Federico Negri

Reduced Basis Methods for Partial Differential Equations. An Introduction

1 Introduction.- 2 Representative problems: analysis and (high-fidelity) approximation.- 3 Getting parameters into play.- 4 RB method: basic principle, basic properties.- 5 Construction of reduced basis spaces.- 6 Algebraic and geometrical structure.- 7 RB method in actions.- 8 Extension to nonaffine problems.- 9 Extension to nonlinear problems.- 10 Reduction and control: a natural interplay.- 11 Further extensions.- 12 Appendix A Elements of functional analysis.

Reduced basis method for parametrized elliptic optimal control problems

We propose a suitable model reduction paradigm -- the certied reduced basis method (RB) -- for the rapid and reliable solution of parametrized optimal control problems governed by partial dierential equations (PDEs). In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as constraint and infinite dimensional control variable. Firstly, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure; an efficient and rigorous a posteriori error estimate on the state, control and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique.

Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations

This paper extends the reduced basis method for the solution of parametrized optimal control problems presented in Negri et?al. (2013) to the case of noncoercive (elliptic) equations, such as the Stokes equations. We discuss both the theoretical properties-with particular emphasis on the stability of the resulting double nested saddle-point problems and on aggregated error estimates-and the computational aspects of the method. Then, we apply it to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three-dimensional flow through boundary control, demonstrating the effectivity of the methodology.

Efficient model reduction of parametrized systems by matrix discrete empirical interpolation

In this work, we apply a Matrix version of the so-called Discrete Empirical Interpolation (MDEIM) for the efficient reduction of nonaffine parametrized systems arising from the discretization of linear partial differential equations. Dealing with affinely parametrized operators is crucial in order to enhance the online solution of reduced-order models (ROMs). However, in many cases such an affine decomposition is not readily available, and must be recovered through (often) intrusive procedures, such as the empirical interpolation method (EIM) and its discrete variant DEIM. In this paper we show that MDEIM represents a very efficient approach to deal with complex physical and geometrical parametrizations in a non-intrusive, efficient and purely algebraic way. We propose different strategies to combine MDEIM with a state approximation resulting either from a reduced basis greedy approach or Proper Orthogonal Decomposition. A posteriori error estimates accounting for the MDEIM error are also developed in the case of parametrized elliptic and parabolic equations. Finally, the capability of MDEIM to generate accurate and efficient ROMs is demonstrated on the solution of two computationally-intensive classes of problems occurring in engineering contexts, namely PDE-constrained shape optimization and parametrized coupled problems.

Heuristic strategies for the approximation of stability factors in quadratically nonlinear parametrized PDEs

In this paper we present some heuristic strategies to compute rapid and reliable approximations to stability factors in nonlinear, inf-sup stable parametrized PDEs. The efficient evaluation of these quantities is crucial for the rapid construction of a posteriori error estimates to reduced basis approximations. In this context, stability factors depend on the problem’s solution, and in particular on its reduced basis approximation. Their evaluation becomes therefore very expensive and cannot be performed prior to (and independently of) the construction of the reduced space. As a remedy, we first propose a linearized, heuristic version of the Successive Constraint Method (SCM), providing a suitable estimate – rather than a rigorous lower bound as in the original SCM – of the stability factor. Moreover, for the sake of computational efficiency, we develop an alternative heuristic strategy, which combines a radial basis interpolant, suitable criteria to ensure its positiveness, and an adaptive choice of interpolation points through a greedy procedure. We provide some theoretical results to support the proposed strategies, which are then applied to a set of test cases dealing with parametrized Navier-Stokes equations. Finally, we show that the interpolation strategy is inexpensive to apply and robust even in the proximity of bifurcation points, where the estimate of stability factors is particularly critical.

Efficient Reduction of PDEs Defined on Domains with Variable Shape

In this work we propose a new, general and computationally cheap way to tackle parametrized PDEs defined on domains with variable shape when relying on the reduced basis method. We easily describe a domain by boundary parametrizations, and generate domain (and mesh) deformations by means of a solid extension, obtained by solving a linear elasticity problem. The proposed procedure is built over a two-stages reduction: (1) first, we construct a reduced basis approximation for the mesh motion problem; (2) then, we generate a reduced basis approximation of the state problem, relying on finite element snapshots evaluated over a set of reduced deformed configurations. A Galerkin-POD method is employed to construct both reduced problems, although this choice is not restrictive. To deal with unavoidable nonaffine parametric dependencies arising in both the mesh motion and the state problem, we apply a matrix version of the discrete empirical interpolation method, allowing to treat geometrical deformations in a non-intrusive, efficient and purely algebraic way. In order to assess the numerical performances of the proposed technique, we address the solution of a parametrized (direct) Helmholtz scattering problem where the parameters describe both the shape of the obstacle and other relevant physical features. Thanks to its easiness and efficiency, the methodology described in this work looks promising also in view of reducing more complex problems.

RB Methods in Action: Setting up the Problem

We show how a parametrized PDE can be transformed into an equivalent problem on a reference (parameter-independent) domain. General mathematical tools for this operation are given, and examples of parametrized PDEs are discussed that are inspired by the four problems of Chap. 2. The primary purpose is to highlight the different role played by physical and geometric parameters. A further critical issue addressed concerns the possible affine parametric dependence of the linear and bilinear forms defined over the reference domain. With the aid of a host of meaningful examples we discuss the way the affine parametric dependence is affected by the problem parametrization.

RB Methods in Action: Computing the Solution

We present a selection of numerical results dealing with the RB approximation of the parametrized problems formulated in the previous chapter. For each problem we highlight the RB method’s computational performance, assess its accuracy by means of a posteriori error bounds, and show various options for the construction of the RB space (either via POD or the greedy algorithm) and different projection criteria (G-RB versus LS-RB methods). Here we focus on linear affine PDEs, and defer nonaffine and nonlinear problems to Chaps. 10 and 11 respectively.

Reduction and Control

We exploit RB methods for the efficient solution of parameter-dependent PDE-constrained optimization problems. According to the optimization strategy adopted, these problems feature a very large size, in case a monolithic strategy is chosen, or huge computational cost due to the need of solving a parametrized PDE many times, when preferring an iterative optimization method.We concentrate on (i) parametric optimization problems, where control variables are described in terms of a vector of parameters, and (ii) parametrized optimal control problems, in which the parameters affect instead the state system and the control variables are functions to be determined. We propose efficient RB strategies to speedup the solution of these problems, pursuing either a (i) state reduction in the former case, or (ii) a simultaneous state and control reduction in the latter.

Extension to Nonaffine Problems

We explain how to set up a RB method for problems not fulfilling the assumption of affine parametric dependence. Since the possibility to devise an offline/online decomposition relies on that assumption, in case of nonaffine problems we recover an approximate affine expansion by means of the so-called empirical interpolation method (EIM). We provide a detailed description of the EIM, focusing on linear problems for the sake of simplicity. A possible alternative formulation, referred to as discrete empirical interpolation method (DEIM), is also presented. As shown in the following chapter, EIM is an essential tool to ensure an offline/online decomposition, under suitable assumptions, also for nonlinear parametrized PDEs.