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Title: A Posteriori Error Analysis of Two-Stage Computation Methods with Application to Efficient Discretization and the Parareal Algorithm

We consider numerical methods for initial value problems that employ a two stage approach consisting of solution on a relatively coarse discretization followed by solution on a relatively fine discretization. Examples include adaptive error control, parallel-in-time solution schemes, and efficient solution of adjoint problems for computing a posteriori error estimates. We describe a general formulation of two stage computations then perform a general a posteriori error analysis based on computable residuals and solution of an adjoint problem. The analysis accommodates various variations in the two stage computation and in formulation of the adjoint problems. We apply the analysis to compute “dual-weighted” a posteriori error estimates, to develop novel algorithms for efficient solution that take into account cancellation of error, and to the Parareal Algorithm. We test the various results using several numerical examples.
Authors:
 [1] ;  [2] ;  [2] ;  [3] ;  [4]
  1. Univ. of New Mexico, Albuquerque, NM (United States)
  2. Colorado State Univ., Fort Collins, CO (United States)
  3. Univ. of Colorado, Denver, CO (United States)
  4. PD&L, DigitalGlobe, Inc., Longmont, CO (United States)
Publication Date:
Grant/Contract Number:
SC0005304; SC0009279; SC0009324; FG02-04ER25620; FG02-05ER25699; FC02-07ER54909; SC0001724
Type:
Accepted Manuscript
Journal Name:
SIAM Journal on Numerical Analysis
Additional Journal Information:
Journal Volume: 54; Journal Issue: 5; Journal ID: ISSN 0036-1429
Publisher:
Society for Industrial and Applied Mathematics
Research Org:
Colorado State Univ., Fort Collins, CO (United States)
Sponsoring Org:
USDOE Office of Science (SC)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING
OSTI Identifier:
1467608

Chaudhry, Jehanzeb Hameed, Estep, Don, Tavener, Simon, Carey, Varis, and Sandelin, Jeff. A Posteriori Error Analysis of Two-Stage Computation Methods with Application to Efficient Discretization and the Parareal Algorithm. United States: N. p., Web. doi:10.1137/16M1079014.
Chaudhry, Jehanzeb Hameed, Estep, Don, Tavener, Simon, Carey, Varis, & Sandelin, Jeff. A Posteriori Error Analysis of Two-Stage Computation Methods with Application to Efficient Discretization and the Parareal Algorithm. United States. doi:10.1137/16M1079014.
Chaudhry, Jehanzeb Hameed, Estep, Don, Tavener, Simon, Carey, Varis, and Sandelin, Jeff. 2016. "A Posteriori Error Analysis of Two-Stage Computation Methods with Application to Efficient Discretization and the Parareal Algorithm". United States. doi:10.1137/16M1079014. https://www.osti.gov/servlets/purl/1467608.
@article{osti_1467608,
title = {A Posteriori Error Analysis of Two-Stage Computation Methods with Application to Efficient Discretization and the Parareal Algorithm},
author = {Chaudhry, Jehanzeb Hameed and Estep, Don and Tavener, Simon and Carey, Varis and Sandelin, Jeff},
abstractNote = {We consider numerical methods for initial value problems that employ a two stage approach consisting of solution on a relatively coarse discretization followed by solution on a relatively fine discretization. Examples include adaptive error control, parallel-in-time solution schemes, and efficient solution of adjoint problems for computing a posteriori error estimates. We describe a general formulation of two stage computations then perform a general a posteriori error analysis based on computable residuals and solution of an adjoint problem. The analysis accommodates various variations in the two stage computation and in formulation of the adjoint problems. We apply the analysis to compute “dual-weighted” a posteriori error estimates, to develop novel algorithms for efficient solution that take into account cancellation of error, and to the Parareal Algorithm. We test the various results using several numerical examples.},
doi = {10.1137/16M1079014},
journal = {SIAM Journal on Numerical Analysis},
number = 5,
volume = 54,
place = {United States},
year = {2016},
month = {10}
}