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On roundoff error growth in elliptic problems

Babuška, Ivo and Söderlind, Gustaf LU (2018) In ACM Transactions on Mathematical Software 44(3).
Abstract

Large-scale linear systems arise in finite-difference and finite-element discretizations of elliptic problems. With increasing computer performance, ever larger systems are solved using direct methods. How large can such systems be without roundoff compromising accuracy? Here we model roundoff dynamics in standard LU and LDLT decompositions with respect to problem size N. For the one-dimensional (1D) Poisson equation with Dirichlet boundary conditions on an equidistant grid, we show that the relative error in the factorized matrix grows like O(ϵN) if roundoffs are modeled as independent, expectation zero random variables. With bias, the growth rate changes to O(ϵN). Subsequent back substitution results in typical error... (More)

Large-scale linear systems arise in finite-difference and finite-element discretizations of elliptic problems. With increasing computer performance, ever larger systems are solved using direct methods. How large can such systems be without roundoff compromising accuracy? Here we model roundoff dynamics in standard LU and LDLT decompositions with respect to problem size N. For the one-dimensional (1D) Poisson equation with Dirichlet boundary conditions on an equidistant grid, we show that the relative error in the factorized matrix grows like O(ϵN) if roundoffs are modeled as independent, expectation zero random variables. With bias, the growth rate changes to O(ϵN). Subsequent back substitution results in typical error growths of O(ϵNN) and O(ϵN2), respectively. Error growth is governed by the dynamics of the computational process and by the structure of the boundary conditions rather than by the condition number. Computational results are demonstrated in several examples, including a few fourth-order 1D problems and second-order 2D problems, showing that error accumulation depends strongly on the solution method. Thus, the same LU solver may exhibit different growth rates for the same 2D Poisson problem, depending on whether the five-point or nine-point FDM operator is used.

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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Roundoff error analysis
in
ACM Transactions on Mathematical Software
volume
44
issue
3
publisher
ACM
external identifiers
  • scopus:85047007397
ISSN
0098-3500
DOI
10.1145/3134444
language
English
LU publication?
yes
id
2b635acf-2f34-4709-8ef4-8eab8ad8b61c
date added to LUP
2018-05-30 15:16:06
date last changed
2018-05-30 15:16:06
@article{2b635acf-2f34-4709-8ef4-8eab8ad8b61c,
  abstract     = {<p>Large-scale linear systems arise in finite-difference and finite-element discretizations of elliptic problems. With increasing computer performance, ever larger systems are solved using direct methods. How large can such systems be without roundoff compromising accuracy? Here we model roundoff dynamics in standard LU and LDL<sup>T</sup> decompositions with respect to problem size N. For the one-dimensional (1D) Poisson equation with Dirichlet boundary conditions on an equidistant grid, we show that the relative error in the factorized matrix grows like O(ϵN) if roundoffs are modeled as independent, expectation zero random variables. With bias, the growth rate changes to O(ϵN). Subsequent back substitution results in typical error growths of O(ϵNN) and O(ϵN<sup>2</sup>), respectively. Error growth is governed by the dynamics of the computational process and by the structure of the boundary conditions rather than by the condition number. Computational results are demonstrated in several examples, including a few fourth-order 1D problems and second-order 2D problems, showing that error accumulation depends strongly on the solution method. Thus, the same LU solver may exhibit different growth rates for the same 2D Poisson problem, depending on whether the five-point or nine-point FDM operator is used.</p>},
  articleno    = {33},
  author       = {Babuška, Ivo and Söderlind, Gustaf},
  issn         = {0098-3500},
  keyword      = {Roundoff error analysis},
  language     = {eng},
  month        = {01},
  number       = {3},
  publisher    = {ACM},
  series       = {ACM Transactions on Mathematical Software},
  title        = {On roundoff error growth in elliptic problems},
  url          = {http://dx.doi.org/10.1145/3134444},
  volume       = {44},
  year         = {2018},
}