High-dimensional problems are crucial in diverse scientific, engineering, and real-world applications. These problems appear in complex, large-scale scenarios. Examples include simulations involving random or stochastic partial differential equations related to climate patterns, weather forecasting, seismic wave propagation in layered media, and machine learning. Such high-dimensional problems are challenging due to the 'curse of dimensionality,' where computational costs increase exponentially with dimension.

This studentship aims to contribute to this field by exploring the challenge of approximating functions governed by stochastic dynamics. The focus of this project is the Dynamical Low-Rank Approximation, a method that captures high-dimensionality of the target quantity at a small cost. It aims not only to answer fundamental questions about Dynamical Low-Rank Approximation but also to position the method as an efficient and theoretically well-founded tool for uncertainty quantification, with broad applications in large-scale problems and data science.