Reduced-Basis approach for homogenization beyond the periodic setting

We consider the computation of averaged coefficients for the homogenization of elliptic partial differential equations. In this problem, like in many multiscale problems, a large number of similar computations parametrized by the macroscopic scale is required at the microscopic scale. This is a framework very much adapted to model order reduction attempts. The purpose of this work is to show how the reduced-basis approach allows to speed up the computation of a large number of cell problems without any loss of precision. The essential components of this reduced-basis approach are the {\it a posteriori} error estimation, which provides sharp error bounds for the outputs of interest, and an approximation process divided into offline and online stages, which decouples the generation of the approximation space and its use for Galerkin projections

Tensor-based multiscale method for diffusion problems in quasi-periodic heterogeneous media

This paper proposes to address the issue of complexity reduction for the numerical simulation of multiscale media in a quasi-periodic setting. We consider a stationary elliptic diffusion equation defined on a domain $D$ such that $\overline{D}$ is the union of cells $\{\overline{D_i}\}_{i\in I}$ and we introduce a two-scale representation by identifying any function $v(x)$ defined on $D$ with a bi-variate function $v(i,y)$, where $i \in I$ relates to the index of the cell containing the point $x$ and $y \in Y$ relates to a local coordinate in a reference cell $Y$. We introduce a weak formulation of the problem in a broken Sobolev space $V(D)$ using a discontinuous Galerkin framework. The problem is then interpreted as a tensor-structured equation by identifying $V(D)$ with a tensor product space $\mathbb{R}^I \otimes V(Y)$ of functions defined over the product set $I\times Y$. Tensor numerical methods are then used in order to exploit approximability properties of quasi-periodic solutions by low-rank tensors.Comment: Changed the choice of test spaces V(D) and X (with regard to regularity) and the argumentation thereof. Corrected proof of proposition 3. Corrected wrong multiplicative factor in proposition 4 and its proof (was 2 instead of 1). Added remark 6 at the end of section 2. Extended remark 7. Added references. Some minor improvements (typos, typesetting

Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems

In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems. It is based on the generalized multiscale finite element method (GMsFEM) and multilevel Monte Carlo (MLMC) methods. The former provides a hierarchy of approximations of different resolution, whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels. The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost, and to efficiently generate samples at different levels. In particular, it is cheap to generate samples on coarse grids but with low resolution, and it is expensive to generate samples on fine grids with high accuracy. By suitably choosing the number of samples at different levels, one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces, while retaining the accuracy of the final Monte Carlo estimate. Further, we describe a multilevel Markov chain Monte Carlo method, which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids, while combining the samples at different levels to arrive at an accurate estimate. The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in \cite{ketelson2013}, and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates.Comment: 29 pages, 6 figure

Coupling reduced basis and numerical homogenization methods for solving quasilinear elliptic problems

The workshop has brought together experts in the broad field of partial differential equations with highly heterogeneous coefficients. Analysts and computational and applied mathematicians have shared results and ideas on a topic of considerable interest both from the theoretical and applied viewpoints. A characteristic feature of the workshop has been to encourage discussions on the theoretical as well as numerical challenges in the field, both from the point of view of deterministic as well as stochastic modeling of the heterogeneities

On Error-Controlled Numerical Model Reduction for Linear Transient FE\ub2 Analysis

Multiscale modeling is of high interest in the engineering community due to its ability to capture the overall response, while still accounting for processes and structures on underlying fine scales. One standard approach to multiscale modeling is the so-called FE\ub2 procedure, where the classic constitutive relation is replaced by a boundary value problem on a Representative Volume Element (RVE) comprising the underlying microscale features. It is well realized that straight-forward use of the FE\ub2-strategy can be computationally intractable for a fine macroscale mesh. Therefore, it is of interest to reduce the cost of solving the individual RVE-problem(s) by introducing some kind of reduced basis, here denoted Numerical Model Reduction (NMR). However, it is important to note that the richness of the reduced basis will determine the accuracy of the solution, which calls for error control.This thesis concerns numerical model reduction for linear transient problems in the FE\ub2 setting, in particular the problems of heat flow and poroelasticity. Two different reduction techniques – Spectral Decomposition and Proper Orthogonal Decomposition – are applied in order to obtain an efficient method of solving and evaluating homogenized quantities on the microscale. For the model problem of linear transient heat flow, the microscale finite element problem reduces to a set of (uncoupled) ordinary differential equations, which, obviously, can be solved more efficiently than the original fully resolved finite element problem.For the error estimation, we focus solely on the error due to the reduced basis and ignore time- and space-discretization errors. We derive guaranteed, explicit bounds on the error in (i) a constructed “energy” norm and (ii) a user-defined quantity of interest (QoI) within the realm of goal-oriented error estimation. As a “workhorse” for the error computation, we introduce an associated (non-physical) symmetrized variational problem in space-time. We obtain low cost estimators, based on the residual, which, in particular, requires no extra modes than the ones used for the reduced basis approximation. The performance of the estimator is demonstrated with numerical examples, and, for both the heat flow problem and the poroelastic problem, we overestimate the error with an order of magnitude, which is deemed acceptable given that the estimate is fully explicit and the extra cost is negligible

A reduced basis localized orthogonal decomposition

In this work we combine the framework of the Reduced Basis method (RB) with the framework of the Localized Orthogonal Decomposition (LOD) in order to solve parametrized elliptic multiscale problems. The idea of the LOD is to split a high dimensional Finite Element space into a low dimensional space with comparably good approximation properties and a remainder space with negligible information. The low dimensional space is spanned by locally supported basis functions associated with the node of a coarse mesh obtained by solving decoupled local problems. However, for parameter dependent multiscale problems, the local basis has to be computed repeatedly for each choice of the parameter. To overcome this issue, we propose an RB approach to compute in an "offline" stage LOD for suitable representative parameters. The online solution of the multiscale problems can then be obtained in a coarse space (thanks to the LOD decomposition) and for an arbitrary value of the parameters (thanks to a suitable "interpolation" of the selected RB). The online RB-LOD has a basis with local support and leads to sparse systems. Applications of the strategy to both linear and nonlinear problems are given

A numerical study of different projection-based model reduction techniques applied to computational homogenisation

Computing the macroscopic material response of a continuum body commonly involves the formulation of a phenomenological constitutive model. However, the response is mainly influenced by the heterogeneous microstructure. Computational homogenisation can be used to determine the constitutive behaviour on the macro-scale by solving a boundary value problem at the micro-scale for every so-called macroscopic material point within a nested solution scheme. Hence, this procedure requires the repeated solution of similar microscopic boundary value problems. To reduce the computational cost, model order reduction techniques can be applied. An important aspect thereby is the robustness of the obtained reduced model. Within this study reduced-order modelling (ROM) for the geometrically nonlinear case using hyperelastic materials is applied for the boundary value problem on the micro-scale. This involves the Proper Orthogonal Decomposition (POD) for the primary unknown and hyper-reduction methods for the arising nonlinearity. Therein three methods for hyper-reduction, differing in how the nonlinearity is approximated and the subsequent projection, are compared in terms of accuracy and robustness. Introducing interpolation or Gappy-POD based approximations may not preserve the symmetry of the system tangent, rendering the widely used Galerkin projection sub-optimal. Hence, a different projection related to a Gauss-Newton scheme (Gauss-Newton with Approximated Tensors- GNAT) is favoured to obtain an optimal projection and a robust reduced model

Numerical Model Reduction and Error Control for Computational Homogenization of Transient Problems

Multiscale modeling is a class of methods useful for numerical simulation of mechanics, in particular, when the microstructure of a material is of importance. The main advantage is the ability to capture the overall response, and, at the same time, account for processes and structures on the underlying fine scales. The FE2 procedure, "finite element squared", is one standard multiscale approach in which the constitutive relation is replaced with a boundary value problem defined on an Representative Volume Element (RVE) which contains the microscale features. The procedure thus involves the solution of finite element problems on two scales: one macroscopic problem and multiple RVE problems, typically one for each quadrature point in the macroscale mesh. While the solution of the independent RVE problems can be trivially parallelized it can still be computationally impractical to solve the two-scale problem, in particular for fine macroscale meshes. It is, therefore, of interest to investigate methods for reducing the computational cost of solving the individual RVE problems, while still having control of the accuracy.In this thesis the concept of Numerical Model Reduction (NMR) is applied for reducing the RVE problems by constructing a reduced spatial basis using Spectral Decomposition (SD) and Proper Orthogonal Decomposition. Computational homogenization of two different transient model problems have been studied: heat flow and consolidation. In both cases the RVE problem reduces to a system of ordinary differential equations, with dimension much smaller than of the finite element system.With the reduced basis and decreased computational time comes also loss of accuracy. Thus, in order to assess results from a reduced computation, it is useful to quantify the error. This thesis focuses solely on estimation of the error stemming from the reduced basis by assuming the fully resolved finite element solution to be exact, thereby ignoring e.g. time- and space-discretization errors. For the linear model problems guaranteed, fully computable, bounds are derived for the error in (i) a constructed "energy" norm and (ii) a user-defined quantity of interest within the realm of goal-oriented error estimation. In the non-linear case approximate, fully computable, bounds are derived based on the linearized error equation.In all cases an associated (non-physical) symmetrized variational problem in space-time is introduced as a "driver" for the estimate. From this residual-based estimates with low computational cost are obtained. In particular, no extra modes than the ones used for the reduced basis approximation are required. The performance of the estimator is demonstrated with numerical examples, and, for both the heat flow problem and the poroelastic problem, the error is overestimated by an order of magnitude, which is deemed acceptable given that the estimate is fully explicit and the extra cost is negligible

Numerical Model Reduction with error estimation for computational homogenization of non-linear consolidation

Numerical Model Reduction (NMR) is adopted for solving the non-linear microscale problem that arises from computational homogenization of a model problem of porous media with displacement and pressure as unknown fields. A reduced basis is obtained for the pressure field using Proper Orthogonal Decomposition and the pertinent displacement basis is obtained using Nonuniform Transformation Field Analysis. An explicit, fully computable, a posteriori error estimator is derived based on the linearized error equation for quantification of the NMR error in terms of a suitably chosen energy norm. The performance of the error estimates is demonstrated via a set of numerical examples with varying load amplitudes