. Moreover, Banach-space norms are advantageous for preservation of important properties, such as sparsity. This fellowship will introduce modern optimisation methods into classical Banach space theory and its successful completion will create novel research opportunities for inverse problems and machine learning.