Research and Publications

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Research papers:

• P.G. Martinsson, Gregorio Quintana-Orti, Nathan Heaver, and R. van de Geijn,
  "Householder QR Factorization With Randomization for Column Pivoting."
  In review. Arxiv.org report #1512.02671.


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


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


• P.G. Martinsson and S. Voronin,
  "A CUR Factorization Algorithm based on the Interpolative Decomposition ."
  In review. Arxiv.org report #1412.8447.


• P.G. Martinsson,
  "Compressing rank-structured matrices via randomized sampling."
  To appear in SIAM Journal on Scientific Computation.
  Arxiv.org report #1503.07152.


• P.G. Martinsson and S. Voronin,
  "A randomized blocked algorithm for efficiently computing rank-revealing factorizations of matrices."
  To appear in SIAM Journal on Scientific Computation.
  Arxiv.org report #1503.07157.


• 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"
  Accepted for publication by IMA Journal of Numerical Analysis.
  arXiv.org report 1402.5168. Local copy.


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


• 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.


• 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, 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, 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, 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 approximation 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. 
   
• 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.
   
• Edo Liberty, Franco Woolfe, Per-Gunnar Martinsson, Vladimir Rokhlin, and Mark Tygert,
  "Randomized algorithms for the low-rank approximation of matrices".
  Proceedings of the National Academy of Sciences 2007 104: 20167-20172. 
   
• 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. 
   
• 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.
   
• 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 scattering problems involving elongated structures". 
  Journal of Computational Physics, 221, pp. 288 - 302, 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, M. Tygert
  "A fast algorithm for the inversion of general Toeplitz matrices".
  Computers and Mathematics with Applications, 50(5), pp 741 - 752, 2005. 
   
• 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. 
   
• 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 I. Babuska,
  "Mechanics of materials with periodic truss or frame micro-structures".
  Archives of Rational Mechanics and Analysis, 185(2), pp. 201-234, 2007. 
   
• 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 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. 
   
• 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. 
   
• 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: 

• 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".
  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.