Applied and Computational Mathematics

The Applied and Computational Mathematics theme develops and applies advanced mathematical techniques to tackle fundamental and applied challenges across science, engineering, and industry. Our research spans numerical analysis, fluid dynamics, mathematical biology, machine learning, and data science, leveraging cutting-edge computational tools and theoretical frameworks to address high-impact problems.

The theme fosters strong interdisciplinary and industrial partnerships, driving impactful research at the intersection of mathematics, computation, and real-world applications. We work closely with interdisciplinary collaborators and industry partners to drive innovation in key areas such as climate modeling, biomedical applications, artificial intelligence, and uncertainty quantification. We welcome collaborations with academia, industry, and research institutions to further explore innovative solutions.

We are part of the Centre for Doctoral Training (CDT) in Sensing, Processing, and AI for Defence and Security (SPADS) focused on generation-after-next technologies for information processing in defence and security, spanning the entire range from hardware development to algorithmic AI development.

Numerical analysis and scientific computing

We develop and analyze state-of-the-art numerical methods for solving partial and stochastic differential equations (PDEs and SPDEs), inverse problems, and uncertainty quantification. Our research includes:

• Finite element, spectral, and boundary element methods
• Adaptive and structure-preserving algorithms
• Fast solvers for large-scale simulations
• Multiscale modelling and high-performance computing

These methods are applied to fluid mechanics, electromagnetics, finance, and materials science.

Particle systems and fluid dynamics

Our work focuses on the mathematical modelling of complex fluids, kinetic theory, and multiscale systems, with applications in:

• Interacting particle systems and mean-field equations
• Traffic and crowd dynamics
• Plasma physics and rarefied gas dynamics
• Renewable energy and wave-structure interactions

We also develop surrogate modelling techniques to enable efficient simulations of multiphase and multicomponent flows.

Mathematical biology

We apply mathematical modelling to address challenges in biology and medicine, developing models that span multiple scales, from cellular to ecosystem levels. Key research areas include:

• Epidemiology: Spatiotemporal modelling of disease spread •
• Cancer and angiogenesis: Simulation of tumour growth and vascular networks
• Biomechanics: Wound healing, intercellular signalling, and transport processes
• Ecology: Population dynamics and plant-soil interactions

These models integrate PDEs, agent-based models, and data-driven methods for predictive insights.

Optimisation, machine learning and stochastic methods

We integrate computational mathematics with data-driven techniques to advance:

• Machine learning and physics-informed neural networks (PINNs)
• Stochastic and non-convex optimisation
• Inverse problems and imaging techniques
• High-dimensional statistical learning

Our methodologies enhance scientific computing, imaging, and data analysis across multiple domains.

Data science and dynamic systems

We develop mathematical frameworks for understanding complex dynamical systems and uncertainty quantification, with applications in:

• Bayesian inference and molecular dynamics
• Koopman operator theory for dynamical analysis
• Scalable computing for high-dimensional problems
• Rare-event simulation and adaptive sampling

These approaches provide robust tools for forecasting, control, and risk assessment in engineering and the sciences.