Thesis
M.Sc. Thesis: Bounds on Mixing of Passive Scalars
Formal thesis title: “Bounds on mixing of passive scalars advected by incompressible energy and enstrophy constrained flows and Anomalous dissipation”. The heading above provides a short descriptive title.
Subsequent Python implementation, numerical report & resolution study
The linked report and resolution study document the later computational work; they are distinct from the original submitted thesis.
A self-contained, open-source Python implementation of the Lin-Thiffeault-Doering (LTD) optimal-mixing scheme for passive scalar transport on the two-dimensional torus, ported from Gautam Iyer's original MATLAB code, with a pseudo-spectral solver and adaptive RK45 time integration. It reproduces the original thesis results exactly and extends them with a resolution study from N=32 to N=512. On fixed time windows the fitted mixing-rate exponent converges at approximately second order; its apparent resolution dependence under the original fitting protocol is primarily a moving-fit-window effect. The effective exponent varies with time and with the tested initial data, so the computations do not establish a single universal or asymptotic exponent.
- Resolution study (GitHub)
- Numerical report (PDF)
- Defence presentation (PDF)
- Code & numerical experiments (GitHub)
B.Sc. Thesis: Application of the Galerkin Finite Element Method to the One-Dimensional Stefan Problem
Applied Galerkin finite-element methods to a one-dimensional Stefan problem, formulating the weak problem and developing computational algorithms for tracking the moving phase boundary of this physically motivated free-boundary PDE.
Open-Source Research Software
Research implementations with public code, each addressing a specific mathematical or computational research question.
- neural-surrogate-burgers: POD-Galerkin ROM, DEIM hyper-reduction, an MLP surrogate, and a Fourier Neural Operator compared on viscous Burgers' equation.
- lorenz96-data-assimilation: stochastic Ensemble Kalman Filter with Gaspari–Cohn localisation for the chaotic Lorenz-96 system, verified against the exact Kalman filter on a linear-Gaussian problem.
- photoacoustic-reconstruction: sparse-view photoacoustic tomography, comparing calibrated time-reversal, a learned U-Net refinement and Tikhonov-regularised inversion, with a model-mismatch check.
- diff-pbr: differentiable physically-based renderer that optimises material maps against synthetic multi-view renderings, with a held-out evaluation and an identifiability study showing which parameters the views determine.
- fem-cylinder-flow: Taylor–Hood FEniCSx/PETSc finite-element solver, verified by manufactured solutions and benchmarked against the DFG Schäfer–Turek cylinder-wake test cases.
- fem-topology-optimization: verified 2D elasticity FEM driving a SIMP compliance-minimisation topology optimiser.
- navier-stokes-2d: verified Fourier pseudo-spectral 2D incompressible Navier–Stokes solver with a POD-Galerkin reduced-order model and DEIM hyper-reduction.
- nerf-tensorf: from-scratch implementations of NeRF and TensoRF, compared on held-out views of a synthetic scene.
- pinn-advection-diffusion: physics-informed neural network for the 1D advection-diffusion equation, trained via automatic differentiation with no labelled data.
- darcy-inverse-problem: deterministic adjoint-based and Bayesian (pCN/MCMC) inversion of a spatially varying Darcy permeability field.
- boiling-phasefield-3d: phase-field solver coupling Allen-Cahn, Navier-Stokes, and an energy equation for multiphase-flow PDEs; a dedicated 1D Stefan solver converges at first order against the analytical benchmark, and the 3D implementation is numerically verified but not yet physically validated.
- abdominal-ct-segmentation: 3D U-Net for liver segmentation from abdominal CT (Medical Segmentation Decathlon Task03), with a leakage-controlled full-volume evaluation pipeline implemented; independent evaluation is pending.
- sss-dipole: differentiable implementation of the Donner & Jensen (2005) dipole BSSRDF for subsurface light transport.