MRes Thesis Project Imperial College London & University of Reading
Author: Joshua Prettyman
Supervisors: Hilary Weller & Phil Browne (Department of Meteorology, University of Reading)
This project develops numerical methods for adaptive mesh generation using solutions to the Monge-Ampere equation from optimal transport theory. The work is motivated by challenges in computational meteorology, where adaptive meshes can dramatically improve efficiency by concentrating computational resources where they are most needed.
Traditional uniform grids waste computational power in regions with smooth solutions while potentially failing to resolve important small-scale features. This research explores r-adaptive methods (mesh redistribution) that move existing mesh points to achieve better resolution without changing the mesh topology.
Left: Random points distributed according to a density function. Right: Optimized mesh after applying Lloyd's algorithm, demonstrating the principle of adaptive mesh concentration.
The main contribution of this thesis is the development of a new Alternative Linearisation (AL) method for solving the Monge-Ampere equation that:
- Requires no tuning parameters - Unlike existing Fixed-Point (FP) and Parabolic Monge-Ampere (PMA) methods which require careful selection of a relaxation parameter gamma
- Provides stable convergence across different monitor functions without adjustment
- Achieves comparable or better performance than existing methods
The thesis provides a systematic comparison of three numerical approaches:
| Method | Relaxation Parameter | Stability | Speed |
|---|---|---|---|
| PMA (Parabolic Monge-Ampere) | Required (gamma) | Good with tuning | Moderate |
| FP (Fixed-Point) | Required (gamma) | Sensitive to monitor function | Fast when stable |
| AL (Alternative Linearisation) | Not required | Robust | Consistent |
The mesh redistribution problem seeks a map F: Omega_C -> Omega_P that transforms a uniform computational mesh into one that equidistributes a monitor function M(x):
M(x) |I + H(phi)| = c (constant)
where H(phi) is the Hessian of the mesh potential. This is the Monge-Ampere equation - a fully nonlinear elliptic PDE.
The AL method introduces a novel linearisation:
|I + H(phi)| = |I + H(phi_bar)| + |H(psi)| + div(A * grad(psi))
where A is a matrix dependent on the previous iterate, avoiding the large nonlinear terms that can destabilize other methods.
MRes/
├── Thesis/
│ ├── thesis.tex # Main LaTeX source
│ ├── thesis.pdf # Compiled thesis
│ ├── thesis.bib # Bibliography
│ └── images/ # Figures and visualizations
│ ├── Bell/ # Bell-shaped monitor function results
│ ├── Ring/ # Ring-shaped monitor function results
│ └── *.pdf # Analysis plots
├── Code/
│ ├── contour.py # Monitor function visualization
│ ├── residplot.py # Residual and convergence analysis
│ └── plots/ # Generated figures
│ ├── Final_lot/ # Final experimental results
│ └── python/ # Analysis plots
├── CourseNotes/ # Supporting course materials
│ ├── VectorCalculus/
│ ├── DynamicalSystems/
│ └── AdvancedPDEs/
└── Presentation/ # Project presentation slides
The experiments use two test monitor functions:
- Ring function: Concentrates mesh points in a ring around the origin
- Bell function: Concentrates mesh points at the center (more challenging)
CPU time comparison showing how convergence speed varies with the relaxation parameter (gamma) across different methods. Lower gamma values are faster but risk instability.
Key findings:
- The AL method converges in ~50 iterations for both monitor functions without parameter tuning
- FP method fails on the bell function without careful gamma selection
- PMA method requires different gamma values for different monitor functions
- The 1/d power law (where d is dimension) significantly improves stability
This work has applications in:
- Numerical Weather Prediction - Resolving weather fronts and convective systems
- Climate Modeling - Adaptive resolution for multi-scale phenomena
- Computational Fluid Dynamics - Tracking shocks and boundary layers
- Astrophysics - Resolving gravitational collapse and accretion
The implementation uses:
- OpenFOAM (finite volume discretization)
- Python (NumPy, Matplotlib, SciPy) for analysis and visualization
Key references from the thesis:
- Budd, C.J., Cullen, M.J.P., Walsh, E.J. (2013). Monge-Ampere based moving mesh methods for numerical weather prediction
- Browne, P.A., Budd, C.J., Piccolo, C., Cullen, M. (2014). Fast three dimensional r-adaptive mesh redistribution
- Weller, H., Browne, P., Budd, C., Cullen, M. (2015). Mesh adaptation on the sphere using optimal transport
This thesis and associated code were produced as part of the Mathematics of Planet Earth CDT programme.
Thesis submitted September 2015 as part of the requirements for the MRes in Mathematics of Planet Earth