Error arising from the separation of input data in PGD: a priori estimates and implementation best practices
Author
Diez, P.; Zlotnik, S.; Huerta, A.
Type of activity
Presentation of work at congresses
Name of edition
3rd Workshop Reduced Basis, POD and PGD Model Reduction Techniques
Date of publication
2015
Presentation's date
2015-11-06
Book of congress proceedings
Book of Abstracts of the 3rd International Workshop on Reduced Basis, POD and PGD Model Reduction Techniques (ROM 2015). November 4-6, 2015, Cachan, France
First page
96
Last page
96
Abstract
Separated representations are commonly used to avoid the so-called curse of dimensionality, that is, an exponential increase in the number of degrees of freedom with the number of dimensions.
In order to use separated functions to approximate the solution of some Boundary Value Problem (BVP), the differential operators defining the BVP must be separable. This is the case, for example, of the Proper Generalised Decomposition (PGD). Separability of operators, in practice, requires separability of...
Separated representations are commonly used to avoid the so-called curse of dimensionality, that is, an exponential increase in the number of degrees of freedom with the number of dimensions.
In order to use separated functions to approximate the solution of some Boundary Value Problem (BVP), the differential operators defining the BVP must be separable. This is the case, for example, of the Proper Generalised Decomposition (PGD). Separability of operators, in practice, requires separability of all the input data. For example, the diffusivity function in the bilinear operator in a Poisson equation must be separable. In the case the original data is not separable, it has to be approximated by a separated approximation. This approximation can be obtained by means of a singular value decomposition, proper orthogonal decomposition, higher-order singular value decomposition, or other similar techniques.
Thus, in addition to the classical error sources in the PGD context, namely the PGD truncation and the underlying finite-element discretisation, the error arising from using a separable approximation of the input data is also affecting the PGD solution.
In this work we study the error in the PGD solution due to the separation of the input data. This error is mainly controlled by two factors: i) the mesh of the parametric dimensions and ii) the number of terms involved in the separation of the input data. In the case of a Poisson problem separated in spatial coordinates, the error due to the separation of the diffusivity behaves linearly with the error of the PGD solution.
This source of error is eventually limiting the convergence or the solution because the data resolution establishes a threshold in the accuracy of the final approximation