Reduced Basis Method for 3D Problems governed by Parametrized PDEs and Applications

In this thesis we will deal with the creation of a Reduced Basis (RB) approximation of parametrized Partial Differential Equations (PDE) for three-dimensional problems. The the idea behind RB is to decouple the generation and projection stages (Ofﬂine/Online computational proce- dures) of the approximation process in order to solve parametrized (PDE) in a fast, cheap and reliable way. The RB method, especially applied to 3D problems, allows great computational savings with respect to the clas- sical Galerkin Finite Element (FE) Method. The standard FE method is typically ill suited to (i) iterative contexts like in optimization, sensitivity analysis and many queries in general and (ii) real time evaluation. We consider both coercive and noncoercive PDEs. For each class we discuss the steps to set up a RB approximation, either from an analytical and a numerical point of view. Then we present the applications of the RB method to three different problems of engineering interest and applica- bility: (i) a steady thermal conductivity problem in heat transfer; (ii) a linear elasticity problem; (iii) Stokes ﬂows with emphasis on geometrical and physical parameters.

Reduced Basis method for Isogeometric Analysis: application to structural problems

Marina Rinaldi

Isogeometric Analysis (IGA) is a computational methodology for the numerical approximation of Partial Differential Equations (PDEs). IGA is based on the isogeometric concept, for which the same basis functions, usually Non-Uniform Rational B-Splines (NURBS), are used both to represent the geometry and to approximate the unknown solutions of PDEs. Compared to the standard Finite Element method, NURBS-based IGA offers several advantages: ideally a direct interface with CAD tools, exact geometrical representation, simple refinement procedures, and smooth basis functions allowing to easily solve higher order problems, including structural shell problems. In these contexts, repeatedly solving a problem for a large set of geometric parameters might lead to high and eventually prohibitive computational cost. To cope with this problem, we consider in this work the Reduced Basis (RB) method for the solution of parameter dependent PDEs, specifically for which the NURBS representation of the computational domain is parameter dependent. RB refers to a technique that enables a rapid and reliable approximation of parametrized PDEs by constructing low dimensional approximation spaces. In this work, for the construction of the reduced spaces we adopt two different strategies, namely the Proper Orthogonal Decomposition and the greedy algorithm. In this thesis we combine RB and IGA for the efficient solution of parametrized problems for all the possible cases of NURBS geometrical parametrizations, which specifically include the NURBS control points, the weights, and both the control points and weights. In particular, we first focus on the solution of second order PDEs on parametrized lower dimensional manifolds, specifically surfaces in the three dimensional space. We consider geometrical parametrizations that entail a nonaffine dependence of the variational forms on the spatial coordinates and the geometric parameters. Thus, depending on the parametrization at hand and in order to ensure a suitable Offline/Online decomposition between the reduced order model construction and solution, we resort to the Empirical Interpolation Method (EIM) or the Matrix Discrete Empirical Interpolation Method (MDEIM), by comparing their performances. As application, we solve a class of benchmark structural problems modeled by Kirchoff-Love shells for which we consider NURBS geometric parametrizations and we apply the RB method to the solution of this class of fourth order PDEs. We highlight by means of numerical tests, the performance of the RB method applied to standard IGA approximation of parametrized shell geometries.

2015

Multi space reduced basis preconditioners for parametrized partial differential equations

Niccolò Dal Santo

The multiquery solution of parametric partial differential equations (PDEs), that is, PDEs depending on a vector of parameters, is computationally challenging and appears in several engineering contexts, such as PDE-constrained optimization, uncertainty quantification or sensitivity analysis. When using the finite element (FE) method as approximation technique, an algebraic system must be solved for each instance of the parameter, leading to a critical bottleneck when we are in a multiquery context, a problem which is even more emphasized when dealing with nonlinear or time dependent PDEs. Several techniques have been proposed to deal with sequences of linear systems, such as truncated Krylov subspace recycling methods, deflated restarting techniques and approximate inverse preconditioners; however, these techniques do not satisfactorily exploit the parameter dependence. More recently, the reduced basis (RB) method, together with other reduced order modeling (ROM) techniques, emerged as an efficient tool to tackle parametrized PDEs. In this thesis, we investigate a novel preconditioning strategy for parametrized systems which arise from the FE discretization of parametrized PDEs. Our preconditioner combines multiplicatively a RB coarse component, which is built upon the RB method, and a nonsingular fine grid preconditioner. The proposed technique hinges upon the construction of a new Multi Space Reduced Basis (MSRB) method, where a RB solver is built at each step of the chosen iterative method and trained to accurately solve the error equation. The resulting preconditioner directly exploits the parameter dependence, since it is tailored to the class of problems at hand, and significantly speeds up the solution of the parametrized linear system. We analyze the proposed preconditioner from a theoretical standpoint, providing assumptions which lead to its well-posedness and efficiency. We apply our strategy to a broad range of problems described by parametrized PDEs: (i) elliptic problems such as advection-diffusion-reaction equations, (ii) evolution problems such as time-dependent advection-diffusion-reaction equations or linear elastodynamics equations (iii) saddle-point problems such as Stokes equations, and, finally, (iv) Navier-Stokes equations. Even though the structure of the preconditioner is similar for all these classes of problems, its fine and coarse components must be accurately chosen in order to provide the best possible results. Several comparisons are made with respect to the current state-of-the-art preconditioning and ROM techniques. Finally, we employ the proposed technique to speed up the solution of problems in the field of cardiovascular modeling.

EPFL2018

A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices

Denis Devaud, Andrea Manzoni, Gianluigi Rozza

We consider a method to efficiently evaluate in a real-time context an output based on the numerical solution of a partial differential equation depending on a large number of parameters. We state a result allowing to improve the computational performance of a three-step RB-ANOVA-RB method. This is a combination of the reduced basis (RB) method and the analysis of variations (ANOVA) expansion, aiming at compressing the parameter space without affecting the accuracy of the output. The idea of this method is to compute a first (coarse) RB approximation of the output of interest involving all the parameter components, but with a large tolerance on the a posteriori error estimate; then, we evaluate the ANOVA expansion of the output and freeze the least important parameter components; finally, considering a restricted model involving just the retained parameter components, we compute a second (fine) RB approximation with a smaller tolerance on the a posteriori error estimate. The fine RB approximation entails lower computational costs than the coarse one, because of the reduction of parameter dimensionality. Our result provides a criterion to avoid the computation of those terms in the ANOVA expansion that are related to the interaction between parameters in the bilinear form, thus making the RB-ANOVA-RB procedure computationally more feasible. (C) 2013 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.