A Duality-Based Optimization Approach for Model Adaptivity in Heterogeneous Multiscale Problems
Abstract
This paper introduces a novel framework for model adaptivity in the context of heterogeneous multiscale problems. The framework is based on the idea to interpret model adaptivity as a minimization problem of local error indicators that are derived in the general context of the dual weighted residual (DWR) method. Based on the optimization approach a postprocessing strategy is formulated that lifts the requirement of strict a priori knowledge about applicability and quality of effective models. This allows for the systematic, “goal-oriented” tuning of effective models} with respect to a quantity of interest. The framework is tested numerically on elliptic diffusion problems with different types of heterogeneous, random coefficients, as well as an advection-diffusion problem with a strong microscopic, random advection field.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
A. Abdulle and A. Nonnenmacher, A posteriori error estimate in quantities of interest for the finite element heterogeneous multiscale method, Numer. Methods Partial Differential Equations, 29 (2013), pp. 1629--1656.
2.
G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), pp. 1482--1518.
3.
W. Bangerth, D. Davydov, T. Heister, L. Heltai, G. Kanschat, M. Kronbichler, M. Maier, B. Turcksin, and D. Wells, The \textttdeal.II Library, Version 8.4, J. Numer. Math., 24 (2016), pp. 135--141.
4.
R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: Basic analysis and examples, East-West J. Numer. Math, 4 (1996), pp. 237--264.
5.
R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10 (2001), pp. 1--102.
6.
M. Braack and A. Ern, A posteriori control of modeling errors and discretization errors, Multiscale Model. Simul., 1 (2003), pp. 221--238.
7.
F. Brezzi, Interacting with the subgrid world, Numer. Anal., 1999, pp. 69--82.
8.
D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Ser. Math. Appl. 17, Oxford University Press, Oxford, 1999.
9.
W. E and B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci., 1 (2003), pp. 87--132.
10.
Y. Efendiev, V. Ginting, and T.Y. Hou, Multiscale finite element methods for nonlinear problems and their applications, Commun. Math. Sci., 2 (2004), pp. 553--589.
11.
P. Henning and M. Ohlberger, A note on homogenization of advection-diffusion problems with large expected drift, Z. Anal. Anwend., 30 (2011), pp. 319--339.
12.
P. Henning, M. Ohlberger, and B. Schweizer, An Adaptive Multiscale Finite Element Method, Technical report 05/12-N, FB 10, Universität Münster, 2012.
13.
T.J.R. Hughes, G.R. Feijóo, L. Mazzei, and J.-B. Quincy, The variational multiscale method---A paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166 (1998), pp. 3--24.
14.
M.G. Larson and A. M\aalqvist, Adaptive variational multiscale methods based on a posteriori error estimation, in Proceedings of the 4th European Congress on Computational Methods in Applied Sciences and Engineering, 2004.
15.
M.G. Larson and A. M\aalqvist, Adaptive variational multiscale methods based on a posteriori error estimation: Duality techniques for elliptic problems, in Multiscale Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng. 44, 2005, pp. 181--193.
16.
M. Maier, Duality-Based Adaptivity of Model and Discretization in Multiscale Finite-Element Methods, Dissertation, Ruprecht--Karls Universität Heidelberg, Germany, 2015, ŭlhttp://archiv.ub.uni-heidelberg.de/volltextserver/18889/.
17.
M. Maier and R. Rannacher, Duality-based adaptivity in heterogeneous multiscale finite element discretization, J. Numer. Math., 24 (2016).
18.
D.W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Indust. Appl. Math., 11 (1963), pp. 431--441.
19.
J.T. Oden and K.S. Vemaganti, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. Part I: Error estimates and adaptive algorithms, J. Comput. Phys., 164 (2000), pp. 22--47.
20.
J.T. Oden and K.S. Vemaganti, Adaptive modeling of composite structures: Modeling error estimation, Intl. J. Civil Struct. Engrg., 1 (2000b), pp. 1--16.
21.
A. Romkes and T.C. Moody, Local goal-oriented estimation of modeling error for multi-scale modeling of heterogeneous elastic materials, Int. J. Comput. Methods Eng. Sci. Mech., 8 (2007), pp. 201--209.
22.
H.-J. Vogel, QuantIm $4.01$b, C/C++ Library for Scientific Image Processing, 2008.
23.
J.E. Warren and H.S. Price, Flow in heterogeneous porous media, Soc. Petroleum Eng. J., 1 (1961), pp. 153--169.
Information & Authors
Information
Published In
Copyright
© 2018, Society for Industrial and Applied Mathematics.
History
Submitted: 29 November 2016
Accepted: 29 November 2017
Published online: 1 March 2018
Keywords
MSC codes
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- A posteriori single- and multi-goal error control and adaptivity for partial differential equationsError Control, Adaptive Discretizations, and Applications, Part 2 | 1 Jan 2024
- Neural network guided adjoint computations in dual weighted residual error estimationSN Applied Sciences, Vol. 4, No. 2 | 31 January 2022
- Reliability and Efficiency of DWR-Type A Posteriori Error Estimates with Smart Sensitivity Weight RecoveringComputational Methods in Applied Mathematics, Vol. 21, No. 2 | 9 January 2021
- Goal-oriented adaptive modeling of random heterogeneous media and model-based multilevel Monte Carlo methodsComputers & Mathematics with Applications, Vol. 78, No. 8 | 1 Oct 2019
- Adaptive multiscale predictive modellingActa Numerica, Vol. 27 | 4 May 2018
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].