PhD Studentship: Sketchy Numerical Approximation of Wave Propagation

Numerical approximation methods like the finite element method (FEM) and the finite difference time domain (FDTD) schemes are at the forefront of engineering design and scientific computing. Ingrained in physical laws and engineering principles, these schemes have played a pivotal role in providing quantitative insights of complex phenomena including electrodynamics, heat transfer, and photonics, but due to their computational complexity their use was restricted exclusively to offline simulation. To serve the needs of modern engineering, like AI, IoT and digital twins, these schemes must be redesigned to become more agile and suitable for online and edge (in-situ) computing. This transformation requires (i) a capability to perform model prediction/calibration at arbitrary scale (up to a few billion parameters), (ii) a capability to operate with low storage/energy resources (embedded within battery operated processing units), (iii) extending their prediction skill by augmenting with non-physics (data driven) model extensions, and (iv) execution in almost real-time to inform decision making processes. To provide robust solutions to these challenges one must exploit the inherent structures of the FEM/FDTD schemes in order to dramatically reduce their computational complexity and allow speed ups of 1000x or more without a significant compromise (< 1%) in their accuracy or increase in prediction uncertainty. An approach to enable such a transformational research is to utilize the framework of randomised numerical linear algebra (aka randomized sketching) whilst leveraging algorithms from deep learning research (e.g. learned continuous operators). Randomised algebra provides a prudent way to “replace” expensive matrix-vector operations of extremely large dimension with simple-to-implement random projections or sample-based estimators that utilize only a tiny fraction of the available data/computational objects. The project will focus on the numerical solution of the wave equation that underpins a multitude of physical phenomena and consider its application in the context of modelling data from a sensor network. This research will lead to significant advances in the areas of edge computing, and machine learning for science/engineering.