Applied Mathematician
at University College London
About me
Matlab/Octave package for efficient evaluation of oscillatory integrals in the class \[ \int_{a}^bf(x)\exp(\mathrm{i}\omega g(x)) \mathrm{d} x \] where $g$ is a polynomial, $\omega>0$ and $f$ is entire. The endpoints $a,b$ can be complex, even infinite.
Julia package for the approximation of integrals and integral equations posed on self-similar fractals $\Gamma$. The main requirement is that $\Gamma$ can be described by a set of affine contraction maps $s_m$, such that \[ \Gamma = \bigcup_{m=1}^M s_m(\Gamma). \] Specifically, these maps are \[ s_m(x) := A_m\rho_mx+\delta_m, \] where $A_m$ is a rotation/reflection matrix, $\delta_m$ is a translation and $\rho_m<1$ is a contraction.
Matlab package for solving the Helmholtz BVP \begin{align} (\Delta+\omega^2)u=0\quad\text{ in }\mathbb{R}^2\setminus\Omega\\ u=0\quad\text{on }\partial\Omega \end{align} where $\Omega$ is polygonal, or a (bounded) screen $\Omega\subset\mathbb{R}.$ The method requirs $O(1)$ DOFs for convex polygons or screens as $\omega\to\infty$, and has $O(1)$ computational cost for multiple aligned screens.
A. M. Caetano, X. Claeys, S. N. Chandler-Wilde, A. Gibbs, D. P. Hewett, A. Moiola. Integral equation methods for acoustic scattering by fractals.Proc. R. Soc. B. (2025)arXiv preprint
A. Gibbs and S. Langdon. An efficient frequency-independent numerical method for computing the far-field pattern induced by polygonal obstacles. SISC (2024). , arXiv preprint
A. M. Caetano, S. N. Chandler-Wilde, A. Gibbs, D. P. Hewett, A. Moiola. A Hausdorff-measure boundary element method for acoustic scattering by fractal screens. Numer. Math. (2024). , arxiv preprint
A. Gibbs, D. P. Hewett, D. Huybrechs. Numerical evaluation of oscillatory integrals via automated steepest descent contour deformation. J. Comput. Phys., 501, (2024), arxiv preprint.
J. R. Ockendon, H. Ockendon, R. H. Tew, D. P. Hewett, A. Gibbs. A caustic terminating at an inflection point. Wave Motion, 125, (2024), arxiv preprint.
A. Gibbs, D. P. Hewett, B. Major. Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets. Numer. Algorithms (2024), arxiv preprint.
A. Gibbs, D. P. Hewett, A. Moiola. Numerical quadrature for singular integrals on fractals. Num. Alg. 1-54, (2022), arxiv preprint.
J. Bannister, A. Gibbs, D. P. Hewett. Acoustic scattering by impedance screens/cracks with fractal boundary: well-posedness analysis and boundary element approximation. Math. Mod. Meth. Appl. Sci. 32(2), (2022), arxiv preprint.
A. Gibbs, D. Hewett, D. Huybrechs, E. Parolin. Fast hybrid numerical-asymptotic boundary element methods for high frequency screen and aperture problems based on least-squares collocation. SN Partial Differ. Equ. Appl. 1(4):21, (2020), arxiv preprint.
A. Gibbs, S. Langdon, A. Moiola and S.N. Chandler-Wilde. A high frequency boundary element method for scattering by a class of multiple obstacles. IMA JNA 41(2), (2021), arxiv preprint.
A. Gibbs, S. Langdon, A. Moiola. Numerically stable computation of embedding formulae for scattering by polygons (technical report).
S.N. Chandler-Wilde, E.A. Spence, A. Gibbs, V. P. Smyshlyaev. High-frequency bounds for the Helmholtz equation under parabolic trapping and applications in numerical analysis.SIAM J. Math. Anal., 52(1), (2020),, arxiv preprint.
J. Bannister, A. Gibbs, D. P. Hewett, and V. Smyshlyaev. Volume integral equations on fractal domains and the Koch snowflake transmission problem. In 15th International Conference on Mathematical and Numerical Aspects of Wave Propagation, ENSTA Polytechnique Paris, (2022).
A. M. Caetano, S. N. Chandler-Wilde, A. Gibbs, D. P. Hewett, and A. Moiola. A Hausdorff-measure boundary element method for scattering by fractal screens. I: Numerical analysis. In 15th International Conference on Mathematical and Numerical Aspects of Wave Propagation, ENSTA Polytechnique Paris, (2022).