to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

a 'sleeper', receiving only modest attention for 50 years before emerging as a cornerstone of communications engineering in more recent times. Roughly one in six of Walsh's 281 publications are included, photographically reproduced. Reproduction is excellent except for one paper from 1918. The book also reproduces three brief papers about Walsh and his work, by W. E. Sewell, D. V. Widder and Morris Marden. The first two were written for a special issue of the SIAM Journal celebrating Walsh's 70th birthday; the third is an obituary.

Matrix iterative analysis (2nd edn), by Richard S. Varga. Springer Series in Computational Mathematics 27. Pp. 358. £55. 2000. ISBN 3 540 66321 5 (Springer Verlag).

In this article, we present a reference case of mean field games. This case can be seen as a reference for two main reasons. First, the case is simple enough to allow for explicit resolution: Bellman functions are quadratic, stationary measures are normal and stability can be dealt with explicitly using Hermite polynomials. Second, in spite of its simplicity, the case is rich enough in terms of mathematics to be generalized and to inspire the study of more complex models that may not be as tractable as this one.

A reference case for mean field games models

The Lagrange-Galerkin method has proved a very successful method in computational fluid dynamics (CFD), in practice, but it has suffered from question marks about its stability when the integrals are approximated by quadratures. In this paper we will prove instability for wider ranges of quadrature formulae than previously considered. We will also introduce on approximate integration technique on triangular grids that cheaply recovers the unconditional stability of the Lagrange-Galerkin method with no loss in accuracy. The same technique can also be used for the exact projection of data from one arbitrary triangular grid to another, different, arbitrary triangular grid. This could be used in grid adaptation algorithms or in triangular multigrid procedures.

Exact projections and the Lagrange-Galerkin method: A realistic alternative to quadrature

Conservative modified Serre–Green–Naghdi equations with improved dispersion characteristics

A parallel implementation of a method of the semi-Lagrangian type for the advection equation on a hybrid architecture computation system is discussed. The difference scheme with variable stencil is constructed on the base of an integral equality between the neighboring time levels. The proposed approach allows one to avoid the Courant-Friedrichs-Lewy restriction on the relation between time step and mesh size. The theoretical results are confirmed by numerical experiments. Performance of a sequential algorithm and several parallel implementations with the OpenMP and CUDA technologies in the C language has been studied.

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A Computational Realization of a Semi-Lagrangian Method for Solving the Advection Equation

The numerical methods are presented for solving economic problems formulated in the Mean Field Game (MFG) form. The mean-field equilibrium (i.e., the Nash equilibrium for an infinite number of players) leads to the coupled system of two parabolic partial differential equations: the Hamilton-Jacobi-Bellman-Isaacs equation and the Fokker-Planck-Kolmogorov one. The description is focused on the discrete approximation of these equations and on the application of the MFG theory directly at discrete level. This approach results in an efficient algorithm for finding the corresponding grid control function. Contrary to difference schemes with directed differences used by other authors, here the semi-Lagrangian approximation is applied which improves some properties of a discrete problem of this type. This implies the fast convergence of an iterative algorithm for the monotone minimization of the cost functional.

Conservative difference schemes for the computation of mean-field equilibria

In this work we describe one of the methods of constructing approximate hydrodynamic models with the supposition of a long wave nature of flow. This method allows us to derive a nonlinear dispersive model to obtain its variational wording. We consider the general case of hydrodynamic equations for a potential ideal fluid flow over an irregular bottom as well as some special cases. In this work we study the nonlinear dispersive equations which describe the propagation of long surface water waves. In the past twenty years great interest has been shown in the nonlinear dispersive equations in various fields of physics and mathematics (see, e.g. the monograph [3] and references therein). A multitude of works on the soliton theory, which is used in nearly all fields of present-day physics, has given impetus to the new development in the long water wave theory. It is interesting to note that the emergence of the nonlinear dispersive equations dates back to the works by Korteweg, de Vries, and Boussinesq (the second half of the last century) and is just associated with the description of the propagation of a water wave (Scott Russell solitary wave). However, these first results were not widely known in those days. In spite of their long history it is only quite recently (see [9,10]) that the nonlinear dispersive models came into use to study the dynamics of long water waves. The application of the nonlinear dispersive models to the computational modelling of the wave processes has shown on the one hand that it is possible to solve a wide class of problems and on the other hand it has shown that both the differential models and the corresponding computational algorithms have not been adequately studied. This primarily applies to the well-defined statement of the initial boundary value problem and conservation laws (see [1,10,12] and other). At the same time, it is common knowledge that the concept of the conservation law is of major importance in analysing the main properties of the solutions to systems of differential equations. If a system arises from a variational principle, each conservation law of this system corresponds to one of the symmetry properties [7]. For example, the invariance of a variational problem under a group of time translations corresponds to the conservation of energy for the solutions of the corresponding Euler equations, and the invariance under space translations corresponds to the conservation of momentum [7]. In this work we obtain the variational wording of a nonlinear dispersive model by a rather formal method, which is for several reasons considered as a 'base' model in the following. On the basis of this model other nonlinear dispersive models used in practice may be constructed. We briefly describe the application of the variational principle to the wave hydrodynamic problems. ' Computer Center, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk 660036, Russia 184 Z.I. Fedotova and E.D. Karepova It is common knowledge that the variational principle is one of the most general principles in mechanics and physics. The idea that 'nature in all its manifestations chooses the simplest way at less cost' dates back to Aristotle [2]. The first precise wordings of the variational principle are due to Pierre de Fermat and Johann Bernoulli. The principle of least action is generally associated with Lagrange, and its modifications, which are widely used nowadays, are known as the HamiltonOstrogradskii and Hamilton-Poincare principles. Continuum mechanics is one of the fields of mechanics, where the variational principles are fruitfully used to solve a great wide class of problems (see, e.g. [2]). However, judging from the publications this approach has not been adequately studied in hydrodynamics, especially in free surface problems. This is also the case with approximate models of hydrodynamics. The first wording of the variational principle for an ideal incompressible fluid is due to Lagrange. It is this wording that gives rise to the flow equations which are called the Euler-Lagrange equations [2]. It was not until 1967 that Luke developed the variational principle for potential flows of an inviscid incompressible fluid over a plane with a free surface as a principle of stationary pressure (see [6]). The variation of the Luke functional with respect to the flow velocity potential and the free surface elevation allows us to obtain a boundary value problem, viz. the Laplace equation and three nonlinear boundary conditions (at the bottom and at the free surface). V. E. Zakharov, the translation editor [13], showed in his comments that the Luke functional may be derived from the Hamilton's principle by adding relations with the aid of indefinite Lagrange multipliers. In this case the difference between the kinetic and potential energies is considered as a functional for the variational principle. The incompressibility conditions and the kinetic condition at the free surface may be used as such relations. Whitham [11,12] gives much attention to the variational wordings which describe the wave motion in the first place and the hydrodynamic models in particular. One of the typical examples of application of the variational principle to wave flows is the study of the models of linear dispersive waves for slowly changing flows (see [12]). These models are, for example, Boussinesq and Korteweg-de Vries linear approximations. Having formulated for these models a variational principle 'averaged' over the wave period Whitham obtained some interesting results. In particular, the Lagrangian in the 'averaged' variational principle for any linear problem is the product of the square of the amplitude by the dispersive relation. Besides, the variational principle in the above case readily leads to the conservation laws of energy and other quantities (for example, Svave effect' and Svave momentum'). Whitham in [11] considers the variational wordings of some approximate models of hydrodynamics and gives the variational wording of the nonlinear dispersive model at a flat bottom with the flow velocity potential taken as the known truncated expansion of the potential in terms of depth. Moreover, the variational wordings of the Whitham model [12] and the nonlinear dispersive model in which the velocity is taken at the wave crest are obtained. In this work we try to find the variational wording that corresponds to the nonlinear dispersive shallow water model in general form. This problem is of interest, because most of the applied problems in long wave hydrodynamics may now be solved numerically, using the approximation considered. However, the available methods of constructing the computational algorithms for the Variational principle for approximate models 185 particular class of problems do not allow us to obtain the efficient numerical models. The construction of the variational principle and the main conservation laws would allow us to develop conservative difference schemes for a variety of nonlinear dispersive models. There exist some approaches to the derivation of shallow water equations. They are generally obtained as the approximation of the problem of a potential fluid flow with the supposition of a long wave nature of flow, i.e. the ratio of the average depth of a basin to the average wavelength is small: β = (Λ0//) is much less than unity. The nonlinear dispersive model in this case is an approximate problem in which only the terms of first-order infinitesimal with respect to β are retained (see [1,10]). In this work we search for the variational wording by expanding the velocity potential in a power series of the vertical coordinate and substituting this expansion in the Luke functional. We further isolate the small parameter that specifies the flow as the long wave one. On the assumption that water is shallow, the approximation of the Luke functional is the sought functional for the approximate model. Recall that its variation leads to a boundary value problem for the Laplace operator with nonlinear boundary conditions. If we consider its variation with retention of the terms up to the second-order infinitesimal, we obtain the equations of the nonlinear dispersive model which is called the 'base' model in the subsequent discussion. We can obtain a variety of nonlinear dispersive models from the 'base' model by one of three methods: retaining the terms that correspond only to weak nonlinearity (models of the Boussinesq type); choosing different values as a 'flow velocity' for a given model (averaging the actual velocity over depth, the velocity at the bottom or the velocity at the wave crest, etc.); changing the form of some terms of order β on the basis of zero-order approximation. We note that some particular results obtained by us are published in [4,5]. 1. POSING THE PROBLEM We consider a flow of an inviscid incompressible fluid. Luke showed (see [6,12]) that the variational principle

Variational principle for approximate models of wave hydrodynamics

Numerical simulation of surface waves in basins is considered, which is described within acceptable accuracy in the context of the shallow water theory taking into account the Earth sphericity and the Coriolis acceleration. The Bubnov–Galerkin method is used for space approximation. A space of functions linear on triangular finite elements of a certain consistent triangulation is used as spaces of trial and test functions. A priori stability estimates are obtained for the discrete analogue, and the second order of approximation in the interior nodes of a uniform grid is demonstrated. Computation results are given for a test problem with an exact solution and for the basin of the Sea of Okhotsk.

https://www.degruyter.com/view/j/rnam.2006.21.issue-4/156939806777973409/156939806777973409.pdf

Simulation of surface waves in basins by the finite element method

In the paper, the semi-Lagrangian method is considered for the numerical solution of the advection problem. A numerical solution is constructed as a piecewise constant function on a rectangular grid. The proposed method is stable and gives an approximate solution with the first order of accuracy. To reduce the effect of smoothing an approximate solution because of numerical viscosity, a mesh refinement is applied in the vicinity of large gradients of the approximate solution. The localization of the smoothing effect is illustrated by a numerical example. In contrast to the traditional Eulerian schemes, semi-Lagrangian algorithms do not involve a time step restriction.

E. Karepova

Papers

A Computational Realization of a Semi-Lagrangian Method for Solving the Advection Equation

Conservative difference schemes for the computation of mean-field equilibria

Variational principle for approximate models of wave hydrodynamics

Simulation of surface waves in basins by the finite element method

Semi-Lagrangian method for advection problem with adaptive grid