Reduced order modeling techniques for numerical homogenization methods applied to linear and nonlinear multiscale problems

The characteristic of effective properties of physical processes in heterogeneous media is a basic modeling and computational problem for many applications. As standard numerical discretization of such multiscale problems (e.g. with classical finite element method (FEM)) is often computationally prohibitive, there is a need for a novel computational algorithm able to capture the effective behavior of the physical system without resolving the finest scale in the system on the whole computational domain. In this thesis we propose and analyze a new class of numerical methods that combine the so-called finite element heterogeneous multiscale method (FE-HMM) with reduced order modeling techniques for linear and nonlinear multiscale problems. In the first part of the thesis we generalize the FE-HMM to elliptic problems with an arbitrary number of well-separated scales. We provide a rigorous a priori error analysis of this method that generalizes previous work restricted to two-scale problems. In the second part of the thesis, we develop our new reduced order multiscale method that combines the FE-HMM with reduced basis (RB) method. This method, the reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) provides a significant improvement in computational efficiency compared to the FE-HMM, specially for high dimensional problems or high order methods. A priori and a posteriori error analyses are derived for linear elliptic problems, as well as goal oriented adaptivity techniques. The RB-FE-HMM is then generalized to a class of nonlinear elliptic and parabolic problems. A priori error analysis and extensive computational results for nonlinear problems are also provided.

Model order reduction for large-scale structures with local nonlinearities

Mengwu Guo, Jan Sickmann Hesthaven, Zhenying Zhang

In solid mechanics, linear structures often exhibit (local) nonlinear behavior when close to failure. For instance, the elastic deformation of a structure becomes plastic after being deformed beyond recovery. To properly assess such problems in a real-life application, we need fast and multi-query evaluations of coupled linear and nonlinear structural systems, whose approximations are not straight forward and often computationally expensive. In this work, we propose a linear-nonlinear domain decomposition, where the two systems are coupled through the solutions on the linear-nonlinear interface. After necessary sensitivity analysis, e.g. for structures with a high dimensional parameter space, we adopt a non-intrusive method, e.g. Gaussian processes regression (GPR), to solve for the solution on the interface. We then utilize different model order reduction techniques to address the linear and nonlinear problems individually. To accelerate the approximation, we employ again the non-intrusive GPR for the nonlinearity, while intrusive model order reduction methods, e.g. the conventional reduced basis (RB) method or the static-condensation reduced- basis-element (SCRBE) method, are employed for the solution in the linear subdomain. We provide several numerical examples to demonstrate the effectiveness of our method.

ELSEVIER SCIENCE SA2019

Adaptive Finite Element Methods for Multiscale Partial Differential Equations

Achim Nonnenmacher

Engineers rely on efficient simulations that provide them with reliable data in order to make proper engineering design decisions. The purpose of this thesis is to design adaptive numerical methods for multiscale problems in this spirit. We consider elliptic homogenization problems discretized by the finite element heterogeneous multiscale method (FE-HMM). Unlike standard (single-scale) finite element methods, our multiscale discretization scheme relies on coupled macro and micro finite elements. The framework of the HMM allows to design an algorithm that follows the classical finite element structure on the macro level. The fine scales of the multiscale problems are taken into account by replacing the element-wise numerical integration over unknown macroscopic data by a numerical integration over suitably averaged micro solutions. These micro solutions are obtained from micro FE problems on sampling domains within the macro elements. This thesis is divided into two parts. In the first part, we discuss a short and versatile FE implementation of the multiscale algorithm. The implementation is flexible, easy to use and to modify and can handle simplicial or quadrilateral FE and various macro-micro coupling conditions for the constrained micro problems. The implementation of time-dependent problems is also discussed. Numerical examples including three dimensional problems are presented and demonstrate the efficiency and the versatility of the computational strategy. In the second part (the main part of this thesis), we present an a posteriori error analysis for the FE-HMM. The a posteriori analysis enables us to estimate the accuracy of a numerical solution (and therefore its reliability) and further it allows for the design of adaptive numerical methods, which are the most efficient. The crucial component for the design of an adaptive multiscale method is the introduction of appropriate error indicators. As the error indicators depend on macroscopic data (such as the macroscopic diffusion tensor) that are not readily available, we construct error indicators that only depend on the available macro and micro FE solutions, available from previous computations. We provide a posteriori estimates for the upper and lower bound in the energy norm. The corresponding macroscopic mesh refinement strategy is therefore both reliable and efficient. The microscopic mesh is refined simultaneously and – under appropriate assumptions – optimally with the macroscopic mesh. This means that the strategy reduces the macro and micro error at the same rate. In the case of a uniformly oscillating tensor and exact micro computations, the standard a posteriori error estimates for the FEM applied to the homogenized problem are recovered. Numerical experiments confirm the efficiency and reliability of the adaptive multiscale method and demonstrate the optimality of the chosen macro-micro coupling. We extend the adaptive FE-HMM to higher order FE. We further derive a posteriori estimates for the error in quantities of interest that are needed to make certain design decisions; the quantity of interest is represented by a linear functional. We derive and analyze a multiscale counterpart to the classical dual-weighted residual method and design a corresponding goal-oriented adaptive multiscale method. The efficiency of the method is shown in numerical experiments.

EPFL2011

Dynamical low rank approximation for uncertainty quantification of time-dependent problems

Eva Vidlicková

The quantification of uncertainties can be particularly challenging for problems requiring long-time integration as the structure of the random solution might considerably change over time. In this respect, dynamical low-rank approximation (DLRA) is very appealing. It can be seen as a reduced basis method, thus solvable at a relatively low computational cost, in which the solution is expanded as a linear combination of a few deterministic functions with random coefficients. The distinctive feature of the DLRA is that both the deterministic functions and random coefficients are computed on the fly and are free to evolve in time, thus adjusting at each time to the current structure of the random solution. This is achieved by suitably projecting the dynamics onto the tangent space of a manifold consisting of all random functions with a fixed rank. In this thesis, we aim at further analysing and applying the DLR methods to time-dependent problems.Our first work considers the DLRA of random parabolic equations and proposes a class of fully discrete numerical schemes.Similarly to the continuous DLRA, our schemes are shown to satisfy a discrete variational formulation.By exploiting this property, we establish the stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a ``parabolic'' type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions. The discrete variational formulation is further applied in our second work, which derives a-priori and a-posteriori error estimates for the discrete DLRA of a random parabolic equation obtained by the three newly-proposed schemes. Under the assumption that the right-hand side of the dynamical system lies in the tangent space up to a small remainder, we show that the solution converges with standard convergence rates w.r.t. the time, spatial, and stochastic discretization parameters, with constants independent of singular values.We follow by presenting a residual-based a-posteriori error estimation for a heat equation with a random forcing term and a random diffusion coefficient which is assumed to depend affinely on a finite number of independent random variables. The a-posteriori error estimate consists of four parts: the finite element method error, the time discretization error, the stochastic collocation error, and the rank truncation error. These estimators are then used to drive an adaptive choice of FE mesh, collocation points, time steps, and time-varying rank.The last part of the thesis examines the idea of applying the DLR method in data assimilation problems, in particular the filtering problem. We propose two new filtering algorithms. They both rely on complementing the DLRA with a Gaussian component. More precisely, the DLR portion captures the non-Gaussian features in an evolving low-dimensional subspace through interacting particles, whereas each particle carries a Gaussian distribution on the whole ambient space. We study the effectiveness of these algorithms on a filtering problem for the Lorenz-96 system.