Alex Bespalov

School of Mathematics

Research

             

My research specialisation is Numerical Analysis.
My research interests concern the design, analysis, and implementation of efficient numerical methods for partial differential and boundary integral equations. The special emphasis is on
     •  adaptive algorithms,
     •  high order methods (p- and hp-versions),
     •  problems with singularities,
     •  problems with random inputs,
     •  software development.
Areas of application include linear elasticity, electromagnetism, groundwater flow modelling, and quantum mechanics.

PhD projects

• Adaptivity and machine learning techniques for PDE problems with uncertain inputs
• Adaptive numerical algorithms for PDE problems with random input data

Funded research projects

• Adaptive multilevel stochastic collocation methods for uncertainty quantification (2021 – 2022; funded by EPSRC)
  This project aims to develop, analyse, implement and test a novel methodology for reliable error estimation and adaptive error control in the framework of the multilevel stochastic collocation finite element method for PDEs with random inputs.
• Numerical analysis of adaptive UQ algorithms for PDEs with random inputs (2017 – 2021; funded by EPSRC)
  This joint project with collaborators at The University of Manchester focused on the development of robust, accurate, and practical numerical methods for solving parameter-dependent PDEs stemming from uncertainty quantification models.
  Postdoc at Birmingham: Dr Feng Xu.

Current PhD students

• Jingye Li (since January 2023)
  Project title: "Machine-learning techniques for PDE-based reliability analysis problems"
  Jointly supervised with Prof Jinglai Li
  
• Andrey Savinov (since September 2022)
  Project title: "Design and analysis of adaptive stochastic collocation finite element methods"
  

Supervised PhD and Master students

• Thomas Round (PhD student, 2019 – 2024)
  PhD Thesis: "Goal-oriented adaptive algorithms for stochastic collocation finite element methods"
  
• Rawin Youngnoi (PhD student, 2016 – 2021)
  Jointly supervised with Dr Daniel Loghin
  PhD Thesis: "Preconditioning techniques for elliptic partial differential equations with random data"
• Leonardo Rocchi (PhD student, 2015 – 2019)
  PhD Thesis: "Adaptive algorithms for partial differential equations with parametric uncertainty"
• Adam Hayes (MSci project, 2021 – 2022 )
  MSci Thesis: "Solving differential equations using neural networks"
• Oliver Hedges (MSci project, 2020 – 2021)
  MSci Thesis: "Function approximation with feedforward neural networks"
• Ryan Sephton (MSc project, 2020)
  MSc Thesis: "Deep artificial neural networks for high-dimensional option pricing"
• Matthew Williams (MSci project, 2014 – 2015)
  MSci Thesis: "Stochastic sampling methods for PDEs with random input data"