Adaptivity and machine learning techniques for PDE problems with uncertain inputs

PhD Studentship in Numerical Analysis and Scientific Computing

The project concerns numerical solution of partial differential equations (PDEs) with parametric uncertainty in input data. It will focus on developing novel algorithms for efficient numerical solution of such problems. Research work on the project will involve rigorous mathematical analysis and implementation of the developed algorithms as well as extensive numerical experimentation.

PDEs are key tools in the mathematical modelling of processes in science and engineering. In practical PDE-based models, precise knowledge of inputs (e.g., material properties, initial conditions, external forces) may not be available, or there might be uncertainty about the inputs. Thus, PDEs with uncertain input data arise in many scientific and industrial contexts when it is essential to accurately model complex processes and perform a reliable risk assessment. One of the major challenges in numerical solution of PDEs with uncertain inputs is the high dimensionality of the resulting discretisations. Therefore, the development of robust and effective numerical methods which make best use of available computational resources is a very active research area.

While adaptive strategies provide an effective mechanism for building approximations, accelerating convergence, and resolving local features of solutions, modern machine-learning techniques (in particular, those based on neural network architectures) have demonstrated impressive results for a wide range of data-intensive applications. Theoretical understanding of the remarkable performance of machine-learning methodologies is an emerging topic in mathematical research at the interface of approximation theory, analysis, probability and statistics. Moreover, there is a growing interest in employing these methodologies for the efficient solution of challenging problems, such as high-dimensional parametric PDEs and PDEs with uncertain inputs.

This project will aim at developing mathematically justified algorithms that combine state-of-the-art adaptive techniques for building approximations and resolving local features of solutions with machine learning techniques for efficient sampling.

The project will provide training in numerical analysis, scientific computing, and uncertainty quantification techniques, thus equipping the student with highly desirable skills for working in either industry or academia.

Entry requirements:
We are looking for an enthusiastic and highly-motivated graduate with
- a 1st class degree in Mathematics or a closely related discipline with strong mathematical component (Master’s level or equivalent);
- a solid background in numerical analysis of PDEs;
- excellent programming skills;
- good communication skills (oral and written).
Good knowledge of probability theory, statistics as well as basic understanding of machine learning techniques and software will be advantageous.

The application procedure and the deadlines for scholarship applications are advertised at https://www.birmingham.ac.uk/schools/mathematics/phd/phd-programme.aspx.

Informal inquiries should be directed to Alex Bespalov, e-mail: [Email Address Removed]