absorbing boundary conditions for difference approximations to the multi dimensional wave equation construction des conditions aux limites absorbantes pour l equation des ondes multidimensionnelle a l aide d approximations aux differences formulation sous forme canonique des conditions aux limites absorbantes analytiques

domain decomposition method for harmonic wave propagation a general presentation in this paper we give a general presentation of non overlapping domain decomposition methods for harmonic wave propagation models our abstract framework leads to concise convergence proofs and contains some recent developments about the use of non local transmission conditions it also includes applications to acoustic electromagnetic or elastic waves as well as the treatment of space discretization

optimized schwarz methods without overlap for the helmholtz equation the classical schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel convergence is achieved through overlap of the subdomains we study in this paper a variant of the schwarz method which converges without overlap for the helmholtz equation we show that the key ingredients for such an algorithm are the transmission conditions we derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps these conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the schwarz method this leads to an algorithm in the class of optimized schwarz methods we present an asymptotic analysis of the optimized schwarz method for two types of transmission conditions robin conditions and transmission conditions with second order tangential derivatives numerical results illustrate the effectiveness of the optimized schwarz method on a model problem and on a problem from industry

a sweeping preconditioner for time harmonic maxwell s equations with finite elements this paper is concerned with preconditioning the stiffness matrix resulting from finite element discretizations of maxwell s equations in the high frequency regime the moving pml sweeping preconditioner first introduced for the helmholtz equation on a cartesian finite difference grid is generalized to an unstructured mesh with finite elements the method dramatically reduces the number of gmres iterations necessary for convergence resulting in an almost linear complexity solver numerical examples including electromagnetic cloaking simulations are presented to demonstrate the efficiency of the proposed method

probability inequalities for empirical processes and a law of the iterated logarithm given a class f of functions on x we can view vn as a stochastic process indexed by f and consider limit theorems for this process to prove such theorems it is often helpful to have bounds on the tail of the r v sup w i n f i see for example dudley 1978 kuelbs and dudley 1980 dudley and phillip 1983 or gine and zinn 1983 our main question of interest is for what classes f can best possible bounds be obtained so first we must ask what are these best possible bounds of course

the method of polarized traces for the 2d helmholtz equation we present a solver for the 2d high frequency helmholtz equation in heterogeneous acoustic media with online parallel complexity that scales optimally as o n l where n is the number of volume unknowns and l is the number of processors as long as l grows at most like a small fractional power of n the solver decomposes the domain into layers and uses transmission conditions in boundary integral form to explicitly define polarized traces i e up and down going waves sampled at interfaces local direct solvers are used in each layer to precompute traces of local green s functions in an embarrassingly parallel way the offline part and incomplete green s formulas are used to propagate interface data in a sweeping fashion as a preconditioner inside a gmres loop the online part adaptive low rank partitioning of the integral kernels is used to speed up their application to interface data the method uses second order finite differences the complexity scalings are empirical but motivated by an analysis of ranks of off diagonal blocks of oscillatory integrals they continue to hold in the context of standard geophysical community models such as bp and marmousi 2 where convergence occurs in 5 to 10 gmres iterations while the parallelism in this paper stems from decomposing the domain we do not explore the alternative of parallelizing the systems solves with distributed linear algebra routines

an iterative method for the helmholtz equation an iterative algorithm for the solution of the helmholtz equation is developed the algorithm is based on a preconditioned conjugate gradient iteration for the normal equations the preconditioning is based on an ssor sweep for the discrete laplacian numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm

sweeping preconditioner for the helmholtz equation hierarchical matrix representation the paper introduces the sweeping preconditioner which is highly efficient for iterative solutions of the variable coefficient helmholtz equation including very high frequency problems the first central idea of this novel approach is to construct an approximate factorization of the discretized helmholtz equation by sweeping the domain layer by layer starting from an absorbing layer or boundary condition given this specific order of factorization the second central idea is to represent the intermediate matrices in the hierarchical matrix framework in two dimensions both the construction and the application of the preconditioners are of linear complexity the generalized minimal residual method gmres solver with the resulting preconditioner converges in an amazingly small number of iterations which is essentially independent of the number of unknowns this approach is also extended to the three dimensional case with some success numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach

on behaviour of solutions for a nonlinear viscoelastic equation with variable exponent nonlinearities abstract this paper deals with a viscoelastic wave equation with variable exponent nonlinearities subject to nonlinear boundary feedback under appropriate conditions a general decay result associated to solution energy is proved it is also shown that regarding arbitrary positive initial energy solutions blow up in a finite time

a uniform central limit theorem for set indexed partial sum processes with finite variance on obtient des resultats de convergence faible qui ameliorent ceux de pyke 1983 et de bass et pyke 1984

sample moduli for set indexed gaussian processes on etudie le comportement de la trajectoire de processus gaussiens wp indexes par des classes c de sous ensembles d un espace de probabilite x 7b a p

stabilization of wave and plate like equations with nonlinear dissipation on the boundary abstract in this paper we consider second order in time equations of petrovsky type wave and plate like with nonlinear dissipative boundary conditions under certain geometric conditions imposed on the domain the results on asymptotic behavior of the solutions are established

a weakly nonlinear analysis of elasto plastic microstructure models at certain critical values of the hardening modulus the governing equations of elasto plastic flow may lose their hyperbolicity and exhibit two modes of ill posedness shear band and flutter ill posedness these modes of ill posedness are characterized by the uncontrolled growth of modes at infinitely fine scales which ultimately violates the continuum assumption in previous work l an and a peirce siam j appl math 54 1994 pp 708 730 a continuum model accounting for microscale deformations was built linear analysis demonstrated the regularizing effect of the microstructure and provided a relationship between the width of the localized instabilities and the microlength scale in this paper a weakly nonlinear analysis is used to explore the immediate post critical behavior of the solutions for both one dimensional and anti plane shear models post critical deformations in the plastic regions are shown to be governed by the boussinesq equation one of the completely integrable pdes having so

estimation of the marginal expected shortfall the mean when a related variable is extreme abstract denote the loss return on the equity of a financial institution as x and that of the entire market as y for a given very small value of p 0 the marginal expected shortfall mes is defined as e x y qy 1 p where qy 1 p is the 1 p th quantile of the distribution of y the mes is an important factor when measuring the systemic risk of financial institutions for a wide nonparametric class of bivariate distributions we construct an estimator of the mes and establish the asymptotic normality of the estimator when p 0 as the sample size n since we are in particular interested in the case p o 1 n we use extreme value techniques for deriving the estimator and its asymptotic behavior the finite sample performance of the estimator and the adequacy of the limit theorem are shown in a detailed simulation study we also apply our method to estimate the mes of three large u s investment banks

second order absorbing boundary conditions for the wave equation a solution for the corner problem the treatment of domains with corners in the problem of absorbing boundary conditions for the wave equation is very important from a practical point of view a technical difficulty appears as soon as conditions of order greater than or equal to 2 are considered a solution is proposed for the two dimensional case when second order conditions are used this solution consists of prescribing an adequate corner condition the problem thus obtained is analyzed theoretically and the condition is proved to be optimal the results obtained here are illustrated by numerical simulations some extensions to higher space dimensions and higher order conditions are proposed

rates of convergence in active learning we study the rates of convergence in generalization error achievable by active learning under various types of label noise additionally we study the general problem of model selection for active learning with a nested hierarchy of hypothesis classes and propose an algorithm whose error rate provably converges to the best achievable error among classifiers in the hierarchy at a rate adaptive to both the complexity of the optimal classifier and the noise conditions in particular we state sufficient conditions for these rates to be dramatically faster than those achievable by passive learning

global attractors and their hausdorff dimensions for a class of kirchhoff models this paper studies the existence regularity and hausdorff dimensions of global attractors for a class of kirchhoff models arising in elastoplastic flow utt div u m 1 u ut 2u h ut g u f x it proves that under rather mild conditions the dynamical system associated with above mentioned models possesses in phase space x a global attractor which has further regularity in x 1 x and has finite hausdorff dimension for application the fact shows that for the concerned elastoplastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space and the dimension of the attractor that is the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow is finite

a quasi optimal non overlapping domain decomposition algorithm for the helmholtz equation this paper presents a new non overlapping domain decomposition method for the helmholtz equation whose effective convergence is quasi optimal these improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the dirichlet to neumann operator a convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions

a preconditioned dual primal finite element tearing and interconnecting method for solving three dimensional time harmonic maxwell s equations a new preconditioned dual primal nonoverlapping domain decomposition method is proposed for the finite element solution of three dimensional large scale electromagnetic problems with the aid of two lagrange multipliers the new method converts the original volumetric problem to a surface problem by using a higher order transmission condition at the subdomain interfaces to significantly improve the convergence of the iterative solution of the global interface equation similar to the previous version a global coarse problem related to the degrees of freedom at the subdomain corner edges is formulated to propagate the residual error to the whole computational domain at each iteration which further increases the rate of convergence in addition a fully algebraic preconditioner based on matrix splitting is constructed to make the proposed domain decomposition method even more robust and scalable perfectly matched layers pmls are considered for the boundary truncation when solving open region problems the influence of the pml truncation on the convergence performance is investigated by examining the convergence of the transmission condition for an interface inside the pml numerical examples including wave propagation and antenna radiation problems truncated with pmls are presented to demonstrate the validity and the capability of this method

global existence and nonexistence in a system of petrovsky in this paper we consider the nonlinearly damped semilinear petrovsky equationutt 2u aut ut m 2 bu u p 2in a bounded domain where a b 0 we prove the existence of a local weak solution and show that this solution blows up in finite time if p m and the energy is negative we also show that the solution is global if m p

the central limit theorem for empirical processes on vapnik cervonenkis classes sufficient conditions which under weak additional assumptions are also necessary are found for the central limit theorem for empirical processes indexed by classes of functions which have the vapnik cervonenkis property this improves an earlier theorem of pollard 1982a and leads to necessary and sufficient conditions for the clt for weighted empirical processes indexed by vapnik cervonenkis classes of sets

convergence analysis for a second order schwarz method for nonoverlapping subdomains abstract a second order schwarz method for elliptic partial differential equations is developed and analyzed at a continuous level which can be parallel implemented with local communications it uses a second order boundary condition in terms of poincare steklov operators as its transmission condition on the interfaces the convergence of the algorithm in h 1 norm is obtained by using energy estimates and convergence results hold on arbitrary geometry of subdomains with no cross points

global attractor for a class of kirchhoff models the paper studies the longtime behavior of solutions to the initial boundary value problem ibvp for a class of kirchhoff models arising in elastoplastic flow utt div u m 1 u ut 2u h ut g u f x by combining the decomposition idea with the operate technique it proves that under rather mild conditions the dynamical system associated with above mentioned ibvp possesses in different phase spaces a global attractor which is connected respectively for application the fact shows that for the concerned elastoplastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space

analysis on block diagonal and triangular preconditioners for a pml system of an electromagnetic scattering problem abstract we shall propose several block triangular preconditioners for a pml system of an electromagnetic wave scattering problem and analyze the spectral behavior of the preconditioned systems when the pml system is discretized by edge element methods it results in a discrete system with its stiffness matrix being complex symmetric but indefinite which can be formulated into a real symmetric but indefinite saddle point system in order to preserve the symmetry of the coefficient matrix we present block triangular preconditioners with two sided preconditioning for the discrete pml system we will estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices respectively on the other hand one may also like to apply some iteration methods for nonsymmetric linear systems in applications although the discrete systems are symmetric to this end we propose a block triangular preconditioner to precondition the systems only from one side and analyze the spectrum of the preconditioned systems in addition we have also established a spectral estimate of the preconditioned system by an effective preconditioner that was recently developed in literature numerical experiments are presented to demonstrate the effectiveness and robustness of these new preconditioners and our theoretical predictions on the spectral bounds of the preconditioned systems

a domain decomposition method for the helmholtz equation and related optimal control problems we present an iterative domain decomposition method to solve the helmholtz equation and related optimal control problems the proof of convergence of this method relies on energy techniques this method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problems in homogeneous and heterogeneous media

global nonexistence for a semilinear petrovsky equation abstract in this paper we consider a semilinear petrovsky equation with damping and source terms it is proved that the solution blows up in finite time if the positive initial energy satisfies a suitable condition moreover for the linear damping case we show that the solution blows up in finite time even for vanishing initial energy this is an important breakthrough since it is only well known that the solution blows up in finite time if the initial energy is negative from all the previous literature

some nonoverlapping domain decomposition methods the purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving second order elliptic problems in two and three dimensions the methods under scrutiny fall into two major categories the substructuring type methods and the neumann neumann type methods the basic framework used for analysis is the parallel subspace correction method or additive schwarz method and other technical tools include local global and global local techniques the analyses for both two and three dimensional cases are carried out simultaneously some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived

global existence asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term abstract the paper studies the global existence asymptotic behavior and blowup of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative term it proves that under rather mild conditions on nonlinear terms and initial data the above mentioned problem admits a global weak solution and the solution decays exponentially to zero as t respectively in the states of large initial data and small initial energy in particular in the case of space dimension n 1 the weak solution is regularized to be a unique generalized solution and if the conditions guaranteeing the global existence of weak solutions are not valid then under the opposite conditions the solutions of above mentioned problem blow up in finite time and an example is given

rates of growth and sample moduli for weighted empirical processes indexed by sets probability inequalities are obtained for the supremum of a weighted empirical process indexed by a vapnik cervonenkis class c of sets these inequalities are particularly useful under the assumption p c c p c t 0 as t 0 they are used to obtain almost sure bounds on the rate of growth of the process as the sample size approaches infinity to find an asymptotic sample modulus for the unweighted empirical process and to study the ratio pn p of the empirical measure to the actual measure

optimized schwarz methods optimized schwarz methods are a new class of schwarz methods with greatly enhanced convergence properties they converge uniformly faster than classical schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical schwarz methods if the overlap is of the order of the mesh parameter which is often the case in practical applications they achieve this performance by using new transmission conditions between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem we analyze in this paper these new methods for symmetric positive definite problems and show their relation to other modern domain decomposition methods like the new finite element tearing and interconnect feti variants

new nonoverlapping domain decomposition methods for the harmonic maxwell system we study a nonoverlapping domain decomposition method for the harmonic maxwell equations with a new kind of interface condition using fourier analysis we derive suitable families of transmission conditions in r3 that involve second order tangential differential operators and that guarantee convergence for both propagative and evanescent modes such families depend upon parameters that are chosen to optimize the convergence rate of the corresponding iterative algorithm we then propose iterative solvers for the maxwell equations based on a domain decomposition procedure where such conditions are enforced on the interface some numerical results for a two domain decomposition show the effectiveness of the optimized interface conditions

on the general decay of a nonlinear viscoelastic plate equation with a strong damping and p x t laplacian abstract in this paper we consider a nonlinear viscoelastic plate equation with a lower order perturbation of a p x t laplacian operator of the form u t t 2 u p x t u 0 t g t s u s d s u t f u 0 x t q t 0 t associated with initial and dirichlet neumann boundary conditions under suitable conditions on g f and the variable exponent of the p x t laplacian operator we prove a general decay result in the presence of a strong damping u t 0 acting in the domain this equation corresponds to a viscoelastic version arising in dynamics of elastoplastic flows and plate vibrations

decay estimates for second order evolution equations with memory abstract this paper develops a unified method to derive decay estimates for general second order integro differential evolution equations with semilinear source terms depending on the properties of convolution kernels at infinity we show that the energy of a mild solution decays exponentially or polynomially as t our approach is based on integral inequalities and multiplier techniques these decay results can be applied to various partial differential equations we discuss three examples a semilinear viscoelastic wave equation a linear anisotropic elasticity model and a petrovsky type system

convergence estimates for an higher order optimized schwarz method for domains with an arbitrary interface optimized schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations optimized schwarz methods employ a first or higher order boundary condition along the artificial interface to accelerate convergence in the literature the analysis of optimized schwarz methods relies on fourier analysis and so the domains are restricted to be regular rectangular in this paper we express the interface operator of an optimized schwarz method in terms of poincare steklov operators this enables us to derive an upper bound of the spectral radius of the operator arising in this method of 1 o h 1 4 on a class of general domains where h is the discretization parameter this is the predicted rate for a second order optimized schwarz method in the literature on rectangular subdomains and is also the observed rate in numerical simulations