Error Estimation and Adaptivity for PGD based on Complementary Solutions
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. 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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Due to technical limitations, full-text HTML conversion of this manuscript could not be completed. However, the manuscript can be downloaded and accessed as a PDF.
Posted 24 May, 2020
On 30 Oct, 2020
On 20 Aug, 2020
Received 17 Jul, 2020
On 05 Jun, 2020
Received 27 May, 2020
On 22 May, 2020
Invitations sent on 21 May, 2020
On 20 May, 2020
On 19 May, 2020
On 19 May, 2020
On 14 May, 2020
Error Estimation and Adaptivity for PGD based on Complementary Solutions
Posted 24 May, 2020
On 30 Oct, 2020
On 20 Aug, 2020
Received 17 Jul, 2020
On 05 Jun, 2020
Received 27 May, 2020
On 22 May, 2020
Invitations sent on 21 May, 2020
On 20 May, 2020
On 19 May, 2020
On 19 May, 2020
On 14 May, 2020
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. 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.
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Due to technical limitations, full-text HTML conversion of this manuscript could not be completed. However, the manuscript can be downloaded and accessed as a PDF.