Note: Dissertations and theses appear at the bottom of the page.

Google scholar page: here

Book: Fast Direct Solvers for Elliptic PDEs, SIAM, 2019.

Research papers:

• Y. Dong, P.G. Martinsson, Y. Nakatsukasa,
  "Efficient Bounds and Estimates for Canonical Angles in Randomized Subspace Approximations".
  SIAM Journal on Matrix Computations. Accepted for publication. Arxiv 2211.04676.


• A. Yesypenko, P.G. Martinsson,
  "SlabLU: A Two-Level Sparse Direct Solver for Elliptic PDEs".
  Advances in Computational Mathematics. Accepted for publication. Arxiv 2211.07572.


• K. Chen, D. Appelö, T. Babb, P.G. Martinsson,
  "Fast and high-order approximation of parabolic equations using hierarchical direct solvers and implicit Runge-Kutta methods".
  Communications on Applied Mathematics and Computation. Accepted for publication. Arxiv 2306.02526.


• T. Liang, C. Chen, P.G. Martinsson, G. Biros,
  "A distributed-memory parallel algorithm for discretized integral equations using Julia".
  IEEE IPDPS'24 Proceedings. Arxiv 2310.15458.


• Levitt, J. and Martinsson, P.G.,
  "Linear-Complexity Black-Box Randomized Compression of Rank-Structured Matrices".
  SIAM Journal on Scientific Computing, 46(3), pp. A1747-A1763, 2024. Arxiv 2205.02990.


• Levitt, J. and Martinsson, P.G.,
  "Randomized compression of rank-structured matrices accelerated with graph coloring".
  Journal of Computational and Applied Mathematics, 451(1), 2024. Arxiv 2205.03406.


• B. Wu, P.G. Martinsson,
  "A Unified Trapezoidal Quadrature Method for Singular and Hypersingular Boundary Integral Operators on Curved Surfaces".
  SIAM Journal on Numerical Analysis, 61(5), 2023.


• N. Heavner, P.G. Martinsson, G. Quintana-Orti,
  "Computing rank-revealing factorizations of matrices stored out-of-core".
  Concurrency and Computation: Practice and Experience, 35(22), 2023. Arxiv report 2002.06960.


• C. Chen, N. Heavner, A. Gopal, P.G. Martinsson,
  "Efficient algorithms for computing rank-revealing factorizations on a GPU".
  Numerical Linear Algebra with Applications, 30(6), 2023. Arxiv report 2106.13402.


• Y. Dong, P.G. Martinsson,
  "Simpler is better: a comparative study of randomized pivoting algorithms for CUR and interpolative decompositions".
  Advances in Computational Mathematics, 49(4), 2023. Arxiv report 2104.05877.


• C. Chen, P.G. Martinsson,
  "Solving linear systems on a GPU with hierarchically off-diagonal low-rank approximations".
  SC '22: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, Article No. 84, pages 1–15, 2022. Arxiv report 2208.06290.


• A. Gopal, P.G. Martinsson,
  "An accelerated, high-order accurate direct solver for the Lippmann--Schwinger equation for acoustic scattering in the plane" .
  Advances in Computational Mathematics, 48(4), pp. 1 - 31, 2022. Arxiv report 2007.12718.


• N. Heavner, F. Igual, G. Quintana-Orti, P.G. Martinsson,
  "Algorithm 1022: Efficient Algorithms for Computing a Rank-Revealing UTV Factorization on Parallel Computing Architectures".
  ACM TOMS, 48(2), pp. 1 - 42, 2022. Arxiv report 2104.05782.


• A.Yesypenko and P.G.Martinsson,
  GPU Optimizations for the Hierarchical Poincare-Steklov Scheme.
  DD '27: International Conference on Domain Decomposition Methods, Springer, pp. 519-528, 2022.


• A. Buluc, T. Kolda, S. Wild, et al
  "Randomized Algorithms for Scientific Computing (RASC)"
  DOE workshop report, arxiv 2104.11079, 2021.


• B. Wu and P.G. Martinsson,
  "Corrected trapezoidal rules for boundary integral equations in three dimensions"
  Numerische Mathematik, 149(4), 2021. Arxiv report 2007.02512.


• B. Wu and P.G. Martinsson,
  "Zeta correction: a new approach to constructing corrected trapezoidal quadrature rules for singular integral operators"
  Advances in Computational Mathematics, 47(45), 2021. Arxiv report 2007.13898.


• P.G. Martinsson and J. Tropp,
  "Randomized Numerical Linear Algebra: Foundations & Algorithms"
  Acta Numerica, 29, pp 403-572, 2020. Arxiv report 2002.01387.


• T. Babb, P.G. Martinsson, and D. Appelo,
  "HPS Accelerated Spectral Solvers for Time Dependent Problems: Part I, Algorithms"
  In collection Spectral and High Order Methods for Partial Differential Equations, book series Lecture notes in computational science and engineering, 134, pp. 131 - 141, Springer, 2020


• T. Babb, P.G. Martinsson, and D. Appelo,
  "HPS Accelerated Spectral Solvers for Time Dependent Problems: Part II, Numerical Experiments"
  In collection Spectral and High Order Methods for Partial Differential Equations, book series Lecture notes in computational science and engineering, 134, pp. 155 - 166, Springer, 2020


• P.G. Martinsson,
  "Randomized Projection Methods in Linear Algebra and Data Analysis."
  SIAM News, December 2018.


• P.G. Martinsson, G. Quintana-Orti, N. Heavner,
  "randUTV: A blocked randomized algorithm for computing a rank-revealing UTV factorization."
  ACM TOMS, 45(1), pp. 4:1--4:26, 2019. arxiv.org report #1703.00998.


• P.G. Martinsson,
  "Randomized methods for matrix computations."
  The Mathematics of Data, IAS/Park City Mathematics Series, 25(4), pp. 187 - 231, 2018. Arxiv.org report #1607.01649


• T. Babb, A. Gillman, S. Hao, P.G.Martinsson,
  "An accelerated Poisson solver based on a multidomain spectral discretization."
  BIT Numerical Analysis, 58(4), pp. 851-879, 2018. Arxiv.org report #1612.02736.


• P.G. Martinsson, G. Quintana-Orti, N. Heavner, and R. van de Geijn,
  "Householder QR Factorization With Randomization for Column Pivoting (HQRRP)."
  SIAM Journal on Scientific Computation, 39(2), pp. C96-C115, 2017. Arxiv.org report #1512.02671.


• P.G. Martinsson and S. Voronin,
  "Efficient algorithms for CUR and interpolative matrix decompositions".
  Advances in Computational Mathematics, 43(3), pp. 495-516, 2017. Arxiv.org report #1412.8447. Local copy.


• P.G. Martinsson,
  "Compressing rank-structured matrices via randomized sampling."
  SIAM Journal on Scientific Computation, 38(4), pp. A1959-A1986, 2016.
  Arxiv.org report #1503.07152. Local copy.


• M.A. Echeverri Bautista, M.A. Francavilla, P.G. Martinsson, F. Vipiana,
  O(N) Nested Skeletonization Scheme for the Analysis of Multiscale Structures Using the Method of Moments"
  IEEE Journal on Multiscale and Multiphysics Computational Techniques, 1, pp. 139--150, 2016. Local copy.


• P.G. Martinsson and S. Voronin,
  "A randomized blocked algorithm for efficiently computing rank-revealing factorizations of matrices."
  SIAM Journal on Scientific Computation., 38(5), S485 - S507, 2016. Local copy.
  Arxiv.org report #1503.07157.


• S. Hao and P.G. Martinsson,
  "A direct solver for elliptic PDEs in three dimensions based on hierarchical merging of Poincare-Steklov operators."
  Journal of Computational and Applied Mathematics, 308, pp. 419 - 434, 2016. Local copy.


• T. Haut, T. Babb, P.G. Martinsson, B. Wingate,
  "A high-order scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator"
  IMA Journal of Numerical Analysis. 36(2), pp. 688 - 716, 2016.
  arXiv.org report 1402.5168. Local copy.


• P.G. Martinsson,
  "Blocked rank-revealing QR factorizations: How randomized sampling can be used to avoid single-vector pivoting."
  Arxiv.org report #1505.08115, 2015.


• J. Bremer, A. Gillman, P.G. Martinsson,
  "A high-order accurate accelerated direct solver for acoustic scattering from surfaces"
  BIT Numerical mathematics, 55(2), pp. 367-397, 2015. arXiv.org report 1308.6643. Local copy.


• S. Hao, P.G. Martinsson, P. Young,
  "An efficient and highly accurate solver for multi-body acoustic scattering problems involving rotationally symmetric scatterers".
  CAMWA (Computers and Mathematics with Applications) 69(4), 2015, pp 304 - 318. arXiv.org report 1402.5168. Local copy.


• E. Corona, P.G. Martinsson, D. Zorin
  "An O(N) Direct Solver for Integral Equations in the Plane"
  Advances in Computational Harmonic Analysis, 38(2), pp. 284-317, 2015. arXiv.org report 1303.5466.


• A. Gillman, A. Barnett, P.G. Martinsson
  "A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media"
  BIT Numerical mathematics, 55(1), pp. 141 - 170, 2015. arXiv.org report 1308.5998. Local copy.


• A. Gillman, 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 on Scientific Computation, 36(4), pp. A2023-A2046, 2014. arXiv.org report 1307.2665. Local copy.


• A. Gillman, 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(4), pp 773 - 796, 2014. arXiv.org report 1302.5995.


• A. Gillman, S. Hao, 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(1), pp. 214--219, 2014. Local copy.


• A. Gillman, P.G. Martinsson
  "A fast solver for Poisson problems on infinite regular lattices"
  Journal of Computational and Applied Mathematics, 258(1), pp. 42-56, 2014. Local copy.


• S. Hao, A.H. Barnett, P.G. Martinsson, and P. Young,
  "High-order accurate Nystrom discretization of integral equations with weakly singular kernels on smooth curves in the plane"
  Advances in Computational Mathematics, 40(1), pp. 245-272, 2014. (arXiv.org report 1112.6262.) Local copy.


• P.G. Martinsson
  "A direct solver for variable coefficient elliptic PDEs discretized via a composite spectral collocation method"
  Journal of Computational Physics, 242(1), 2013, pp. 460-479. Local preprint.
  An early version of this manuscript was published on arxiv as 1206.4136, which in turn is based on 1101.3383.


• P.G. Martinsson
  "Fast Multipole Methods"
  Entry for Springer Encyclopedia of Applied and Computational Mathematics, 2013.


• P. Young, S. Hao, and P.G. Martinsson
  "A high-order Nystrom discretization scheme for boundary integral equations defined on rotationally symmetric surfaces"
  Journal of Computational Physics, 231(11), 2012, pp. 4142-4159.
  An early version appeared as arXiv.org report 1201.0077. Citation info: bibtex.


• A. Gillman, P. Young, P.G. Martinsson
  "A direct solver with O(N) complexity for integral equations on one-dimensional domains". Local pdf.
  Frontiers of Mathematics in China., 7(2), pp. 217--247, 2012. Citation info: bibtex.


• N. Halko, P.G. Martinsson, J. Tropp,
  "Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions."
  SIAM Review, 53(2), 2011, pp. 217-288. Tech report. Citation info: bibtex.


• P.G. Martinsson,
  "A fast randomized algorithm for computing a Hierarchically Semi-Separable representation of a matrix". Local pdf.
  SIAM Journal on Matrix Analysis and Applications., 32(4), 2011, pp. 1251-1274.
  An early version of this paper was published as arXiv report 0806.2339. A somewhat later tech report. Citation info: bibtex.


• 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, pp 187-216. Springer Verlag.


• N. Halko, P.G. Martinsson, Y. Shkolnisky, M. Tygert,
  "An Algorithm for the Principal Component Analysis of large Data Sets"
  SIAM Journal on Scientific Computation, 33(5), 2011, pp. 2580-2594. Citation info: bibtex.


• P. Young and P.G. Martinsson,
  "A high order Nystrom method for boundary integral equations on axisymmetric surfaces".
  arXiv.org report number1002.2001.


• P.G. Martinsson, V. Rokhlin, and M. Tygert,
  A randomized algorithm for the decomposition of matrices".
  Applied and Computational Harmonic Analysis, 30(1), pp. 47--68, 2011. Local copy.
  An early version was published as Yale CS research report YALEU/DCS/RR-1361.


• A. Gillman and P.G. Martinsson,
  "Fast and accurate numerical methods for solving elliptic difference equations defined on lattices".
  Journal of Computational Physics, 229(24), pp. 9026--9041, 2010.


• P.G. Martinsson, G.J.Rodin
  "Boundary Algebraic Equations for Lattice Problems"
  Proceedings of the Royal Society A, 465(2108), pp. 2489-2503, 2009. Correction.
  


• L. Greengard, D. Gueyffier, P.G. Martinsson, V. Rokhlin,
  "Fast direct solvers for integral equations in complex three-dimensional domains".
  Acta Numerica, 18, pp. 243-275, 2009.
  
  
• P.G. Martinsson,
  "A fast direct solver for a class of elliptic partial differential equations"
  Journal of Scientific Computation, pp. 316-330, 38(3), 2009.


• E. Liberty, F. Woolfe, P.G. Martinsson, V. Rokhlin, and M. Tygert,
  "Randomized algorithms for the low-rank approximation of matrices".
  Proceedings of the National Academy of Sciences 2007 104: 20167-20172.


• P.G. Martinsson and V. Rokhlin,
  "A fast direct solver for scattering problems involving elongated structures".
  Journal of Computational Physics, 221, pp. 288 - 302, 2007.
  


• P.G. Martinsson and I. Babuska,
  "Mechanics of materials with periodic truss or frame micro-structures".
  Archives of Rational Mechanics and Analysis, 185(2), pp. 201-234, 2007. Local copy.
  


• P.G. Martinsson and I. Babuska,
  "Homogenization of materials with periodic truss or frame micro-structures".
  Mathematical Models and Methods in Applied Sciences, 17(5), pp. 805-832, 2007.
  


• P.G. Martinsson and V. Rokhlin,
  "An accelerated kernel-independent fast multipole method in one dimension" .
  SIAM Journal of Scientific Computing, 29(3), pp. 1160-1178, 2007.


• P.G. Martinsson,
  "Rapid evaluation of electro-static interactions in multi-phase dielectric media".
  Journal of Computational Physics, 211, pp. 289 - 299, 2006.
  


• P.G. Martinsson, V. Rokhlin, and M. Tygert,
  "On interpolation and integration in finite-dimensional spaces of bounded functions".
  Communications in Applied Mathematics and Computational Science, 1, Jan. 2006. (Local copy.)


• P.G. Martinsson and V. Rokhlin
  "A fast direct solver for boundary integral equations in two dimensions".
  Journal of Computational Physics, 205(1), pp. 1-23, 2005.
  


• P.G. Martinsson, V. Rokhlin, M. Tygert
  "A fast algorithm for the inversion of general Toeplitz matrices".
  Computers and Mathematics with Applications, 50(5), pp 741 - 752, 2005.


• H. Cheng, Z. Gimbutas, P.G. Martinsson, V. Rokhlin,
  "On the compression of low rank matrices".
  SIAM Journal of Scientific Computing, 26(4), pp. 1389-1404, 2005. Clarifications / corrections.
  


• P.G. Martinsson and A.B. Movchan,
  "Vibrations of lattice structures and phononic bandgaps".
  The Quarterly Journal of Mechanics and Applied Mathematics , 56 , 2003, pp. 45-64.


• P.G. Martinsson and G. Rodin,
  "Boundary algebraic equations for lattice problems".
  IUTAM proceedings, Liverpool, 2002.
  


• P.G. Martinsson and G. Rodin,
  "Asymptotic expansions of lattice Green's functions".
  Proceedings of the Royal Society A , 458 , 2002, pp. 2609--2622.


• E. Cornea, R. Howard and P.G. Martinsson,
  "Solutions near singular points to the Eikonal and related first-order nonlinear partial differential equations in two dimensions".
  Differential and Integral Equations , 14 , 2001, pp. 1441-1468.
  

Dissertations:

• 2023 - Anna Yesypenko - "Randomized Algorithms for the Efficient Solution of Elliptic PDEs on Modern Architectures".


• 2023 - Yijun Dong - "Randomized Dimension Reduction with Statistical Guarantees".


• 2022 - James Levitt - "Building rank-revealing factorizations with randomization".


• 2019 - Nathan Heavner - "Building rank-revealing factorizations with randomization".


• 2019 - Tracy Babb - "Accelerated time-stepping of parabolic and hyperbolic PDEs via fast direct solvers for elliptic problems".


• 2015 - Sijia Hao - "Numerical methods for solving linear elliptic PDEs: Direct solvers and high order accurate discretizations".


• 2012 - Nathan Halko - "Randomized methods for computing low-rank approximations of matrices".


• 2011 - Adrianna Gillman - "Fast direct solvers for elliptic partial differential equations".


• 2011 - Patrick Young - "Numerical Techniques for the Solution of Partial Differential and Integral Equations on Irregular Domains with Applications to Problems in Electrowetting".

My own theses:

• P.G. Martinsson, (under the supervision of Professors Ivo Babuska and Gregory Rodin)
  "Fast Multiscale Methods for Lattice Equations".
  Doctoral Thesis, Computational and Applied Mathematics, University of Texas at Austin, June 2002.
  
  
• P.G. Martinsson, (under the supervision of Professor Vidar Thomee)
  "Discretisation of Certain Evolution Equations with Memory using Convolution Quadrature". (pdf).
  Licentiate Thesis, Dept. of Mathematics, Chalmers University of Technology, 1998.
  
  
• P.G. Martinsson, (under the supervision of Dr William McLean)
  "Separation of Variables Techniques in Particle Simulations and Integral Equations".
  Honours Thesis, Dept. of Mathematics, University of New South Wales, 1995.