On a Goal-Oriented Version of the Proper Generalized Decomposition Method
In this paper, we introduce, analyze, and numerically illustrate a goal-oriented version of the Proper Generalized Decomposition method. The objective is to derive a reduced-order formulation such that the accuracy in given quantities of interest is increased when compared to a standard Proper Generalized Decomposition method. Traditional goal-oriented methods usually compute the solution of an adjoint problem following the calculation of the primal solution for error estimation and adaptation. In the present work, we propose to solve the adjoint problem first, based on a reduced approach, in order to extract estimates of the quantities of interest and use this information to constrain the reduced primal problem. The resulting reduced-order constrained solution is thus capable of delivering more accurate estimates of the quantities of interest. The performance of the proposed approach is illustrated on several numerical examples.
KeywordsModel reduction Quantities of interest Constrained problem Mixed formulation Penalization Lagrange multiplier
SP is grateful for the support by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. He also acknowledges the support by KAUST under Award Number OCRF-2014-CRG3-2281. Moreover, the authors gratefully acknowledge Olivier Le Maître for fruitful discussions on the subject.
- 17.Karush, W.: Minima of functions of several variables with inequalities as side constraints. Master’s thesis, Department of Mathematics, University of Chicago, Chicago, Illinois (1939)Google Scholar
- 20.Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–492. University of California Press, Berkeley (1951)Google Scholar
- 22.Ladevèze, P., Chamoin, L.: Toward guaranteed PGD-reduced models. Bytes and Science, pp. 143–154. CIMNE, Barcelona (2013)Google Scholar
- 28.Venturi, L., Torlo, D., Ballarin, F., Rozza, G.: A weighted POD method for elliptic PDEs with random inputs. ArXiv e-prints (2018)Google Scholar
- 29.Venturi, L., Torlo, D., Ballarin, F., Rozza, G.: Weighted reduced order methods for parametrized partial differential equations with random inputs. ArXiv e-prints (2018)Google Scholar