using underapproximations for sparse nonnegative matrix factorization nonnegative matrix factorization consists in approximately factorizing a nonnegative data matrix by the product of two low rank nonnegative matrices it has been successfully applied as a data analysis technique in numerous domains e g text mining image processing microarray data analysis collaborative filtering etc we introduce a novel approach to solve nmf problems based on the use of an underapproximation technique and show its effectiveness to obtain sparse solutions this approach based on lagrangian relaxation allows the resolution of nmf problems in a recursive fashion we also prove that the underapproximation problem is np hard for any fixed factorization rank using a reduction of the maximum edge biclique problem in bipartite graphs we test two variants of our underapproximation approach on several standard image datasets and show that they provide sparse part based representations with low reconstruction error our results are comparable and sometimes superior to those obtained by two standard sparse nonnegative matrix factorization techniques

a viscosity solutions approach to shape from shading the problem of recovering a lambertian surface from a single two dimensional image may be written as a first order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions a new approach based on hamilton jacobi bellman equations and viscosity solutions theories enables one to study non uniqueness phenomenon and thus to characterize the surface among the various solutions a consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle and numerical results are presented

numerical schemes for hamilton jacobi equations on unstructured meshes in this paper a numerical scheme is presented by applying the finite element method to the viscosity equations of the hamilton jacobi equations on unstructured meshes by improving the finite element scheme another numerical scheme is constructed under certain limitations the numerical solutions of the two schemes converge to the viscosity solutions of the hamilton jacobi equations the latter numerical scheme has weaker restrictions than the former scheme for convergence numerical examples are provided to test the stability convergence and sensitivity to different meshes

analysis of in plane wave propagation in periodic structures with sierpinski carpet unit cells abstract in this study the dispersion relationships and characteristics of directional periodic structure waveguides with sierpinski carpet unit cells were investigated finite element method was used to analyze the sierpinski carpet periodic structures with square pb cylinders inserted in the rubber background the dispersion curves of unit cells with different stages were investigated first then group velocities of periodic structures at given frequencies were calculated and used to analyze the directional waveguide characteristics in the sierpinski carpet periodic structures the dynamic responses of finite periodic structures were investigated to observe the directional features and the results were used to support the fact that sierpinski carpet unit cells with different stages can control the propagation behavior of elastic waves in specific ways the dispersion curves grew compressed as the stage increased multiple band characteristics were observed in the case of the second stage fractal configuration the directional propagation characteristics in sierpinski carpet periodic structures can be utilized to further extend the range of vibration reduction zones the results of this study have significant potential in terms of the application of sierpinski carpet periodic structures for vibration isolation

cav u and the dual game we give an alternative proof of a theorem of aumann and maschler that characterizes the limit of the values of finitely repeated games with lack of information on one side as the concavification of the value of the game where none of the players has any information our proof is based on fenchel duality techniques

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

a rate of convergence for monotone finite difference approximations to fully nonlinear uniformly elliptic pdes we obtain an algebraic rate of convergence for monotone and consistent finite difference approximations to lipschitz continuous viscosity solutions of uniformly elliptic partial differential equations 2007 wiley periodicals inc

fractal dimensions of random attractors for stochastic benjamin bona mahony equation on unbounded domains abstract this paper considers random attractor and its fractal dimension for benjamin bona mahony equation driven by additive white noise on unbounded domains firstly we investigate the existence of random attractor for the random dynamical system defined on an unbounded domain secondly we present criterion for estimating an upper bound of the fractal dimension of a random invariant set of a random dynamical system on a separable banach space finally we apply expectations of some random variables and these conditions to prove the finiteness of fractal dimension of the random attractors for stochastic benjamin bona mahony equation driven by additive white noise

randomized algorithms for the low rank approximation of matrices we describe two recently proposed randomized algorithms for the construction of low rank approximations to matrices and demonstrate their application inter alia to the evaluation of the singular value decompositions of numerically low rank matrices being probabilistic the schemes described here have a finite probability of failure in most cases this probability is rather negligible 10 17 is a typical value in many situations the new procedures are considerably more efficient and reliable than the classical deterministic ones they also parallelize naturally we present several numerical examples to illustrate the performance of the schemes

evaluating information in zero sum games with incomplete information on both sides we study zero sum games with incomplete information and analyze the impact that the information players receive has on the payoffs it turns out that the functions that measure the value of information share two properties the first is blackwell monotonicity which means that each player gains from knowing more the second is concavity on the space of conditional probabilities we prove that any function satisfying these two properties is the value function of a zero sum game

differential games with asymmetric information we investigate a two player zero sum differential game in which the players have asymmetric information on the random terminal payoff we prove that the game has a value and characterize this value in terms of dual solutions of some hamilton jacobi equation we also explain how to adapt the results to differential games where the initial position is random

an eulerian method for capturing caustics a robust numerical method for the localization of caustics is proposed for general hamiltonians it is based on the direct resolution of a system of partial differential equations obtained through a local change of the time variable in the hamilton jacobi equation and complemented by a set of transport equations numerical results 1 to 3 d are presented

egalitarian solutions of large games ii the asymptotic approach this is the second of two papers devoted to the study of egalitarian solutions for nontransferable utility ntu games with a large number of players this paper is concerned with the egalitarian solutions of finite games as the number of players increases we show that these converge to the egalitarian solution of the limit game with a continuum of players as defined in our previous paper the same convergence holds for the underlying potential functions these asymptotic results are particularly significant since they provide the definitive justification for our definitions in the limit continuum case

a hierarchy of rational timoshenko dispersion relations abstract this paper shows that the conventional timoshenko dispersion relation for bending waves is the 0 1 member of a two parameter family m n of approximations to the exact dispersion relation obtained from linear elasticity higher members of the family are shown to converge rapidly to the exact dispersion relation in particular any number of branches can be captured accurately over their entire length i e up to arbitrarily high frequencies and wavenumbers the theory admits a rigorous accuracy analysis and demonstrates that the conventional timoshenko dispersion relation is a completely rational approximation thus answering some previous doubts about this matter lower members of the two parameter family include the euler bernoulli dispersion relation and a rayleigh type inertia correction especially useful is the timoshenko 1 2 dispersion relation which extends the conventional timoshenko 0 1 dispersion relation by capturing the first four branches of the exact dispersion relation rather than merely the first two and is accurate over a wide range an inertia corrected timoshenko dispersion relation is also derived and is shown to include many previous approximations as special cases

a continuous time approach for the asymptotic value in two person zero sum repeated games we consider the asymptotic value of two person zero sum repeated games with general evaluations of the stream of stage payoffs we show existence for incomplete information games splitting games and absorbing games the technique of proof consists of embedding the discrete repeated game into a continuous time game and to use viscosity solution tools

experimental study of the timoshenko beam theory predictions abstract the theory of flexural vibrations proposed by timoshenko almost 90 years ago has been the subject of several recent papers in the timoshenko beam theory a critical frequency f c is expected and for frequencies f larger than f c some authors argue that a second spectrum exists this is still the matter of much debate in order to settle this question we have performed experimental measurements of flexural normal mode frequencies and amplitudes which are much more precise than what has been obtained up to now for some purposes cylindrical rods are more convenient whereas in other cases rectangular beams are easier to deal with so we analyze rods and beams of different shapes and sizes we show that below f c all observed resonant peaks can be associated with one of the compressional torsional or timoshenko resonances and that no resonant peaks are left undetected the validity of timoshenko beam theory below f c and that only one spectrum appears in this regime is therefore proved the comparison between theory and experiment for rectangular beams also shows that our experimental setup is appropriate so we can use it confidently for f f c the measurements we report here in this regime show the existence of a second spectrum at least for free free boundary conditions

an error estimate for a new scheme for mean curvature motion in this work we propose a new numerical scheme for the anisotropic mean curvature equation the solution of the scheme is not unique but for all numerical solutions we provide an error estimate between the continuous solution and the numerical approximation our scheme is also applicable to computing the solution to dislocation dynamics equations we also provide some numerical simulations

on hopf s formulas for solutions of hamilton jacobi equations soit h r n r on considere l equation de hamilton jacobi u t h du 0 dans r n 1 r n 0 on etudie les proprietes des formules de hopf pour les solutions

repeated games with lack of information on one side we introduce the dual differential game of a repeated game with lack of information on one side as the natural continuous time version of the dual game introduced by de meyer 1996 a traditional way to study the value of differential games is through discrete time approximations here we follow the opposite approach we identify the limit value of a repeated game in discrete time as the value of a differential game namely we use the recursive structure for the finitely repeated version of the dual game to construct a differential game for which the upper values of the uniform discretization satisfy precisely the same property the value of the dual differential game exists and is the unique viscosity solution of a first order derivative equation which appears implicitely in de meyer and rosenberg 1999 with a limit condition we identify the solution by translating viscosity properties in the primal

fast sweeping methods for static hamilton jacobi equations we propose a new sweeping algorithm which discretizes the legendre transform of the numerical hamiltonian using an explicit formula this formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically the minimization that is related to the legendre transform in our sweeping scheme can either be solved analytically or numerically we illustrate the efficiency and accuracy approach with several numerical examples in two and three dimensions

convergence analysis for a class of high order semi lagrangian advection schemes the convergence properties of a class of high order semi lagrangian schemes for pure advection equations are studied here in the framework of the theory of viscosity solutions we review the general convergence results for discrete time approximation schemes belonging to that class and we prove some a priori estimates in l infty and l2 for the rate of convergence of fully discrete schemes we prove then that a careful coupling of time and space discretizations can allow large time steps in the numerical integration still preserving the accuracy of the solutions several examples of schemes and numerical tests are presented

extensions of the cavu theorem for repeated games with incomplete information on one side this work is devoted to extend several asymptotic results concerning repeated games with incomplete information on one side the model we consider is a generalization of the classical model of aumann and maschler aumann et al aumann rj maschler m stearns re 1995 repeated games with incomplete information mit press cambridge ma to infinite action spaces and partial information we prove an extension of the classical cavu theorem in this model for both the lower and upper value functions using two different methods respectively a probabilistic method based on martingales and a functional one based on approximation schemes for viscosity solutions of hamilton jacobi equations similar to the dual differential approach of laraki laraki r 2002 repeated games with lack of information on one side the dual differential approach math oper res 272 419 440 moreover we show that solutions of these two asymptotic problems provide asymptotically optimal strategies for both players in any game of length n all these results are based on a compact approach which consists in identifying a continuous time problem defined on the time interval 0 1 representing the limit of a sequence of finitely repeated games as the number of repetitions is going to infinity finally our results imply the existence of the uniform value of the infinitely repeated game whenever the value of the non revealing game exists

a geometrical optics based numerical method for high frequency electromagnetic fields computations near fold caustics part i this paper presents an eulerian numerical method for the computation of a bi valued solution of hamilton jacobi type equation in a particular geometric setting more precisely we consider high frequency electromagnetic fields in the vicinity of fold caustics

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

error bounds for monotone approximation schemes for parabolic hamilton jacobi bellman equations we obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation numerical schemes for parabolic hamilton jacobi bellman equations by introducing a new notion of consistency our results are robust and general they improve and extend earlier results by krylov barles and jakobsen we apply our general results to various schemes including crank nicholson type finite difference schemes splitting methods and the classical approximation by piecewise constant controls in the first two cases our results are new and in the last two cases the results are obtained by a new method which we develop here

the value of markov chain games with incomplete information on both sides we consider zero sum repeated games with incomplete information on both sides where the states privately observed by each player follow independent markov chains it generalizes the model introduced by aumann and maschler in the sixties and solved by mertens and zamir in the seventies where the private states of the players were fixed it also includes the model introduced in renault 19 renault j 2006 the value of markov chain games with lack of information on one side math oper res 31 3 490 512 of markov chain repeated games with lack of information on one side where only one player privately observes the sequence of states we prove here that the limit value exists and we obtain a characterization via the mertens zamir system where the nonrevealing value function plugged in the system is now defined as the limit value of an auxiliary nonrevealing dynamic game this nonrevealing game is defined by restricting the players not to reveal any information on the limit behavior of their

matrix compression by common subexpression elimination in this report a method for common subexpression elimination in matrices isexplored the method is applied to several types of matrices occurring in numericalsimulations in all cases the cost of

numerical eulerian method for linearized gas dynamics in the high frequency regime we study the propagation of an acoustic wave in a moving fluid in the high frequency regime we calculate the asymptotic approximation of the solution around a mean flow of this problem using an eulerian method by introducing the stretching matrix deformation tensor for the geometrical optics rays of the linearized euler system we deduce the geometrical spreading this quantity is the key tool for computing the leading order term of the asymptotic expansion thanks to a conservation equation along the group velocity the main contribution is to construct and implement a numerical scheme in the eulerian framework for the eikonal equation and for the transport equation on the stretching matrix we present numerical results for several test cases to study the convergence and validate our approach

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

mindlin plate theory best shear coefficient and higher spectra validity mindlin plate theory predicts three frequency spectra or equivalently three branches to a phase velocity dispersion diagram the lowest of which thew1mode provides rotatory inertia and shear deformation corrections to classical thin plate theory employing consistent truncation procedures to both the mindlin and the exact rayleigh lamb frequency equations valid for long wavelength and low phase velocity one finds thatw1mode agreement is achieved when the shear coefficient takes the valuek 5 6 v the mindlin prediction is then less than 0 5 in error when the wavelength is equal to the plate thickness and less than 1 in error as wavelength approaches zero the previously dismissed mindlinhmode is seen to be in exact frequency or phase velocity agreement with the second slowestshwave in the infinite plate as long as the shear coefficient for this mode takes the valuek 2 12 however thew2mode as with the second frequency spectrum of timoshenko beams should be regarded as the inevitable but meaningless consequence of an otherwise remarkable approximate engineering theory

stochastic games with a single controller and incomplete information we study stochastic games with incomplete information on one side in which the transition is controlled by one of the players we prove that if the informed player also controls the transitions the game has a value whereas if the uninformed player controls the transitions the max min value as well as the min max value exist but they may differ we discuss the structure of the optimal strategies and provide extensions to the case of incom plete information on both sides

finite difference methods for the infinity laplace and p laplace equations abstract we build convergent discretizations and semi implicit solvers for the infinity laplacian and the game theoretical p laplacian the discretizations simplify and generalize earlier ones we prove convergence of the solution of the wide stencil finite difference schemes to the unique viscosity solution of the underlying equation we build a semi implicit solver which solves the laplace equation as each step it is fast in the sense that the number of iterations is independent of the problem size this is an improvement over previous explicit solvers which are slow due to the cfl condition

convergent numerical scheme for singular stochastic control with state constraints in a portfolio selection problem we consider a singular stochastic control problem with state constraints that arises in problems of optimal consumption and investment under transaction costs numerical approximations for the value function using the markov chain approximation method of kushner and dupuis are studied the main result of the paper shows that the value function of the markov decision problem mdp corresponding to the approximating controlled markov chain converges to that of the original stochastic control problem as various parameters in the approximation approach suitable limits all our convergence arguments are probabilistic the main assumption that we make is that the value function be finite and continuous in particular uniqueness of the solutions of the associated hjb equations is neither needed nor available in the generality under which the problem is considered specific features of the problem that make the convergence analysis nontrivial include unboundedness of the state and control space and the cost function degeneracies in the dynamics mixed boundary dirichlet neumann conditions and presence of both singular and absolutely continuous controls in the dynamics finally schemes for computing the value function and optimal control policies for the mdp are presented and illustrated with a numerical study

numerical shape from shading for discontinuous photographic images the height u x y of a continuous lambertian surface of known albedo i e grayness is related to u x y information recoverable from a black and white flash photograph of the surface by the partial differential equation sqrt u 2 x u 2 y n 0 we review the notion of a unique viscosity solution for this equation when n is continuous and a recent unique extension of the viscosity solution when n is discontinuous we prove convergence to this extension for a wide class of the numerical algorithms that converge when n is continuous after discussing the properties of the extension and the order of error in the algorithms simulating the extension we point out warning signs which when observed in the numerical solution usually indicate that the surface is not continuous or that the viscosity solution or its extension does not correspond to the actual surface finally we discuss a method that in some of these cases allows us to correct the simulation and recover the actual surface again

spectral viscosity approximations to hamilton jacobi solutions the spectral viscosity approximate solution of convex hamilton jacobi equations with periodic boundary conditions is studied it is proved in this paper that the approximation and its gradient remain uniformly bounded formally spectral accurate and converge to the unique viscosity solution the l1 convergence rate of the order 1 varepsilon forall varepsilon 0 is obtained

on the valid frequency range of timoshenko beam theory abstract the frequency equation of timoshenko beam theory factorises for hinged hinged end conditions leading to a first and second spectrum of natural frequencies the latter is largely inaccurate and can be isolated and disregarded for the majority of other end conditions when the frequency equation does not factorise one may think in terms of pseudo second spectrum contributions arising when evanescent waves become propagating above the cut off frequency co ag i and it is conjectured that these may have a corrupting effect on the frequency predictions comparisons with measured and simulated frequencies lead to the conclusion that timoshenko predictions above the cut off frequency should be disregarded for those end conditions for which the frequency equation does not factorise

the finite product method in the theory of waves and stability this paper presents a method of analysing the dispersion relation and field shape of any type of wave field for which the dispersion relation is transcendental the method involves replacing each transcendental term in the dispersion relation by a finite product polynomial the finite products chosen must be consistent with the low frequency low wavenumber limit but the method is nevertheless accurate up to high frequencies and high wavenumbers full details of the method are presented for a non trivial example that of anti symmetric elastic waves in a layer the method gives a sequence of polynomial approximations to the dispersion relation of extraordinary accuracy over an enormous range of frequencies and wavenumbers it is proved that the method is accurate because certain gamma function expressions which occur as ratios of transcendental terms to finite products largely cancel out nullifying runge s phenomenon the polynomial approximations which are unrelated to taylor series introduce no spurious branches into the dispersion relation and are ideal for numerical computation the method is potentially useful for a very wide range of problems in wave theory and stability theory

the second spectrum of timoshenko beam theory further assessment a review of contributions and views on the second spectrum of timoshenko beam theory tbt over the past two decades together with some new results are presented it is shown that the timoshenko frequency equation factorises not solely for hinged hinged end conditions as is often claimed but also for guided guided and guided hinged these new cases may be regarded as portions of a multi span hinged hinged beam a higher derivative lagrangian that leads directly to the well known fourth order timoshenko beam equation is reviewed a simple relationship between the so called ostrogradski energy and the mechanical energy is derived for hinged hinged end conditions it is shown that the ostrogradski energy is positive for the first spectrum but negative for the second within some branches of physics this would be sufficient to conclude that the second spectrum is unphysical a numerical example presented by levinson and cooke is re examined using both tbt and exact plane stress elastodynamic theory agreement is excellent for the first spectrum however the second spectrum predictions are not in consistent agreement with any single mode of vibration for long wavelength it is very close to the second asymmetric mode but as wavelength shortens it becomes closer to the second symmetric then the third asymmetric modes the conclusion remains unchanged the second spectrum predictions of tbt should be disregarded

orthogonality relations for fluid structural waves in a three dimensional rectangular duct with flexible walls an exact expression for the fluid coupled structural waves that propagate in a three dimensional rectangular waveguide with elastic walls is presented in terms of the non separable eigenfunctions n y z it is proved that these eigenfunctions are linearly dependent and that an eigenfunction expansion representation of a suitably smooth function f y z converges point wise to that function orthogonality results for the derivatives n y a z are derived which together with a partial orthogonality relation for n y z enable the solution of a wide range of acoustic scattering problems two prototype problems of the type typically encountered in two part scattering problems are solved and numerical results showing the displacement of the elastic walls are presented

a simple filter for detecting low rank submatrices we present a simple algorithm for detecting low rank submatrices from within a much larger matrix this algorithm relies on a basic geometric property of high dimensional space random 2 d projections of eccentric gaussian distributions are typically concentrated in opposite quadrants of the plane

asymptotic equivalence of homogenisation procedures and fine tuning of continuum theories abstract long wave models obtained in the process of asymptotic homogenisation of structures with a characteristic length scale are known to be non unique the term non uniqueness is used here in the sense that various homogenisation strategies may lead to distinct governing equations that usually for a given order of the governing equation approximate the original problem with the same asymptotic accuracy a constructive procedure presented in this paper generates a class of asymptotically equivalent long wave models from an original homogenised theory the described non uniqueness manifests itself in the occurrence of additional parameters characterising the model a simple problem of long wave propagation in a regular one dimensional lattice structure is used to illustrate important criteria for selecting these parameters the procedure is then applied to derive a class of continuum theories for a two dimensional square array of particles applications to asymptotic structural theories are also discussed in particular we demonstrate how to improve the governing equation for the rayleigh love rod and explain the reasons for the well known numerical accuracy of the mindlin plate theory

discrete time high order schemes for viscosity solutions of hamilton jacobi bellman equations a general method for constructing high order approximation schemes for hamilton jacobi bellman equations is given the method is based on a discrete version of the dynamic programming principle we prove a general convergence result for this class of approximation schemes also obtaining under more restrictive assumptions an estimate in l infty of the order of convergence and of the local truncation error the schemes can be applied in particular to the stationary linear first order equation in bbb r n we present several examples of schemes belonging to this class and with fast convergence to the solution

existence of viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games recently m g crandall and p l lions trans amer math soc 277 1983 1 42 introduced the class of viscosity solutions of these equations and proved uniqueness within this class this paper discusses the existence of these solutions under assumptions closely related to the ones which guarantee the uniqueness

a class of reduced order models in the theory of waves and stability this paper presents a class of approximations to a type of wave field for which the dispersion relation is transcendental the approximations have two defining characteristics i they give the field shape exactly when the frequency and wavenumber lie on a grid of points in the frequency wavenumber plane and ii the approximate dispersion relations are polynomials that pass exactly through points on this grid thus the method is interpolatory in nature but the interpolation takes place in frequency wavenumber space rather than in physical space full details are presented for a non trivial example that of antisymmetric elastic waves in a layer the method is related to partial fraction expansions and barycentric representations of functions an asymptotic analysis is presented involving stirling s approximation to the psi function and a logarithmic correction to the polynomial dispersion relation

detecting low rank clusters via random sampling we present an algorithm for detecting a low rank cluster of vectors from within a much larger group of vectors this algorithm relies on a basic geometric property of high dimensional space most of the volume of a typical eccentric ellipsoid is confined to relatively few orthants within the ambient space this simple fact can be used to quickly detect a collection of vectors with low numerical rank from amongst a larger group of vectors with higher numerical rank

existence of value for differential games with incomplete information and signals on initial states and payoffs abstract this article is devoted to proving the existence of the value for a two players differential game with incomplete information and a signal structure on the data initial positions dynamics and payoffs before the game starts the data are chosen randomly in a finite set the players are not informed on the exact data but they receive a signal the main result of the paper says that the value of the game does exist for suitably defined nonanticipative random strategies to obtain such a result we introduce and study a new type of second order double obstacle hamilton jacobi isaacs equation we also prove that the value is the unique viscosity solution of this partial differential equation

hamilton jacobi isaacs equations for differential games with asymmetric information on probabilistic initial condition abstract we investigate hamilton jacobi isaacs equations associated to a two players zero sum differential game with incomplete information the first player has complete information on the initial state of the game while the second player has only information of a possibly uncountable probabilistic nature he knows a probability measure on the initial state such differential games with finite type incomplete information can be viewed as a generalization of the famous aumann maschler theory for repeated games the main goal and novelty of the present work consists in obtaining and investigating a hamilton jacobi isaacs equation satisfied by the upper and the lower values of the game since we obtain a uniqueness result for such hamilton jacobi equation as a byproduct this gives an alternative proof of the existence of a value of the differential game which has been already obtained in the literature by different technics since the hamilton jacobi equation is naturally stated in the space of probability measures we use the wasserstein distance and some tools of optimal transport theory

efficient algorithms for globally optimal trajectories we present serial and parallel algorithms for solving a system of equations that arises from the discretization of the hamilton jacobi equation associated to a trajectory optimization problem of the following type a vehicle starts at a prespecified point x sub o and follows a unit speed trajectory x t inside a region in spl rscr sup m until an unspecified time t that the region is exited a trajectory minimizing a cost function of the form spl int sub 0 sup t r x t dt q x t is sought the discretized hamilton jacobi equation corresponding to this problem is usually solved using iterative methods nevertheless assuming that the function r is positive we are able to exploit the problem structure and develop one pass algorithms for the discretized problem the first algorithm resembles dijkstra s shortest path algorithm and runs in time o n log n where n is the number of grid points the second algorithm uses a somewhat different discretization and borrows some ideas from a variation of dial s shortest path algorithm 1969 that we develop here it runs in time o n which is the best possible under some fairly mild assumptions finally we show that the latter algorithm can be efficiently parallelized for two dimensional problems and with p processors its running time becomes o n p provided that p o spl radic n log n

a multilevel matrix decomposition algorithm for analyzing scattering from large structures a multilevel algorithm is presented for analyzing scattering from electrically large surfaces the algorithm accelerates the iterative solution of integral equations that arise in computational electromagnetics the algorithm permits a fast matrix vector multiplication by decomposing the traditional method of moment matrix into a large number of blocks with each describing the interaction between distant scatterers the multiplication of each block by a trial solution vector is executed using a multilevel scheme that resembles a fast fourier transform fft and that only relies on well known algebraic techniques the computational complexity and the memory requirements of the proposed algorithm are o n log sup 2 n

covariance control problems over martingales with fixed terminal distribution arising from game theory we study several aspects of covariance control problems over martingale processes in mathbb r d with constraints on the terminal distribution arising from the theory of repeated games with incomplete information we show that these control problems are the limits of discrete time stochastic optimization problems called problems of maximal variation of martingales meaning that sequences of optimizers for problems of length n seen as piecewise constant processes on the uniform partition of 0 1 define relatively compact sequences having all their limit points in the set of optimizers of the control problem optimal solutions of this limit problem are then characterized using convex duality techniques and the dual problem is shown to be an unconstrained stochastic control problem characterized by a second order nonlinear pde of hjb type we deduce from this dual relationship that solutions of the control problem are the images by the spatial gradient of the solution of the hjb equation of the

efficient methods for grouping vectors into low rank clusters we present a few practical algorithms for sorting vectors into low rank clusters these algorithms rely on a subdivision scheme applied to the space of projections from d dimensions to 1 dimension this subdivision scheme can be thought of as a higher dimensional generalization of quicksort given the ability to quickly sort vectors into low rank clusters one can efficiently search a matrix for low rank sub blocks of large diameter the ability to detect large diameter low rank sub blocks has many applications ranging from data analysis to matrix compression

fast computation of fourier integral operators we introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation general hyperbolic equations and curvilinear tomography the problem is to numerically evaluate a so called fourier integral operator fio of the form int e 2 pi i phi x xi a x xi hat f xi d xi at points given on a cartesian grid here xi is a frequency variable hat f xi is the fourier transform of the input f a x xi is an amplitude and phi x xi is a phase function which is typically as large as xi hence the integral is highly oscillatory because a fio is a dense matrix a naive matrix vector product with an input given on a cartesian grid of size n by n would require o n 4 operations this paper develops a new numerical algorithm which requires o n 2 5 log n operations and as low as o sqrt n in storage space the constants in front of these estimates are small it operates by localizing the integral over polar wedges with small angular aperture in the frequency plane on each wedge the algorithm factorizes the kernel e 2 pi i phi x xi a x xi into two components 1 a diffeomorphism which is handled by means of a nonuniform fft and 2 a residual factor which is handled by numerical separation of the spatial and frequency variables the key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology we also discuss the potential of our ideas for various applications such as reflection seismology

determination of lamb mode eigenvalues an original method is presented to determine the complex lamb wave spectrum by using a numerical spectral method applied to the elasticity equations this method presents the advantage to directly determine complex wave numbers for a given frequency via a classical matricial eigenvalue problem and allows the wave numbers to be determined at relatively high frequencies i e corresponding to many propagating modes it does not need initial guess values for the wave numbers contrary to the usual method of root finding of the rayleigh lamb frequency equations dispersion relation in the complex plane results are presented and the method is discussed

finite volume schemes for hamilton jacobi equations we introduce two classes of monotone finite volume schemes for hamilton jacobi equations the corresponding approximating functions are piecewise linear defined on a mesh consisting of triangles the schemes are shown to converge to the viscosity solution of the hamilton jacobi equation

elastic waves in timoshenko beams the lost and found of an eigenmode this paper considers propagating waves in elastic bars in the spirit of asymptotic analysis and shows that the inclusion of shear deformation amounts to singular perturbation in the eulerbernoulli

a probabilistic representation for the value of zero sum differential games with incomplete information on both sides we prove that for a class of zero sum differential games with incomplete information on both sides the value admits a probabilistic representation as the value of a zero sum stochastic differential game with complete information where both players control a continuous martingale a similar representation as a control problem over discontinuous martingales was known for games with incomplete information on one side see cardaliaguet rainer 8 and our result is a continuous time analog of the so called splitting game introduced in laraki 20 and sorin 27 in order to analyze discrete time models it was proved by cardaliaguet 4 5 that the value of the games we consider is the unique solution of some hamilton jacobi equation with convexity constraints our result provides therefore a new probabilistic representation for solutions of hamilton jacobi equations with convexity constraints as values of stochastic differential games with unbounded control spaces and unbounded volatility

a convergent difference scheme for a class of partial integro differential equations modeling pricing under uncertainty in this paper we present a finite difference scheme to approximate viscosity solutions of a class of partial integro differential equations describing pricing under model uncertainty we establish

a hierarchy of high order theories for symmetric modes in an elastic layer abstract a hierarchy of high order theories for symmetric modes in an infinitely long elastic layer of the constant thickness is derived by means of the inertia corrected polynomial approximations for each member of the hierarchy boundary conditions for layers of the finite length are formulated the forcing problems at several approximation levels are solved with the use of the bi orthogonality conditions accuracy of these approximations is assessed by comparison of results with the exact solution of the rayleigh lamb problem special attention is paid to the power flow analysis in alternative modal excitation cases and to the applicability of the saint venant s principle in stationary elasto dynamics

games with coupled propagated constraints in optical networks with multi link topologies we consider games in optical networks in the class of m player games with coupled utilities and constraints nash equilibria of such games can be computed based on recent extension of duality to a game theoretical framework this work extends previous results on games with coupled constraints in optical links to multi link topologies coupled constraints in optical networks are propagated along links which introduces additional complexities for analysis specifically convexity of the propagated constraints is no longer automatically ensured we show that convexity is satisfied for single sink multi link topologies the general case of multi links with arbitrary sources and sinks is dealt with by a partitioned game with stages we exploit the single sink structure of each stage and the ladder nested form of the game and we discuss iterative computation of equilibria based on a three level hierarchical algorithm and prove its convergence under certain conditions

approximation of the flexural velocity branch in plates the fitting of the flexural velocity branch by means of the pade approximation is examined its use for free isotropic plates is elaborated and compared to other approaches the flexural velocity expansion into power series which underlies the pade implementation is detailed the generalization for fluid loaded plates and inhomogeneous plates is outlined and the impact of anisotropy is discussed in some detail

the wkb approximation for analysis of wave propagation in curved rods of slowly varying diameter the wentzelkramersbrillouin wkb approximation is applied for asymptotic analysis of time harmonic dynamics of corrugated elastic rods a hierarchy of three models namely the rayleigh and timosh

an asymptotic decoupling method for waves in layered media this paper presents a technique asymptotic decoup ling for analysing wave propagation in multi layered media the technique leads to a hierarchy of approximations to the exact dispersion relatio

on a continuous time game with incomplete information for zero sum two player continuous time games with integral payoff and incomplete information on one side the authors show that the optimal strategy of the informed player can be computed through an auxiliary optimization problem over some martingale measures the authors also characterize the optimal martingale measures and compute them explicitly in several examples

new high resolution semi discrete central schemes for hamilton jacobi equations we introduce a new high resolution central scheme for multidimensional hamilton jacobi equations the scheme retains the simplicity of the non oscillatory central schemes developed by c t lin and e tadmor in press siam j sci comput yet it enjoys a smaller amount of numerical viscosity independent of 1 t by letting t 0 we obtain a new second order central scheme in the particularly simple semi discrete form along the lines of the new semi discrete central schemes recently introduced by the authors in the context of hyperbolic conservation laws fully discrete versions are obtained with appropriate runge kutta solvers the smaller amount of dissipation enables efficient integration of convection diffusion equations where the accumulated error is independent of a small time step dictated by the cfl limitation the scheme is non oscillatory thanks to the use of nonlinear limiters here we advocate the use of such limiters on second discrete derivatives which is shown to yield an improved high resolution when compared to the usual limitation of first derivatives numerical experiments demonstrate the remarkable resolution obtained by the proposed new central scheme

repeated games and partial differential equations let vnp denote the value of the n times repeated zero sum game with incomplete information on one side and full monitoring and let up be the value of the average game gp the error term enp vnp cavup is then converging to zero at least as rapidly as 1 n in this paper we analyze the convergence of np nenp in the games with square payoff matrices such that the optimal strategy of the informed player in the average game gp is unique is completely mixed and does not depend on p our main result is that the existence of a solution to a partial differential equation with appropriate boundary conditions and regularity properties implies the uniform convergence of n to the fenchel conjugate of in particular cases the p d e problem is linear and its solution is then related to the multidimensional normal distribution

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

repeated games duality and the central limit theorem this paper is concerned with the repeated zero sum games with one sided information and standard signaling we introduce here dual games that allow us to analyze the markovian behavior of the uninformed player and to explicitly compute his optimal strategies we then apply our results on the dual games to explain the appearance of the normal density in the n 1 2 term of the asymptotic expansion of vn as a consequence of the central limit theorem

error bounds for monotone approximation schemes for hamilton jacobi bellman equations we obtain error bounds for monotone approximation schemes of hamilton jacobi bellman equations these bounds improve previous results of krylov and the authors the key step in the proof of these new estimates is the introduction of a switching system which allows the construction of approximate almost smooth supersolutions for the hamilton jacobi bellman equation

some estimates for finite difference approximations some estimates for the approximation of optimal stochastic control problems by discrete time problems are obtained in particular an estimate for the solutions of the continuous time versus the discrete time hamilton jacobi bellman equations is given the technique used is more analytic than probabilistic

convergence of muscl and filtered schemes for scalar conservation laws and hamilton jacobi equations this paper considers the questions of convergence of i muscl type i e second order tvd finite difference approximations towards the entropic weak solution of scalar one dimensional conservation laws with strictly convex flux and ii higher order schemes filtered to preserve an upper bound on some weak second order finite differences towards the viscosity solution of scalar multi dimensional hamilton jacobi equations with convex hamiltonians

experimental study of the timoshenko beam theory predictions further results abstract in a previous paper 2012 we presented experimental results proving that the critical frequency f c predicted by timoshenko beam theory indeed exists we also showed that for frequencies f smaller than f c the spectrum is formed by almost equally spaced levels whereas for f f c the spectrum consists of pairs of eigenvalues very close to each other as predicted by numerical solutions of timoshenko s equation we shall refer to them as timoshenko doublets in this work we measure for the first time experimental dispersion relations for this purpose it was necessary to obtain normal mode amplitudes with a high precision which was done with a new experimental setup developed by us we found that experimental dispersion relations coincide very well with theoretical predictions furthermore we provide an explanation of timoshenko doublets

a long wave model for the surface elastic wave in a coated half space the paper deals with the three dimensional problem in linear isotropic elasticity for a coated half space the coating is modelled via the effective boundary conditions on the surface of the substrate initially established on the basis of an ad hoc approach and justified in the paper at a long wave limit an explicit model is derived for the surface wave using the perturbation technique along with the theory of harmonic functions and radon transform the model consists of three dimensional quasi static elliptic equations over the interior subject to the boundary conditions on the surface which involve relations expressing wave potentials through each other as well as a two dimensional hyperbolic equation singularly perturbed by a pseudo differential or integro differential operator the latter equation governs dispersive surface wave propagation whereas the elliptic equations describe spatial decay of displacements and stresses as an illustration the dynamic response is calculated for impulse and moving surface loads the explicit analytical solutions obtained for these cases may be used for the non destructive testing of the thickness of the coating and the elastic moduli of the substrate