Encapsulated PGD Algebraic Toolbox Operating with High-Dimensional Data

Dı́ez, Pedro; Zlotnik, Sergio; García, A.; Huerta, Antonio

doi:10.1007/s11831-019-09378-0

Cited by 26 publications

(55 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The variables

\bar{u}

and

\bar{s}

are the displacement and stress vectors describing the spatial modes. Matrices

U (μ)

and

S (μ)

correspond to sets of nodal functions of each parameter, which are obtained using the algebraic PGD approach described in Reference 16. Note that

\overset{bold-italic^}{v}

has a similar approximation, which will not be used in this article and, therefore, is not presented here.…”

Section: Parametric Problem and Matrix Separationmentioning

confidence: 99%

Error estimation for proper generalized decomposition solutions: A dual approach

Reis

Almeida

Dı́ez

et al. 2020

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

Summary The proper generalized decomposition is a well‐established reduced order method, used to efficiently obtain approximate solutions of multi‐dimensional problems in a procedure that controls the effects of the “curse of dimensionality.” The question of assessing the quality of the solutions obtained and adapting the approximations assumed, for example, the finite element meshes used, so that the best result is obtained at minimal cost, remains a relevant challenge. This article deals with finite element solutions for solid mechanics problems, using the error obtained from a dual analysis, the difference between complementary solutions, to bound the error in the solutions and to drive an optimal adaptivity process, which obtains meshes with errors significantly lower than those obtained using a uniform refinement.

show abstract

“…The variables

\bar{u}

and

\bar{s}

are the displacement and stress vectors describing the spatial modes. Matrices

U (μ)

and

S (μ)

correspond to sets of nodal functions of each parameter, which are obtained using the algebraic PGD approach described in Reference 16. Note that

\overset{bold-italic^}{v}

has a similar approximation, which will not be used in this article and, therefore, is not presented here.…”

Section: Parametric Problem and Matrix Separationmentioning

confidence: 99%

Error estimation for proper generalized decomposition solutions: A dual approach

Reis

Almeida

Dı́ez

et al. 2020

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

“…We can then average Equations (26) and (27) to write the error of the output associated with an averaged solution:…”

Section: Local Outputs and Their Boundsmentioning

confidence: 99%

“…One way to perform compression is by doing a least-square projection of the solution into the same approximation space using the same PGD procedure. 27,30,31 Although this does not impose orthogonality between the terms, in practice it usually reduces the number of terms of a separated function.…”

Section: Large Number Of Modesmentioning

confidence: 99%

See 1 more Smart Citation

Error estimation for proper generalized decomposition solutions: Dual analysis and adaptivity for quantities of interest

Reis

Almeida

Dı́ez

et al. 2020

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

When designing a structure or engineering a component, it is crucial to use methods that provide fast and reliable solutions, so that a large number of design options can be assessed. In this context, the proper generalized decomposition (PGD) can be a powerful tool, as it provides solutions to parametric problems, without being affected by the “curse of dimensionality.” Assessing the accuracy of the solutions obtained with the PGD is still a relevant challenge, particularly when seeking quantities of interest with guaranteed bounds. In this work, we compute compatible and equilibrated PGD solutions and use them in a dual analysis to obtain quantities of interest and their bounds, which are guaranteed. We also use these complementary solutions to compute an error indicator, which is used to drive a mesh adaptivity process, oriented for a quantity of interest. The corresponding solutions have errors that are much lower than those obtained using a uniform refinement or an indicator based on the global error, as the proposed approach focuses on minimizing the error in the quantity of interest.

show abstract

“…Several reduced order method (ROM) techniques were developed in the last decades, all with the common goal of finding low-order models, described by a reduced order basis, which are able to capture the essential behaviour of a complex system. In this work, the encapsulated proper generalized decomposition (Encapsulated-PGD) toolbox, based on the PGD Least-Squares approximation [1] is proposed. This tool is able to provide, with only one offline computation, an explicit separable solution in terms of an a-priori unknown number of parametric and mechanic modes or snapshots.…”

Section: Introductionmentioning

confidence: 99%

Nonintrusive Proper Generalized Decomposition Method for the Design Optimization of a Car

Cavaliere¹,

Zlotnik²,

Sevilla³

et al. 2021

10th International Conference on Adaptative Modeling and Simulation

Self Cite

View full text Add to dashboard Cite

Encapsulated PGD Algebraic Toolbox Operating with High-Dimensional Data

Cited by 26 publications

References 18 publications

Error estimation for proper generalized decomposition solutions: A dual approach

Error estimation for proper generalized decomposition solutions: A dual approach

Error estimation for proper generalized decomposition solutions: Dual analysis and adaptivity for quantities of interest

Nonintrusive Proper Generalized Decomposition Method for the Design Optimization of a Car

Contact Info

Product

Resources

About