A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems

Summary In this paper, we address the effective and accurate solution of problems with parameterized geometry. Considering the attractive framework of isogeometric analysis, which enables a natural and flexible link between computer‐aided design and simulation tools, the parameterization of the geometry is defined on the mapping from the isogeometric analysis parametric space to the physical space. From the subsequent multidimensional problem, model reduction based on the proper generalized decomposition technique with off‐line/online steps is introduced in order to describe the resulting manifold of parametric solutions with reduced CPU cost. Eventually, a posteriori estimation of various error sources inheriting from discretization and model reduction is performed in order to control the quality of the approximate solution, for any geometry, and feed a robust adaptive algorithm that optimizes the computational effort for prescribed accuracy. The overall approach thus constitutes an effective and reliable numerical tool for shape optimization analyses. Its performance is illustrated on several two‐ and three‐dimensional numerical experiments.

Section: Error Estimation and Adaptive Processmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Certified real‐time shape optimization using isogeometric analysis, PGD model reduction, and a posteriori error estimation

Chamoin

Thai

2019

“…In particular, the growing computation cost of the maps composition for sequential updating could be improved as mentioned in the work of Spantini and Bigoni . Consideration of the modeling error in the Bayesian formulation could also be investigated following the works on PGD error estimation . These aspects will be the topics of future research activities.…”

Section: Conclusion and Prospectsmentioning

confidence: 99%

“…36 Consideration of the modeling error in the Bayesian formulation could also be investigated following the works on PGD error estimation. 55,56 These aspects will be the topics of future research activities.…”

Section: Conclusion and Prospectsmentioning

confidence: 99%

Transport Map sampling with PGD model reduction for fast dynamical Bayesian data assimilation

Rubio

Louf

Chamoin

2019

Self Cite

Summary The motivation of this work is to address real‐time sequential inference of parameters with a full Bayesian formulation. First, the proper generalized decomposition (PGD) is used to reduce the computational evaluation of the posterior density in the online phase. Second, Transport Map sampling is used to build a deterministic coupling between a reference measure and the posterior measure. The determination of the transport maps involves the solution of a minimization problem. As the PGD model is quasi‐analytical and under a variable separation form, the use of gradient and Hessian information speeds up the minimization algorithm. Eventually, uncertainty quantification on outputs of interest of the model can be easily performed due to the global feature of the PGD solution over all coordinate domains. Numerical examples highlight the performance of the method.

Randomized residual‐based error estimators for the proper generalized decomposition approximation of parametrized problems

Smetana

Zahm

2020

This paper introduces a novel error estimator for the Proper Generalized Decomposition (PGD) approximation of parametrized equations. The estimator is intrinsically random: It builds on concentration inequalities of Gaussian maps and an adjoint problem with random right-hand side, which we approximate using the PGD. The effectivity of this randomized error estimator can be arbitrarily close to unity with high probability, allowing the estimation of the error with respect to any user-defined norm as well as the error in some quantity of interest. The performance of the error estimator is demonstrated and compared with some existing error estimators for the PGD for a parametrized time-harmonic elastodynamics problem and the parametrized equations of linear elasticity with a high-dimensional parameter space.in general only as costly as the computation of the PGD approximation or even significantly less expensive, which makes our error estimator strategy attractive from a computational viewpoint. To build this estimator, we first estimate the norm of the error with a Monte Carlo estimator using Gaussian random vectors whose covariance is chosen according to the desired error measure, e.g., user-defined norms or quantity of interest. Then, we introduce a dual problem with random right-hand side the solution of which allows us to rewrite the error estimator in terms of the residual of the original equation. Approximating the random dual problems with the PGD yields a fast-to-evaluate error estimator that has low marginal cost.Next, we relate our error estimator to other stopping criteria that have been proposed for the PGD solution. In [2] the authors suggest letting an error estimator in some linear QoI steer the enrichment of the PGD approximation. In order to estimate the error they employ an adjoint problem with the linear QoI as a right-hand side and approximate this adjoint problem with the PGD as well. It is important to note that, as observed in [2], this type of error estimator seems to require in general a more accurate solution of the dual than the primal problem. In contrast, the framework we present in this paper often allows using a dual PGD approximation with significantly less terms than the primal approximation. An adaptive strategy for the PGD approximation of the dual problem for the error estimator proposed in [2] was suggested in [23] in the context of computational homogenization. In [1] the approach proposed in [2] is extended to problems in nonlinear solids by considering a suitable linearization of the problem.An error estimator for the error between the exact solution of a parabolic PDE and the PGD approximation in a suitable norm is derived in [34] based on the constitutive relation error and the Prager-Synge theorem. The discretization error is thus also assessed, and the computational costs of the error estimator depend on the number of elements in the underlying spatial mesh. Relying on the same techniques an error estimator for the error between the exact solution of a parabolic PDE and a...