Error estimation and adaptivity for PGD based on complementary solutions applied to a simple 1D problem

Author (s): Reis, J.; Moitinho de Almeida, J.P.; Díez, P. and Zlotnik, S.
Journal: Advanced Modeling and Simulation in Engineering Sciences

Volume: 7
Date: 2020

Abstract:
Reduced order methods are powerful tools for the design and analysis of sophisticated
systems, reducing computational costs and speeding up the development process.
Among these reduced order methods, the Proper Generalized Decomposition is a
well-established one, commonly used to deal with multi-dimensional problems that
often suffer from the curse of dimensionality. Although the PGD method has been
around for some time now, it still lacks mechanisms to assess the quality of the
solutions obtained. This paper explores the dual error analysis in the scope of the PGD,
using complementary solutions to compute error bounds and drive an adaptivity
process, applied to a simple 1D problem. The energy of the error obtained from the
dual analysis is used to determine the quality of the PGD approximations. We define a
new adaptivity indicator based on the energy of the error and use it to drive parametric
h- and p- adaptivity processes. The results are positive, with the indicator accurately
capturing the parameter that will lead to lowest errors.

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Bibtex:

@article{2020-AMSES-RMDZ,
        Author = {Reis, Jonatha
and Moitinho de Almeida, Jose Paulo
and D{\'i}ez, Pedro
and Zlotnik, Sergio},
        Title = {Error estimation and adaptivity for PGD based on complementary solutions applied to a simple 1D problem},
        Fjournal = {Advanced Modeling and Simulation in Engineering Sciences},
        Volume = {7},
        Number = {1},
        Pages = {752-776 },
        Year = {2020},
         Doi = {https://doi.org/10.1186/s40323-020-00180-3},
        }