# Master's Theses offered by James Jackaman

*Språk:* Veiledningen vil foregå på engelsk.

## Information

I am a numerical analysis and work primarily on approximations of partial differential equations which preserve some interesting qualitative features. Here are some of my current research interests:

- Finite element methods (with a focus on studying nonlinear PDEs).
- Geometric numerical integration.
- Convolutional neural networks as projection operators.
- "Structure preserving" machine learning.

If you want to read more about my interests, check out my website.

As a postdoctoral fellow, any student working with me will be co-supervised by a permanent member of faculty in the numerics group. The member of faculty depends on the project, but would likely be either Elena Celledoni or Brynjulf Owren.

I am happy for students to come to me to discuss project topics, and also offer the project described below.

## Physics constrained iterative solvers

The (numerical) approximation of a linear PDE ultimately boils down to solving a linear system `Ax=b`

, but to solve the PDE accurately, one requires the dimension of the matrix `A`

to be very large and solving the system exactly becomes prohibitively expensive. This motivates the use of iterative linear solvers, but there is a problem: We cannot expect the physical properties of a PDE to be preserved before our iterative solver converges. This problem can be solved through constraining the iterative linear solver by the desired physics, as described in this paper.

This project will generalise the results from the linear case and develop a conservative nonlinear iterative solver for discretisations of nonlinear PDEs.

### Requirements for the project

- Basic knowledge of finite elements
- Basic knowledge of linear and nonlinear iterative solvers.
- Experience with coding, preferably in Python.

## Other project ideas

- Parallel-in-time implementation of space-time finite elements for fluid dynamics