During ICIAM 1995, in Hamburg, David Tranah approached Jacques-Louis Lions and myself and asked us if we were interested in publishing in book form our two-part article “Exact and approximate controllability for distributed parameter systems” which had appeared in Acta Numerica 1994 and 1995. The length of the article (almost 300 pages) was a justification, among several others, for such an initiative. While I was very enthusiastic about this project, J.L. Lions was more cautious, without being against it. Actually, his reservation concerning this book project was stemming from recent important developments on controllability related issues, justifying, in his opinion an in-depth revision of our article. Both of us being quite busy, the project was practically forgotten. As everyone knows in the Scientific Community, and elsewhere, Jacques-Lions passed away in June 2001, while still active scientifically. He largely contributed in making the Control of Distributed Parameter Systems a most important field where sophisticated mathematical and computational techniques meet with advanced applications. Therefore, when David Tranah renewed his 1995 suggestion during a conference of the European Mathematical Society held in Nice in February 2003, we thought that it would be a very nice way to pay to J.L. Lions the tribute he fully deserves. The idea was to respect as much as possible the original text, since it largely reflects J.L. Lions’ inspired scientific vision, and also its inimitable way at making simple complicated notions. On the other hand, it was also agreed that additional material should be included to make the text more up to date.

Introduction. Synopsis

We will conclude this volume by discussing some aspects (mostly computational) of the optimal control of systems governed by the Navier-Stokes equations modeling unsteady incompressible Newtonian viscous fluids. This chapter can be viewed as a sequel of Chapter 3, where we addressed the controllability of Stokes How. The methods and results presented hereafter were not available at the time of Glowinski and J.L. Lions (1995), explaining thus the need for a new chapter.

To begin with, let us say that engineers have not waited for mathematicians to successfully address flow control problems (sec, for example. Gad Hcl Hak, 1989: Busehnell and Hefner, 1990 for a review of How control from the Engineering point of view); indeed Prandtl as early as 1915 was concerned with flow control and was designing ingenious systems to suppress or delay boundary layer separation (sec Prandtl, 1925). The last two decades have seen an explosive growth of investigations and publications of mathematical nature concerning various aspects of the control of viscous flow, good examples of these publications being Gun/burger (1995) and Sritharan (1998). Actually, the above two references also contain articles related to the computational aspects of the optimal control of viscous flow, but, usually, the geometry of the flow region is fairly simple and Reynolds numbers rather low. Some publications of computational nature arc llou and Ravindran (1996), Ghattas and Bark (1997), and Ito and Ravindran (1998); however, in those articles, once again the geometry is simple and/or the Reynolds number is low (more references will be given at the end of this section).

Introduction

Stealth technologies have enjoyed a considerable growth of interest during the last two decades both for aircraft and space applications. Due to the very high frequencies used by modern radars the computation of the Radar Cross Section (RCS) of a full aircraft using the Maxwell equations is still a great challenge (see Talflove, 1992). From the fact that boundary integral methods (see Nedelec, 2001 and the references therein for a discussion of boundary integral methods) are not well suited to general heterogeneous media and coated materials, field approaches seem to provide an alternative which is worth exploring.

In this chapter (which follows closely Section 6.13 of the original Acta Numerica article and Bristeau, Glowinski, and Periaux, 1998), we consider a particular application of controllability methods to the solution of the Helmholtz equations obtained when looking for the monochromatic solutions of linear wave problems. The idea here is to go back to the original wave equation and to apply techniques, inspired by controllability studies, in order to find its time-periodic solutions. Indeed, this method (introduced in Bristeau, Glowinski, and Périaux, 1993a,b) is a competitor – and is related – to the one in which the wave equation is integrated from 0 to +∞in order to obtain asymptotically a time-periodic solution; it is well-known (from, for example, Lax and Phillips, 1989) that if the scattering body is convex, then the solution will converge exponentially to the time-periodic solution as t → +∞.

Generalities. Synopsis

In the original text (that is, Acta Numerica 1995), the content of this chapter was considered as a preliminary step to a more ambitious goal, namely, the control of systems governed by the Navier–Stokes equations modeling incompressible viscous flow. Indeed, substantial progress concerning this objective took place in the late 1990s (some of them to be reported in Part III of this book), making – in some sense – this chapter obsolete. We decided to keep it since it addresses some important issues that will not be considered in Part III (and also because it reflects some of the J.L. Lions scientific concerns at the time).

Back to the original text, let us say that the control problems and methods which have been discussed so far in this book have been mostly concerned with systems governed by linear diffusion equations of the parabolic type, associated with second-order elliptic operators. Indeed, these methods have been applied in, for example, Berggren (1992) and Berggren, Glowinski, and J.L. Lions (1996b), to the solution of approximate boundary controllability problems for systems governed by strongly advection dominated linear advection–diffusion equations. These methods can also be applied to systems of linear advection–diffusion equations and to higher-order parabolic equations (or systems of such equations). Motivated by the solution of controllability problems for the Navier–Stokes equations modeling incompressible viscous flow, we will discuss now controllability issues for a system of partial differential equations which is not of the Cauchy–Kowalewska type, namely, the classical Stokes system.

While addressing in this book the numerical solution of controllability problems for systems governed by partial differential equations, we had the opportunity to encounter a variety of concepts and methods whose applicability goes much beyond the solution of genuine control problems. Among these concept and methods let us mention convex duality, space–time discretization of partial differential equations, numerical methods for the solution of large linear systems, least-squares formulations, optimization algorithms, and so on. In Chapter 7, we have shown while formulating a given problem as a controllability that one may gain access to powerful solution methods. Such a situation is not unique as shown by the following example inspired from work in progress by R. Azencott, A.M. Ramos and the first author (see Azencott, Glowinski, and Ramos, 2007). The (relatively simple) problem that we consider is part of a large research program on shape identification and pattern recognition (largely motivated by medical applications); it can be described as follows:

Let Γ0 be a rectifiable (or piece of) bounded curve in ℝ2; suppose that one wishes to know how close is Γ0 to another curve ΓR (the curve of reference) which is also rectifiable and bounded. The idea here is to introduce a distance between Γ0 and ΓR (rigid displacement and similarity invariant in general, but these conditions can be relaxed if necessary, or more conditions can be added).

In this chapter we discuss numerical methods based on structured grids for direct numerical simulations of multiphase flows involving solid particles. We will focus on numerical approaches that are designed to solve the governing equations in the fluid and the interaction between the phases at their interfaces.

In methods employing structured grids, there are two distinct possibilities for handling the geometric complexities imposed by the phase boundaries. The first approach is to precisely define the phase boundaries and use a body-fitted grid in one or both phases, as necessary. The curvilinear grid that conforms to the phase boundaries greatly simplifies the specification of interaction processes that occur across the interface. Furthermore, numerical methods on curvilinear grids are well developed and a desired level of accuracy can be maintained both within the different phases and along their interfaces. The main difficulty with this approach is grid generation; for example, for the case of flow in a pipe with several hundred suspended particles, obtaining an appropriate structured body-fitted grid around all the particles is a nontrival task. There are several structured grid generation packages that are readily available. Nevertheless, the computational cost of grid generation increases rapidly as geometric complexity increases. This can be severely restrictive, particularly in cases where the phase boundaries are in time-dependent motion. The cost of grid generation can then overwhelm the cost of computing the flow.

The alternative to a body-fitted grid that is increasingly becoming popular are Cartesian grid methods. Here, irrespective of the complex nature of the internal and external boundaries between the phases, only a regular Cartesian mesh is employed and the governing equations are solved on this simple mesh.

Broadly speaking, the goal of (mainstream) learning theory is to approximate a function (or some function features) from data samples, perhaps perturbed by noise. To attain this goal, learning theory draws on a variety of diverse subjects. It relies on statistics whose purpose is precisely to infer information from random samples. It also relies on approximation theory, since our estimate of the function must belong to a prespecified class, and therefore the ability of this class to approximate the function accurately is of the essence. And algorithmic considerations are critical because our estimate of the function is the outcome of algorithmic procedures, and the efficiency of these procedures is crucial in practice. Ideas from all these areas have blended together to form a subject whose many successful applications have triggered its rapid growth during the past two decades.

This book aims to give a general overview of the theoretical foundations of learning theory. It is not the first to do so.Yet we wish to emphasize a viewpoint that has drawn little attention in other expositions, namely, that of approximation theory. This emphasis fulfills two purposes. First, we believe it provides a balanced view of the subject. Second, we expect to attract mathematicians working on related fields who find the problems raised in learning theory close to their interests.

While writing this book, we faced a dilemma common to the writing of any book in mathematics: to strike a balance between clarity and conciseness. In particular, we faced the problem of finding a suitable degree of self-containment for a book relying on a variety of subjects.

In Chapter 4 we described finite difference methods for the direct numerical simulations of fluid–solid systems in which the local flow field around moving particles is resolved numerically without modeling. Here we describe methods of the finite element type for the same purpose.

The first method is termed the ALE (arbitrary Lagrangian-Eulerian) particle mover. The ALE particle mover uses a technique based on a combined formulation of the fluid and particle momentum equations, together with an arbitrary Lagrangian–Eulerian moving unstructured finite element mesh technique to deal with the movement of the particles. It was first developed by Hu et al. (1992, 1996, 2001). The method has been used to solve particle motions in both Newtonian and viscoelastic fluids in two- and three-dimensional flow geometries. It also handles particles of different sizes, shapes, and materials.

The second method is based on a stabilized space-time finite element technique to solve problems involving moving boundaries and interfaces. In this method, the temporal coordinate is also discretized using finite elements. The deformation of the spatial domain with time is reflected simply in the deformation of the mesh in the temporal coordinate. Using this technique, Johnson and Tezduyar (1999) were able to simulate the sedimentation of 1000 spheres in a Newtonian fluid at a Reynolds number of 10.

The third numerical method is termed the DLM (distributed Lagrange multiplier) particle mover. The basic idea of the DLM particle mover is to extend the problem from a time-dependent geometrically complex domain to a stationary fictitious domain, which is larger but simpler.