Data assimilation (DA) techniques allow users to estimate the state of a dynamical system at a given time by combining information from measurements with numerical models. A key application of DA is within numerical weather prediction, and elsewhere in geosciences, where estimates of the current state of the atmosphere and ocean are used as initial conditions for e.g. weather forecasts. DA also forms an important research field within applied mathematics. One important subproblem within variational DA involves the solution of extremely high-dimensional linear systems within a very limited computational budget/wallclock time, typically using iterative methods. These extreme time constraints also lead to an unusual mathematical setting compared to general iterative methods - as very few iterations are performed, improvement in early iterations is vital.    

This project will focus on a particular DA method (weak-constraint 4D-Var) which can be reformulated in a number of ways to reveal different aspects of the underlying structure of the problem. In this project you will:

• Develop new theory for different problem formulations e.g. bound eigenvalues of linear systems, prove convergence results for novel iterative methods.
• Use this improved theoretical understanding to design novel and more reliable implementations of iterative solvers for problems coming from data assimilation.
• Develop practical methods and adapt existing algorithms, before testing them numerically. The supervisor can provide an existing code testbed, and you could also work on community code or write your own test problems. 

The balance of these subprojects will depend on the interest of the student.

Possible specific research directions include:

• Designing new preconditioners for varied iterative solvers and problem formulations.
•  Extending results from standard Krylov solvers to matrix-oriented solvers
•  Reformulation of the problem in a tensor framework to exploit modern multilinear algebra approaches.

It is not expected that you know much about data assimilation yet, but you should have an interest in learning more. There may be opportunities for you to attend external data assimilation training, and engage with researchers at meteorological agencies (e.g. the Met Office, Cerfacs). You will join the Computational Science group (https://casa.win.tue.nl/home/computational-science/), and be an active member of the Centre for Analysis, Scientific Computing and Applications (https://casa.win.tue.nl/home/).

• A master's degree (or an equivalent university degree) in Mathematics or a numerate degree.
• A strong background and interest in (numerical) linear algebra. Some knowledge of Krylov methods is advantageous but not required.
• Programming experience in either Matlab or Python.  
• Ability to work in an interdisciplinary team and interested in collaborating with industrial partners.
• Initiative and motivation to work independently.
• Interest in developing your teaching skills and coaching students.
• Fluent in spoken and written English (C1 level).

A meaningful job in a dynamic and ambitious university, in an interdisciplinary setting and within an international network. You will work on a beautiful, green campus within walking distance of the central train station. In addition, we offer you:

• Full-time employment for four years, with an intermediate evaluation (go/no-go) after nine months. You will spend 10% of your employment on teaching tasks.
• Salary and benefits (such as a pension scheme, paid pregnancy and maternity leave, partially paid parental leave) in accordance with the Collective Labour Agreement for Dutch Universities, PhD scale (min. €2,770 max. €3,539).
• A year-end bonus of 8.3% and annual vacation pay of 8%.
• High-quality training programs and other support to grow into a self-aware, autonomous scientific researcher. At TU/e we challenge you to take charge of your own learning process.
• An excellent technical infrastructure, on-campus children's day care and sports facilities.
• An allowance for commuting, working from home and internet costs.
• A Staff Immigration Team and a tax compensation scheme (the 30% facility) for international candidates.