Duality-based adaptivity in finite element discretization of heterogeneous multiscale problems

Maier, Matthias; Rannacher, Rolf

doi:10.1515/jnma-2014-0074

Cited by 6 publications

(3 citation statements)

References 36 publications

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“…Braack and Ern (2003) presented methods for estimation and control of both modelling and discretization errors. In more recent work, Maier and Rannacher (2016, 2018) described a post-processing procedure for selecting optimal models for estimating error in quantities of interest. Extensions of the adaptive modelling approaches in Oden and Vemaganti (2000) and Vemaganti and Oden (2001) to stochastic systems encountered in the modelling of random heterogeneous materials are discussed in Section 9.5.…”

Section: Adaptive Control Of Multiscale Modelsmentioning

confidence: 99%

Adaptive multiscale predictive modelling

Oden

2018

Acta Numerica

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The use of computational models and simulations to predict events that take place in our physical universe, or to predict the behaviour of engineered systems, has significantly advanced the pace of scientific discovery and the creation of new technologies for the benefit of humankind over recent decades, at least up to a point. That ‘point’ in recent history occurred around the time that the scientific community began to realize that true predictive science must deal with many formidable obstacles, including the determination of the reliability of the models in the presence of many uncertainties. To develop meaningful predictions one needs relevant data, itself possessing uncertainty due to experimental noise; in addition, one must determine model parameters, and concomitantly, there is the overriding need to select and validate models given the data and the goals of the simulation.This article provides a broad overview of predictive computational science within the framework of what is often called the science of uncertainty quantification. The exposition is divided into three major parts. In Part 1, philosophical and statistical foundations of predictive science are developed within a Bayesian framework. There the case is made that the Bayesian framework provides, perhaps, a unique setting for handling all of the uncertainties encountered in scientific prediction. In Part 2, general frameworks and procedures for the calculation and validation of mathematical models of physical realities are given, all in a Bayesian setting. But beyond Bayes, an introduction to information theory, the maximum entropy principle, model sensitivity analysis and sampling methods such as MCMC are presented. In Part 3, the central problem of predictive computational science is addressed: the selection, adaptive control and validation of mathematical and computational models of complex systems. The Occam Plausibility Algorithm, OPAL, is introduced as a framework for model selection, calibration and validation. Applications to complex models of tumour growth are discussed.

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Section: Adaptive Control Of Multiscale Modelsmentioning

confidence: 99%

Adaptive multiscale predictive modelling

Oden

2018

Acta Numerica

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“…The construction of surrogate models with error control within a stochastic framework is also described in [9] for atomic-to-continuum modeling and [40] for a stochastic-deterministic coupling method. More recently, Maier and Rannacher [21,20] continued a duality-based approach to model adaptivity for heterogeneous materials that involves a post-processing procedure for selecting optimal models for estimating errors in QoIs.…”

Section: Introductionmentioning

confidence: 99%

Goal-oriented adaptive modeling of random heterogeneous media and model-based multilevel Monte Carlo methods

Scarabosio

Wohlmuth

Oden

et al. 2019

Computers & Mathematics with Applications

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Methods for generating sequences of surrogates approximating fine scale models of two-phase random heterogeneous media are presented that are designed to adaptively control the modeling error in key quantities of interest (QoIs). For specificity, the base models considered involve stochastic partial differential equations characterizing, for example, steady-state heat conduction in random heterogeneous materials and stochastic elastostatics problems in linear elasticity. The adaptive process involves generating a sequence of surrogate models defined on a partition of the solution domain into regular subdomains and then, based on estimates of the error in the QoIs, assigning homogenized effective material properties to some subdomains and full random fine scale properties to others, to control the error so as to meet a preset tolerance. New model-based Multilevel Monte Carlo (mbMLMC) methods are presented that exploit the adaptive sequencing and are designed to reduce variances and thereby accelerate convergence of Monte Carlo sampling. Estimates of cost and mean squared error of the method are presented. The results of several numerical experiments are discussed that confirm that substantial saving in computer costs can be realized through the use of controlled surrogate models and the associated mbMLMC algorithms.Keywords: Adaptive control of model error, goal-oriented a posteriori error estimation, Multilevel Monte Carlo, random heterogeneous media.

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“…The proposed approach, devoted to the Multiscale Finite Element Method and which fulfills the above two objectives, is an alternative to the goal-oriented adaptive strategy recently proposed in [20] within the Generalized Multiscale Finite Element framework, which uses residual-based techniques and which mainly focuses on goal-oriented adaptivity alone. We also mention the approach [44] proposed in the VMS framework, and the approach [50] proposed in the HMM framework where the overall error is also split between different contributions each related to a discretization parameter.…”

Section: Introduction and Objectivesmentioning

confidence: 99%

Goal-oriented error estimation and adaptivity in MsFEM computations

Chamoin¹,

Legoll²

2019

Preprint

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We introduce a goal-oriented strategy for multiscale computations performed using the Multiscale Finite Element Method (MsFEM). In a previous work, we have shown how to use, in the MsFEM framework, the concept of Constitutive Relation Error (CRE) to obtain a guaranteed and fully computable a posteriori error estimate in the energy norm (as well as error indicators on various error sources). Here, the CRE concept is coupled with the solution of an adjoint problem to control the error and drive an adaptive procedure with respect to a given output of interest. Furthermore, a local and non-intrusive enrichment technique is proposed to enhance the accuracy of error bounds. The overall strategy, which is fully automatic and robust, enables to reach an appropriate trade-off between certified reliability and computational cost in the MsFEM context. The performances of the proposed method are investigated on several illustrative numerical test cases. In particular, the error estimation is observed to be very accurate, yielding a very efficient adaptive procedure.

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Duality-based adaptivity in finite element discretization of heterogeneous multiscale problems

Abstract: This paper introduces an framework for adaptivity for a class of heterogeneous multiscale finite element methods for elliptic problems, which is suitable for

Cited by 6 publications

References 36 publications

Adaptive multiscale predictive modelling

Adaptive multiscale predictive modelling

Goal-oriented adaptive modeling of random heterogeneous media and model-based multilevel Monte Carlo methods

Goal-oriented error estimation and adaptivity in MsFEM computations

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