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Improving the Finite Difference Approximation in the Jacobian-Free Newton–Krylov Method

Van Heyningen, Robert LU (2016) In Bachelor's Theses in Mathematical Sciences FMNL01 20161
Mathematics (Faculty of Engineering)
Abstract
The Jacobian-free Newton–Krylov (JFNK) method is designed to solve a linear system of equations that appears in Newton’s method. It uses the generalized minimal residual (GMRES) method to solve the linear system and a simple function to approximate the matrix-vector multiplications required in GMRES. An advantage of GMRES is the ability to check the residual of a potential solution without doing any extra computations. A previous bachelor’s thesis discovered that the residual seen by the algorithm and the actual residual differ across many test cases. This puts the validity of the solution into question and makes it difficult to implement any sort of error-checking in the algorithm. The purpose of this thesis is to investigate the... (More)
The Jacobian-free Newton–Krylov (JFNK) method is designed to solve a linear system of equations that appears in Newton’s method. It uses the generalized minimal residual (GMRES) method to solve the linear system and a simple function to approximate the matrix-vector multiplications required in GMRES. An advantage of GMRES is the ability to check the residual of a potential solution without doing any extra computations. A previous bachelor’s thesis discovered that the residual seen by the algorithm and the actual residual differ across many test cases. This puts the validity of the solution into question and makes it difficult to implement any sort of error-checking in the algorithm. The purpose of this thesis is to investigate the discrepancy between these residuals. Tests were run on the various problems and although they were unable to determine a concrete explanation for the behavior of the residuals, they provided valuable insight into the potential causes to investigate in the future. (Less)
Please use this url to cite or link to this publication:
author
Van Heyningen, Robert LU
supervisor
organization
course
FMNL01 20161
year
type
M2 - Bachelor Degree
subject
keywords
GMRES, Jacobian-free, Fluid
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUTFNA-4003-2016
ISSN
1654-6229
other publication id
2016:K9
language
English
id
8877342
date added to LUP
2016-08-23 11:22:59
date last changed
2017-05-01 04:09:08
@misc{8877342,
  abstract     = {The Jacobian-free Newton–Krylov (JFNK) method is designed to solve a linear system of equations that appears in Newton’s method. It uses the generalized minimal residual (GMRES) method to solve the linear system and a simple function to approximate the matrix-vector multiplications required in GMRES. An advantage of GMRES is the ability to check the residual of a potential solution without doing any extra computations. A previous bachelor’s thesis discovered that the residual seen by the algorithm and the actual residual differ across many test cases. This puts the validity of the solution into question and makes it difficult to implement any sort of error-checking in the algorithm. The purpose of this thesis is to investigate the discrepancy between these residuals. Tests were run on the various problems and although they were unable to determine a concrete explanation for the behavior of the residuals, they provided valuable insight into the potential causes to investigate in the future.},
  author       = {Van Heyningen, Robert},
  issn         = {1654-6229},
  keyword      = {GMRES,Jacobian-free,Fluid},
  language     = {eng},
  note         = {Student Paper},
  series       = {Bachelor's Theses in Mathematical Sciences},
  title        = {Improving the Finite Difference Approximation in the Jacobian-Free Newton–Krylov Method},
  year         = {2016},
}