Publications

Preprints and In Review

librla: A library of randomized linear algebra routines

A. Gillman, Z. Gimbutas. (2026). "librla: A library of randomized linear algebra routines." arXiv preprint. arXiv:2607.20732. doi:10.48550/arXiv.2607.20732

Construction and profiling of a fast direct solver for distributed finite-element computations

P. Munch, A. Gillman. (2026). "Construction and profiling of a fast direct solver for distributed finite-element computations." Manuscript.

An efficient boundary integral equation solution technique for solving aperiodic scattering problems from two-dimensional, periodic boundaries

R. Fisher, F. Agocs, A. Gillman. (2025). "An efficient boundary integral equation solution technique for solving aperiodic scattering problems from two-dimensional, periodic boundaries." arXiv preprint. arXiv:2511.04424. doi:10.48550/arXiv.2511.04424

A fast direct solver for two dimensional quasi-periodic multilayered media scattering problems, Part II

Y. Zhang, A. Gillman. (2022). "A fast direct solver for two dimensional quasi-periodic multilayered media scattering problems, Part II." arXiv preprint. arXiv:2204.06629. doi:10.48550/arXiv.2204.06629

Journal Articles

Generating entangled steady states in multistable open quantum systems via initial state control

D. Fallas Padilla, R. Kaubruegger, A. Gillman, S. Becker, A. M. Rey. (2026). "Generating entangled steady states in multistable open quantum systems via initial state control." Physical Review Research. 8(1), 013244. doi:10.1103/s8kc-crf3

An Iterative Solver for the HPS Discretization Applied to Three Dimensional Helmholtz Problems

J. P. Lucero Lorca, N. Beams, D. Beecroft, A. Gillman. (2024). "An Iterative Solver for the HPS Discretization Applied to Three Dimensional Helmholtz Problems." SIAM Journal on Scientific Computing. 46(1), A80-A104. doi:10.1137/21m1463380

A fast direct solver for integral equations on locally refined boundary discretizations and its application to multiphase flow simulations

Y. Zhang, A. Gillman, S. Veerapaneni. (2022). "A fast direct solver for integral equations on locally refined boundary discretizations and its application to multiphase flow simulations." Advances in Computational Mathematics. 48(5), 63. doi:10.1007/s10444-022-09974-y

An alternative extended linear system for boundary value problems on locally perturbed geometries

Y. Zhang, A. Gillman. (2021). "An alternative extended linear system for boundary value problems on locally perturbed geometries." Journal of Computational Physics. 433, 110182. doi:10.1016/j.jcp.2021.110182

A fast direct solver for two dimensional quasi-periodic multilayered media scattering problems

Y. Zhang, A. Gillman. (2021). "A fast direct solver for two dimensional quasi-periodic multilayered media scattering problems." BIT Numerical Mathematics. 61(1), 141-171. doi:10.1007/s10543-020-00818-z

A parallel shared-memory implementation of a high-order accurate solution technique for variable coefficient Helmholtz problems

N. N. Beams, A. Gillman, R. J. Hewett. (2020). "A parallel shared-memory implementation of a high-order accurate solution technique for variable coefficient Helmholtz problems." Computers & Mathematics with Applications. 79(4), 996-1011. doi:10.1016/j.camwa.2019.08.019

An Adaptive High Order Direct Solution Technique for Elliptic Boundary Value Problems

P. Geldermans, A. Gillman. (2019). "An Adaptive High Order Direct Solution Technique for Elliptic Boundary Value Problems." SIAM Journal on Scientific Computing. 41(1), A292-A315. doi:10.1137/17m1156320

An accelerated Poisson solver based on multidomain spectral discretization

T. Babb, A. Gillman, S. Hao, P.-G. Martinsson. (2018). "An accelerated Poisson solver based on multidomain spectral discretization." BIT Numerical Mathematics. 58(4), 851-879. doi:10.1007/s10543-018-0714-0

A fast direct solver for boundary value problems on locally perturbed geometries

Y. Zhang, A. Gillman. (2018). "A fast direct solver for boundary value problems on locally perturbed geometries." Journal of Computational Physics. 356, 356-371. doi:10.1016/j.jcp.2017.12.013

An integral equation technique for scattering problems with mixed boundary conditions

A. Gillman. (2017). "An integral equation technique for scattering problems with mixed boundary conditions." Advances in Computational Mathematics. 43(2), 351-364. doi:10.1007/s10444-016-9488-6

High Resolution Inverse Scattering in Two Dimensions Using Recursive Linearization

C. Borges, A. Gillman, L. Greengard. (2017). "High Resolution Inverse Scattering in Two Dimensions Using Recursive Linearization." SIAM Journal on Imaging Sciences. 10(2), 641-664. doi:10.1137/16m1093562

A Fast Algorithm for Simulating Multiphase Flows Through Periodic Geometries of Arbitrary Shape

G. R. Marple, A. Barnett, A. Gillman, S. Veerapaneni. (2016). "A Fast Algorithm for Simulating Multiphase Flows Through Periodic Geometries of Arbitrary Shape." SIAM Journal on Scientific Computing. 38(5), B740-B772. doi:10.1137/15m1043066

A high-order accurate accelerated direct solver for acoustic scattering from surfaces

J. Bremer, A. Gillman, P.-G. Martinsson. (2015). "A high-order accurate accelerated direct solver for acoustic scattering from surfaces." BIT Numerical Mathematics. 55(2), 367-397. doi:10.1007/s10543-014-0508-y

A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media

A. Gillman, A. H. Barnett, P.-G. Martinsson. (2015). "A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media." BIT Numerical Mathematics. 55(1), 141-170. doi:10.1007/s10543-014-0499-8

An O(N) algorithm for constructing the solution operator to 2D elliptic boundary value problems in the absence of body loads

A. Gillman, P.-G. Martinsson. (2014). "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(4), 773-796. doi:10.1007/s10444-013-9326-z

A fast solver for Poisson problems on infinite regular lattices

A. Gillman, P. Martinsson. (2014). "A fast solver for Poisson problems on infinite regular lattices." Journal of Computational and Applied Mathematics. 258, 42-56. doi:10.1016/j.cam.2013.09.003

A simplified technique for the efficient and highly accurate discretization of boundary integral equations in 2D on domains with corners

A. Gillman, S. Hao, P. Martinsson. (2014). "A simplified technique for the efficient and highly accurate discretization of boundary integral equations in 2D on domains with corners." Journal of Computational Physics. 256, 214-219. doi:10.1016/j.jcp.2013.08.049

A Direct Solver with O(N) Complexity for Variable Coefficient Elliptic PDEs Discretized via a High-Order Composite Spectral Collocation Method

A. Gillman, P. G. Martinsson. (2014). "A Direct Solver with O(N) Complexity for Variable Coefficient Elliptic PDEs Discretized via a High-Order Composite Spectral Collocation Method." SIAM Journal on Scientific Computing. 36(4), A2023-A2046. doi:10.1137/130918988

A fast direct solver for quasi-periodic scattering problems

A. Gillman, A. Barnett. (2013). "A fast direct solver for quasi-periodic scattering problems." Journal of Computational Physics. 248, 309-322. doi:10.1016/j.jcp.2013.04.015

A direct solver with O(N) complexity for integral equations on one-dimensional domains

A. Gillman, P. M. Young, P.-G. Martinsson. (2012). "A direct solver with O(N) complexity for integral equations on one-dimensional domains." Frontiers of Mathematics in China. 7(2), 217-247. doi:10.1007/s11464-012-0188-3

Fast and accurate numerical methods for solving elliptic difference equations defined on lattices

A. Gillman, P. Martinsson. (2010). "Fast and accurate numerical methods for solving elliptic difference equations defined on lattices." Journal of Computational Physics. 229(24), 9026-9041. doi:10.1016/j.jcp.2010.07.024

A mixed hybrid formulation based on oscillated finite element polynomials for solving Helmholtz problems

A. Gillman, R. Djellouli, M. Amara. (2007). "A mixed hybrid formulation based on oscillated finite element polynomials for solving Helmholtz problems." Journal of Computational and Applied Mathematics. 204(2), 515-525. doi:10.1016/j.cam.2006.03.032

Book Chapters

Study of Discrete Scattering Operators for Some Linear Kinetic Models

Y. Chen, Z. Chen, Y. Cheng, A. Gillman, F. Li. (2016). "Study of Discrete Scattering Operators for Some Linear Kinetic Models." Topics in Numerical Partial Differential Equations and Scientific Computing, IMA Volumes in Mathematics and its Applications, vol. 160. pp. 99-136. doi:10.1007/978-1-4939-6399-7_5

Numerical Homogenization via Approximation of the Solution Operator

A. Gillman, P. Young, P.-G. Martinsson. (2012). "Numerical Homogenization via Approximation of the Solution Operator." Numerical Analysis of Multiscale Computations, Lecture Notes in Computational Science and Engineering, vol. 82. pp. 187-216. doi:10.1007/978-3-642-21943-6_9

Theses

Fast direct solvers for elliptic partial differential equations

A. Gillman. (2011). "Fast direct solvers for elliptic partial differential equations." Ph.D. dissertation, University of Colorado Boulder.

On the numerical performance of a mixed-hybrid type solution methodology for solving high-frequency Helmholtz problems

A. Gillman. (2006). "On the numerical performance of a mixed-hybrid type solution methodology for solving high-frequency Helmholtz problems." M.S. thesis, California State University, Northridge.

Software

librla: A randomized linear algebra library

A. Gillman, Z. Gimbutas. (2026). "librla: A randomized linear algebra library." Software.