Publications

In Preparation or Submitted

• Y. Zhang and A. Gillman. " An adaptive discretization technique for boundary integral equations on the plane," In preparation.

• P. Geldermans and A. Gillman. "A robust method for evaluating the discrete prolate spheroidal wave functions and the corresponding eigenvalues," In preparation.

• JPL Lorca, N Beams, D Beecroft, A Gillman, "An iterative solver for the HPS discretization applied to three dimensional Helmholtz problems," In review.

• Y. Zhang and A. Gillman. "Part II: A fast direct solver for quasi-periodic scattering in multi-layered medium," In review.

Publications

• Y. Zhang, A. Gillman and S. Veerapaneni, "A fast direct solver for integral equations on locally refined boundary discretizations and its application to multiphase flow simulations," Advances in Computational Mathematics, 48, Article number: 63, 2022.

• Y. Zhang and A. Gillman, "An alternative extended linear system for boundary value problems on locally perturbed geometries," 433(15), pp. 110182, 2021.

• Y. Zhang and A. Gillman. "A fast direct solver for quasi-periodic scattering in multi-layered medium," BIT Numerical Mathematics, 61, pp. 141–171, 2021.

• N. Beams, A. Gillman, and R. Hewett. "A parallel implementation of a high order accurate solution technique for variable coefficient Helmholtz problems," Computers and Mathematics with Applications, 79(14), pp. 996-1011, 2020.

• P. Geldermans and A. Gillman. "An adaptive high order direct solution technique for elliptic boundary value problems," SIAM Journal of Scientific Computing , 41(1), pp. A292-A315, 2019.

• T. Babb, A. Gillman, S. Hao, and P.G. Martinsson. "An accelerated Poisson solver based on multidomain spectral discretization," BIT Numerical Analysis, 58(4), 851-879, 2018.

• Y. Zhang, and A. Gillman. "A fast direct solver for boundary value problems on locally perturbed geometries," Journal of Computational Physics, 356(1), pp. 356-371, 2018.

• C. Borges, A. Gillman and L. Greengard. "High resolution inverse scattering in two dimensions using recursive linearization," SIAM Journal on Imaging Sciences, 10(2), pp. 641–664, 2017.

• A. Gillman. "An integral equation technique for scattering problems with mixed boundary conditions" Advances in Computational Mathematics, 43(2), pp. 351–364, 2017.

• G. Marple, A. Barnett, A. Gillman and S. Veerapaneni. "A Fast Algorithm for Simulating Multiphase Flows Through Periodic Geometries of Arbitrary Shape" SIAM Journal of Scientific Computing, 38(5), pp. B740-B772, 2016.

• Y. Chen, Z. Chen, Y. Cheng, A. Gillman and F. Li. "Study of Discrete Scattering Operators for Some Linear Kinetic Models" Springer IMA Volume, the Proceedings for WhAM! A Research Collaboration Workshop for Women in Numerical Analysis and Scientific Computing, IMA, Minneapolis, MN, August 11-15, 2014. Accepted, 2015.

• J. Bremer, A. Gillman, and P.G. Martinsson. "A high-order accurate accelerated direct solver for acoustic scattering from surfaces" BIT Numerical Mathematics, 55, pp. 367--397, 2015, doi:10.1007/s10543-014-0508-y.

• A. Gillman, A. Barnett, and P.G. Martinsson. "A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media" BIT Numerical Mathematics, 55, pp. 141--170, 2015, doi:10.1007/s10543-014-0499-8.

• A. Gillman and P.G. Martinsson. "A direct solver with O(N) complexity for variable coefficient elliptic PDEs discretized via a high-order composite spectral collocation method'' SIAM Journal of Scientific Computing, 36, pp. A2023-A2046, 2014.

• A. Gillman, and P.G. Martinsson. "An O(N) algorithm for constructing the solution operator to 2D elliptic boundary value problems in the absence of body loads" Advances in Computational Mathematics 40, pp. 773-796, 2014, doi:10.1007/s10444-013-9326-z.

• A. Gillman and P.G. Martinsson. "A fast solver for Poisson problems on infinite regular lattices" Journal of Computational and Applied Mathematics, 258, pp. 42-56, 2014.

• A. Gillman, S. Hao, and P.G. Martinsson. "A simplified technique for the efficient and high-order accurate discretization of boundary integral equations in 2D on domains with corners" Journal of Computational Physics, 256, pp. 214–219, 2014.

• A. Gillman, and A. Barnett. "A fast direct solver for quasi-periodic scattering problems" Journal of Computational Physics, 248, pp. 309–322, 2013.

• A. Gillman, P. Young, and P.G. Martinsson. "A direct solver with O(N) complexity for integral equations on one-dimensional domains" Frontiers of Mathematics in China, 7, no. 2, pp. 217-247, 2012. Correction

• A. Gillman, P. Young, P.G. Martinsson. "Numerical homogenization via approximation of the solution operator" In B. Engquist, O. Runborg, R. Tsai, editors, Numerical Analysis of Multiscale Computations, volume 82 of Lecture Notes in Computational Science and Engineering, Heidelberg, 2011. Springer Verlag.

• A. Gillman and P.G. Martinsson. "Fast and accurate numerical methods for solving elliptic difference equations defined on lattices" Journal of Computational Physics, 229, no. 24, pp. 9026–9041, 2010.

• A. Gillman, R. Djellouli, and M. Amara. "A Mixed Hybrid Formulation Based on Oscillated Finite Element Polynomials for Solving Helmholtz Problems" Journal of Computational and Applied Mathematics 204, pp. 515-525, 2007.

Theses

• A. Gillman, Fast direct solvers for elliptic partial differential equations, PhD Thesis, CU Boulder, 2011.

• A. Gillman, On the numerical performance of a mixed-hybrid type solution methodology for solving high-frequency Helmholtz problems, Masters Thesis, CSUN, 2006.