Research and Software

My software projects can be found on my Github.

adaptive partition of unity approximations

For PUchebfun we built on the ideas of Chebfun, a package that provides a way to represent mathematical functions with high-accuracy approximations which can then be manipulated and analyzed numerically with a high level toolset. With the use of a partition of unity and Chebyshev polynomials it is possible to create an adaptive and highly accurate approximation for a given 2D or 3D function. We developed a tree based method to quickly construct the approximations, as well as developed fast methods for numerical computation.

partition of unity approximations for arbitrary domains

PUFunLS provides a way to construct spectral partition of unity approximations on arbitrary domains. The idea of embedding an irregular domain inside a rectangular one is common in computing. For spectrally accurate approximation this idea has seen a surge of recent interest going by the name of Fourier continuation or Fourier extension. We do something analogous by enclosing the domain with the smallest inscribing box and using a least squares approximation by tensor products of Chebyshev polynomials. We employ the same tree based method to adapt both to the features of the function as well as the geometry of the domain.

Nonlinear Preconditioning with Additive Schwarz

We created a new preconditioned nonlinear solver for PDEs that can be implemented in parallel. With our method, the nonlinear residual is preconditioned with a nonlinear additive Schwarz method. Nonlinear preconditioning can reduce the number of nonlinear iterations within Newton's method, as well as increase stability when faced with increased nonlinearity.

Numerical methods for blinking eye models

I am part of a research group that uses models to understanding the dynamic properties of the tear ocular film. With the use of lubrication theory it is possible to develop nonlinear PDE's that model the free surface of the tear film. We will develop a parallel Schwarz method to solve PDE's and apply the method to solve models with a blinking eye (i.e. a time dependent PDE with a moving boundary).

Publications

1. Aiton, K. W., & Driscoll, T. A. (2018). An adaptive partition of unity method for Chebyshev polynomial interpolation. SIAM Journal on Scientific Computing, 40(1), A251-A265.

2. Aiton, K. W., & Driscoll, T. A. (2018). An adaptive partition of unity method for multivariate Chebyshev polynomial approximations. arXiv preprint arXiv:1805.00423.

3. Aiton, K. W., & Driscoll, T. A. (2019). Preconditioned nonlinear iterations for overlapping Chebyshev discretizations with independent grids. arXiv preprint arXiv:1902.01310.