## Stable Parallel Algorithms For Two-Point Boundary Value Problems (1992)

Venue: | SIAM J. Sci. Statist. Comput |

Citations: | 25 - 1 self |

### BibTeX

@ARTICLE{Wright92stableparallel,

author = {Stephen J. Wright},

title = {Stable Parallel Algorithms For Two-Point Boundary Value Problems},

journal = {SIAM J. Sci. Statist. Comput},

year = {1992},

volume = {13},

pages = {742--764}

}

### Years of Citing Articles

### OpenURL

### Abstract

. Some of the most widely used algorithms for two-point boundaryvalue ODEs, namely finite difference and collocation methods and standard multiple shooting, proceed by setting up and solving a structured system of linear equations. It is well known that the linear system can be set up efficiently in parallel; we show here that a structured orthogonal factorization technique can be used to solve this system, and hence the overall problem, in an efficient, parallel, and stable way. Key words. parallel algorithms, two-point boundaryvalue problems, error analysis and stability AMS(MOS) subject classifications. 65F05, 65G05, 65L10, 65L20, 65W05 1. Introduction. Many numerical algorithms for solving the linear two-point boundary value problem y 0 = M (t)y + q(t); t 2 [a; b]; y 2 R n ; (1) B a y(a) + B b y(b) = d; (2) have been proposed and studied over the last 30 years. Many of these methods require a structured linear algebraic system (for example, a block-tridiagonal or staircase ...

### Citations

1171 |
The Algebraic Eigenvalue Problem
- Wilkinson
- 1965
(Show Context)
Citation Context ...ount the structure of the matrix. In addition, to allay any possible concerns about instability at the level of the O(u 2 ) terms, we have removed these terms using the style of analysis in Wilkinson =-=[20]-=-. The relevant results are stated and proved in Appendix B. Here, we summarize the analysis in the following Lemma and Theorem: Lemma 4.1. Let s = (s T 1 ; s T 2 ; : : : ; s T k+1 ) T denote the true ... |

304 | Numerical Solution of Boundary Value Problems for Ordinary Dierential Equations (Prenctice-Hall, Englewood Clis - Ascher, Mattheij, et al. - 1988 |

53 |
An estimate for the condition number of a matrix
- Cline, Moler, et al.
- 1979
(Show Context)
Citation Context ... estimate of the conditioning of the discrete system. Such an estimate can be obtained, simultaneously with the factorization and solution process, by adapting the procedure described in Cline et al. =-=[6]-=- to our situation. We aim to compute an estimate of the quantitys= kAk1 kR \Gamma1 k1 ; where A is one of the coefficient matrices from (4),(7),(5),(8), and R is the upper triangular factor produced b... |

21 |
An adaptive finite difference solver for nonlinear two-point boundary problems with mild boundary layers
- Lentini, Peyrera
- 1977
(Show Context)
Citation Context ...Efficient factorization schemes, based on structured Gaussian elimination, have been implemented and are widely available (see x2, and Varah [19], Diaz, Fairweather and Keast [7], Lentini and Pereyra =-=[13]-=-, and Keller [9].) During the last 10 years, the question of stability of algorithms for (1),(2) has received a great deal of attention (see, for example, Mattheij [15].) It has been recognized that i... |

16 |
FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination
- Diaz, Fairweather, et al.
- 1983
(Show Context)
Citation Context ...t matrix of this system. Efficient factorization schemes, based on structured Gaussian elimination, have been implemented and are widely available (see x2, and Varah [19], Diaz, Fairweather and Keast =-=[7]-=-, Lentini and Pereyra [13], and Keller [9].) During the last 10 years, the question of stability of algorithms for (1),(2) has received a great deal of attention (see, for example, Mattheij [15].) It ... |

11 |
On parallel methods for boundary value ODEs
- Ascher, Chan
- 1991
(Show Context)
Citation Context ...ices. Ascher and Mattheij [2] develop a "theoretical multiple shooting" framework in which they show how boundary values for the sub-BVPs should be chosen to ensure well-conditioning. Ascher=-= and Chan [1]-=- suggest how to implement this in a parallel environment. 8 STEPHEN J. WRIGHT Another possibility, which leads to near-perfect speedup on two processors (but cannot be generalized to a higher order of... |

9 |
Solving almost block diagonal systems on parallel computers
- Paprzycki, Gladwell
- 1991
(Show Context)
Citation Context ...e, and discard it in favor of a more stable method if the kZ i k are too large. In many applications (such as the one described in [21]) the lack of stability is not a problem. Paprzycki and Gladwell =-=[17]-=- describe a partitioned elimination algorithm in which (8) is torn into P submatrices, each of which has the same form as the original ADP , and alternate row and column elimination is applied to each... |

9 |
Iterated Deferred Correction for Nonlinear Boundary Value Problems
- Pereyra
- 1968
(Show Context)
Citation Context ...2 6 6 6 6 6 6 6 4 d a f 1 f 2 . . . f k d b 3 7 7 7 7 7 7 7 5 : (8) The accuracy of finite difference schemes is often enhanced by the use of deferred correction techniques (see, for example, Pereyra =-=[18]-=-). As has been observed, the two algorithms are closely related, in the sense that for a reasonable choice of the approximation (6), \GammaC \Gamma1 i A i should be close to Y i (t i+1 ). Hence, the c... |

8 |
Adaptation of a two-point boundary value problem solver to a vector-multiprocessor environment
- Wright, Pereyra
- 1990
(Show Context)
Citation Context ..., have recently been proposed for (4),(5),(7),(8). In general, they suffer either from poor stability properties or from limitations in the amount of parallelism which is possible. Wright and Pereyra =-=[21]-=- describe variants of a block factorization algorithm applied to (7). In the most highly vectorized variant, a factorization of the form 2 6 6 6 6 6 4 ~ A 1 ~ A 2 . . . ~ A k Z 1 Z 2 : : : Z k ~ A k+1... |

7 |
Alternate row and column elimination for solving certain linear systems
- Varah
- 1976
(Show Context)
Citation Context ...he factorization of the coefficient matrix of this system. Efficient factorization schemes, based on structured Gaussian elimination, have been implemented and are widely available (see x2, and Varah =-=[19]-=-, Diaz, Fairweather and Keast [7], Lentini and Pereyra [13], and Keller [9].) During the last 10 years, the question of stability of algorithms for (1),(2) has received a great deal of attention (see,... |

6 |
The close relationships between methods for solving two-point boundary-value problems
- Lentini, Osborne, et al.
- 1985
(Show Context)
Citation Context ...of the problem (1),(2) by a boundson its Green's function (see Ascher, Mattheij, and Russell [3, x3.2]), it has been shown by Osborne [16] (and also by Mattheij [14] and Lentini, Osborne, and Russell =-=[12]-=-) that the inverse of AS from (4) satisfies the bound kA \Gamma1 S k1sk: Hence if we define fl by fl = 1 + max i=1;:::;k kY i (t i+1 )k 1 4 STEPHEN J. WRIGHT and assume that B a and B b are scaled suc... |

3 |
Bounds for rounding errors in the Gaussian elimination for band systems
- Bohte
- 1975
(Show Context)
Citation Context ...ithms are applied to the matrices in (4),(5),(7),(8). In the partially separated cases (5),(8), similar results can be proved, without referring to the dichotomy at all, by using the results of Bohte =-=[4]-=-. Bohte shows that for banded linear systems, the bound on element growth in partial pivoting algorithms is exponential only in the bandwidth. In the simplest case of Gaussian elimination with row par... |

3 |
Parallel solution of special large block tridiagonal systems: TPBVP, manuscript
- Lentini
- 1989
(Show Context)
Citation Context ...multaneously from both ends of the matrix (either ADP or ASP ). When the factorizations meet in the center, a small reduced system is formed and factored. This is analogous to the approach of Lentini =-=[11]-=-. Finally, we mention the approach of Ascher and Chan [1], who form the normal equations for (5) and (8), and factorize the resulting symmetric, positive definite, block-tridiagonal system using cycli... |

2 |
General Framework, Stability and Error Analysis for Numerical Stiff Boundary Value Methods, Numerische 22
- Ascher, Mattheij
- 1988
(Show Context)
Citation Context ...o ensure well-conditioning of the sub-BVPs. In a linear algebra sense, well-conditioning of the whole matrix ADP does not guarantee well-conditioning of each of the P submatrices. Ascher and Mattheij =-=[2] develop a-=- "theoretical multiple shooting" framework in which they show how boundary values for the sub-BVPs should be chosen to ensure well-conditioning. Ascher and Chan [1] suggest how to implement ... |

2 |
A high order method for stiff boundary value problems with turning points
- Brown, Lorenz
- 1987
(Show Context)
Citation Context ..., DECOMP is known to be unstable. To test the relative speed of the linear solvers, two further problems from the literature were used in addition to problem 1. These were Problem 2 (Brown and Lorenz =-=[5]-=-) a = \Gamma1, b = 1, n = 4, \Gammaffly 00 \Gamma t 2 y 0 + t 2 z 0 + z = ffl 2 cos t + 1 2 t sin t; fflz 00 = z y(\Gamma1) = \Gamma1 y(1) = e \Gamma2= p ffl z(\Gamma1) = 1 z(1) = e \Gamma2= p ffl : P... |

2 |
Accurate difference methods for two-point boundary value problems
- Keller
- 1974
(Show Context)
Citation Context ...zation schemes, based on structured Gaussian elimination, have been implemented and are widely available (see x2, and Varah [19], Diaz, Fairweather and Keast [7], Lentini and Pereyra [13], and Keller =-=[9]-=-.) During the last 10 years, the question of stability of algorithms for (1),(2) has received a great deal of attention (see, for example, Mattheij [15].) It has been recognized that in a well-conditi... |

2 |
The conditioning of linear boundary value problems
- Mattheij
- 1982
(Show Context)
Citation Context ...small. If we quantify the conditioning of the problem (1),(2) by a boundson its Green's function (see Ascher, Mattheij, and Russell [3, x3.2]), it has been shown by Osborne [16] (and also by Mattheij =-=[14]-=- and Lentini, Osborne, and Russell [12]) that the inverse of AS from (4) satisfies the bound kA \Gamma1 S k1sk: Hence if we define fl by fl = 1 + max i=1;:::;k kY i (t i+1 )k 1 4 STEPHEN J. WRIGHT and... |

1 |
Aspects of the numerical solution of boundary value problems with separated boundary conditions
- Osborne
- 1978
(Show Context)
Citation Context ...e similar when the h i are small. If we quantify the conditioning of the problem (1),(2) by a boundson its Green's function (see Ascher, Mattheij, and Russell [3, x3.2]), it has been shown by Osborne =-=[16]-=- (and also by Mattheij [14] and Lentini, Osborne, and Russell [12]) that the inverse of AS from (4) satisfies the bound kA \Gamma1 S k1sk: Hence if we define fl by fl = 1 + max i=1;:::;k kY i (t i+1 )... |