convergence and stability of distributed stochastic iterative processes a mathematical framework is proposed for convergence analysis of stochastic iterative processes arising in applications of pseudogradient optimization algorithms the framework is based upon the concepts of stochastic difference inequalities and vector lyapunov functions which are ideally suited for reducing the dimensionality problem arising in testing convergence of distributed parallel schemes by applying the m matrix conditions to a test matrix having a dimension equal to the number of the processor in the scheme one can use the framework to select suitable scaling factors for each individual processor producing a satisfactory convergence rate of the overall iterative process the results provide new convergence tests for distributed iterative processes arising in decentralized extremal regulation adaptation and parameter estimation schemes

diffractive geometric optics for bloch wave packets we study for times of order 1 varepsilon solutions of wave equations which are fancyscript o varepsilon 2 modulations of an varepsilon periodic wave equation the solutions are of slowly varying amplitude type built on bloch plane waves with wavelength of order varepsilon we construct accurate approximate solutions of the three scale wkb type the leading profile is both transported at the group velocity and dispersed by a schrodinger equation given by the quadratic approximation of the bloch dispersion relation at the plane wave a ray average hypothesis of the small divisor type guarantees stability we introduce techniques related to those developed in nonlinear geometric optics which lead to new results even on time scales t fancyscript o 1 a pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems the leading term yields h1 asymptotics when the envelopes are only h1

a bishop phelps bollobas type theorem for uniform algebras abstract this paper is devoted to showing that asplund operators with range in a uniform banach algebra have the bishop phelps bollobas property i e they are approximated by norm attaining asplund operators at the same time that a point where the approximated operator almost attains its norm is approximated by a point at which the approximating operator attains it to prove this result we use the weak to norm fragmentability of weak compact subsets of the dual of asplund spaces and we need to observe a urysohn type result producing peak complex valued functions in uniform algebras that are small outside a given open set and whose image is inside a stolz region

the bishop phelps bollobas theorem for operators we prove the bishop phelps bollobas theorem for operators from an arbitrary banach space x into a banach space y whenever the range space has property of lindenstrauss we also characterize those banach spaces y for which the bishop phelps bollobas theorem holds for operators from 1 into y s everal examples of classes of such spaces are provided for instance the bishop phelps bollobas theorem holds when the range space is finite dimensional an l1 space for a finite measure a c k space for a compact hausdorff space k or a uniformly convex banach space

finite difference schemes for the parabolic equation in a variable depth environment with a rigid bottom boundary condition we consider a linear schrodinger type partial differential equation the parabolic equation of underwater acoustics in a layer of water bounded below by a rigid bottom of variable topography using a change of depth variable technique we transform the problem into one with horizontal bottom for which we establish an a priori h1 estimate and prove an optimal order error bound in the maximum norm for a crank nicolson type finite difference approximation of its solution we also consider the same problem with an alternative rigid bottom boundary condition due to abrahamsson and kreiss and prove again a priori h1 estimates and optimal order error bounds for a crank nicolson scheme

asymptotic analysis of weakly nonlinear bessel gaus beams abstract in this paper we investigate the propagation of conical waves in nonlinear media in particular we are interested in the effects resulting from applying a gaussian apodization to an ideal nondiffracting wave first we present a multiple scales approach to derive amplitude equations for weakly nonlinear conical waves from a governing equation of cubic nonlinear schrodinger type from these equations we obtain asymptotic solutions for the linear and the weakly nonlinear problem for which we state several uniform estimates that describe the deviation from the ideal nondiffracting solution moreover we show numerical simulations based on an implementation of our amplitude equations to support and illustrate our analytical results

a new iterative solver for the time harmonic wave equation the time harmonic wave equation also known as the helmholtz equation is obtained if the constant density acoustic wave equation is transformed from the time domain to the frequency domain its discretization results in a large sparse linear system of equations in two dimensions this system can be solved efficiently by a direct method in three dimensions direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large iterative methods are an alternative these methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence the iterative solution of the time harmonic wave equation has long been a notoriously difficult problem in numerical analysis recently a new preconditioner based on a strongly damped wave equation has heralded a breakthrough the solution of the linear system associated with the preconditioner is approximated by another iterative method the multigrid method the multigrid method fails for the original wave equation but performs well on the damped version the performance of the new iterative solver is investigated on a number of 2d test problems the results suggest that the number of required iterations increases linearly with frequency even for a strongly heterogeneous model where earlier iterative schemes fail to converge complexity analysis shows that the new iterative solver is still slower than a time domain solver to generate a full time series we compare the time domain numeric results obtained using the new iterative solver with those using the direct solver and conclude that they agree very well quantitatively the new iterative solver can be applied straightforwardly to 3d problems

general decentralized stabilization of large scale linear continuous and discrete time delay systems we consider the general stabilization problem of linear time invariant large scale multivariable decentralized systems with delays we have established sufficient stabilization conditions for a general class of decentralized systems namely in the continuous case the requirement is that the matrix measure of the closed loop subsystem matrices can be made sufficiently negative by appropriate local feedback control whereas in the discrete case the requirement is that the norm of the closed loop subsystem matrices can be made arbitrarily small by feedback control

bishop phelps bollobas property for certain spaces of operators abstract we characterize the banach spaces y for which certain subspaces of operators from l 1 into y have the bishop phelps bollobas property in terms of a geometric property of y namely ahsp this characterization applies to the spaces of compact and weakly compact operators new examples of banach spaces y with ahsp are provided we also obtain that certain ideals of asplund operators satisfy the bishop phelps bollobas property

an inclusion principle for dynamic systems the purpose of this paper is to present a detailed study of the inclusion concept in dynamic systems which is a suitable mathematical framework for comparing systems with different dimensions the framework offers immediate results in reduced order modeling and the overlapping decentralized control of complex systems the presentation which is limited to linear constant systems relies on both the matrix algebra computations and the geometric elements structure to provide a balanced view of the issues involved in the concept of inclusion the framework is quite broad and has been used to consider nonlinear and time varying systems as well as systems with hereditary and stochastic effects

error estimates for finite difference methods for a wide angle parabolic equation we consider a model initial and boundary value problem for a third order partial differential equation pde a wide angle parabolic equation frequently used in underwater acoustics with depth and range dependent coefficients in the presence of horizontal interfaces and dissipation after commenting on the existence uniqueness theory of solution of the equation we discretize the problem by a second order finite difference method of crank nicolson type for which we prove stability and optimal order error estimates in suitable discrete l2 h1 and maximum norms we also prove under certain conditions that the forward euler scheme is also stable and convergent for the problem at hand

absorbing boundary conditions for second order hyperbolic equations abstract a uniform approach to construct absorbing artificial boundary conditons for second order linear hyperbolic equations is proposed the nonlocal boundary condition is given by a pseudodifferential operator that annihilates travelling waves it is obtained through the dispersion relation of the differential equation by requiring that the initial boundary value problem admits the wave solutions travelling in one direction only local approximations of this global boundary condition yield an nth order differential operator it is shown that the best approximations must be in the canonical forms which can be factorized into first order operators these boundary conditions are perfectly absorbing for wave packets propagating at certain group velocities a hierarchy of absorbing boundary conditions is derived for transonic small perturbation equations of unsteady flows these examples illustrate that the absorbing boundary conditions are easy to derive and the effectiveness is demonstrated by the numerical experiments

the stabilization of high order multistep schemes for the laguerre one way wave equation solver abstract this paper considers spectral finite difference methods of a high order of accuracy for solving the one way wave equation using the laguerre integral transform with respect to time as the base in order to provide a high spatial accuracy and stability the richardson method can be employed however such an approach requires high computer costs therefore we consider alternative algorithms based on the adams multistep schemes to reach the stability for the one way equation the stabilizing procedures using the spline interpolation were developed this made it possible to efficiently implement a predictor corrector type method thus decreasing computer costs the stability and accuracy of the procedures proposed have been studied based on the implementation of the migration algorithm within a problem of seismic prospecting

galerkin methods for parabolic and schrodinger equations with dynamical boundary conditions and applications to underwater acoustics in this paper we consider galerkin finite element methods that approximate the solutions of initial boundary value problems in one space dimension for parabolic and schrodinger evolution equations with dynamical boundary conditions error estimates of optimal rates of convergence in l 2 and h 1 are proved for the associated semidiscrete and fully discrete crank nicolson galerkin approximations the problem involving the schrodinger equation is motivated by considering the standard parabolic paraxial approximation to the helmholtz equation used in underwater acoustics to model long range sound propagation in the sea in the specific case of a domain with a rigid bottom of variable topography this model is contrasted with alternative ones that avoid the dynamical bottom boundary condition and are shown to yield qualitatively better approximations in the real parabolic case numerical approximations are considered for dynamical boundary conditions of reactive and dissipative type

the bishop phelps bollobas theorem for l l1 μ l 0 1 abstract we show that the bishop phelps bollobas theorem holds for all bounded operators from l 1 into l 0 1 where is a finite measure

simultaneously continuous retraction and bishop phelps bollobas type theorem abstract the dual space x of a banach space x is said to admit a uniformly simultaneously continuous retraction if there is a retraction r from x onto its unit ball b x which is uniformly continuous in norm topology and continuous in weak topology we prove that if a banach space resp complex banach space x has a normalized unconditional schauder basis with unconditional basis constant 1 and if x is uniformly monotone resp uniformly complex convex then x admits a uniformly simultaneously continuous retraction it is also shown that x admits such a retraction if x x i c 0 or x x i l 1 where x i is a family of separable banach spaces whose duals are uniformly convex with moduli of convexity i e with inf i i e 0 for all 0 e 1 let k be a locally compact hausdorff space and let c 0 k be the real banach space consisting of all real valued continuous functions vanishing at infinity as an application of simultaneously continuous retractions we show that a pair x c 0 k has the bishop phelps bollobas property for operators if x admits a uniformly simultaneously continuous retraction as a corollary c 0 s c 0 k has the bishop phelps bollobas property for operators for every locally compact metric space s

generalized decompositions of dynamic systems and vector lyapunov functions the notion of decomposition is generalized to provide more freedom in constructing vector lyapunov functions for stability analysis of nonlinear dynamic systems a generalized decomposition is defined as a disjoint decomposition of a system which is obtained by expanding the state space of a given system an inclusion principle is formulated for the solutions of the expansion to include the solutions of the original system so that stabliity of the expansion implies stability of the original system stability of the expansion can then be established by standard disjoint decompositions and vector lyapunov functions the applicability of the new approach is demonstrated using the lotka voiterra equations

arbitrarily wide angle wave equations for complex media by combining various ideas related to one way wave equations owwes half space stiffness relation special finite element discretization and complex coordinate stretching a systematic procedure is developed for deriving a series of highly accurate space domain versions of owwes the resulting procedure is applicable to complex media where the governing equation full wave equation is a second order differential system making the procedure applicable for general heterogeneous anisotropic porous viscoelastic media owing to their high accuracy in representing waves propagating in an arbitrarily wide range of angles the resulting equations are named arbitrarily wide angle wave equations awwes in order to illustrate the proposed procedure awwes are derived for one way propagation in acoustic as well as elastic media while acoustic awwes can be considered as modified versions of well known space domain owwes based on rational approximations of the square root operator the elastic awwes are significantly different from the existing elastic owwes unlike the existing elastic owwes elastic awwes are displacement based and are applicable to general anisotropic media furthermore awwes are simple in their form and appear amenable to easy numerical implementation

the bishop phelps bollobas property for operators from c0 into some banach spaces abstract we exhibit a new class of banach spaces y such that the pair c 0 y has the bishop phelps bollobas property for operators this class contains uniformly convex banach spaces and spaces with the property of lindenstrauss we also provide new examples of spaces in this class

parabolic wave equation approximations in heterogenous media the properties of different variants of parabolic approximations of scalar wave equations are analyzed these equations are of general form which includes those used in seismology underwater acoustics and other applications a new version of the parabolic approximation is derived for heterogeneous media it has optimal properties with respect to wave reflection at material interfaces the amplitudes of the reflected and transmitted waves depend continuously on the interface existence uniqueness and energy estimates are proved

crank nicolson finite element discretizations for a two dimensional linear schrodinger type equation posed in a noncylindrical domain first published in mathematics of computation online 2014 84 2015 1571 1598 published by the american mathematical society

on the bishop phelps bollobas theorem for multilinear mappings abstract we study the bishop phelps bollobas property and the bishop phelps bollobas property for numerical radius our main aim is to extend some known results about norm or numerical radius attaining operators to multilinear and polynomial cases we characterize the pair l 1 x y to have the bpbp for bilinear forms and prove that on l 1 the numerical radius and the norm of a multilinear mapping are the same we also show that l 1 fails the bpbp nu for multilinear mappings although l 1 satisfies it in the operator case for every measure

a note on the derivation of paraxial equation in nonhomogeneous media the paraxial approximation of the two dimensional and three dimensional wave equations in nonhomogeneous media is discussed it is shown that the preferred direction of propagation is along the bicharacteristics of the wave equation the paraxial equation is obtained by assuming that in this direction the variation of the propagating wave is slow enough to make negligible its second derivative the best way to get bicharacteristics is to consider them as the null geodesics of some riemannian space

discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics this paper is concerned with transparent boundary conditions tbcs for wide angle parabolic equations wapes in the application to underwater acoustics assuming cylindrical symmetry existing discretizations of these tbcs introduce slight numerical reflections at this artificial boundary and also render the overall crank nicolson finite difference method only conditionally stable here a novel discrete tbc is derived from the fully discretized whole space problem that is reflection free and yields an unconditionally stable scheme while we shall assume a uniform discretization in range the interior depth discretization i e in the water column may be nonuniform and we shall discuss strategies for the best exterior discretization i e in the sea bottom the superiority of the new discrete tbc over existing discretizations is illustrated on several benchmark problems in the literature different wapes or wape and the standard parabolic equation have been coupled in the water and the sea bottom we analyze under which conditions this yields a hybrid model that is conservative for the acoustic field

efficient numerical simulation for long range wave propagation we develop an efficient algorithm for simulating wave propagation over long distances with both weak and strong scatterers in domains with weak heterogeneities the wave field is decomposed into forward propagating and back scattered modes using two coupled parabolic equations in the region near strong scatterers the helmholtz equation is used to capture the strong scattering events the key idea in our method is to combine these two regimes using a combined domain decomposition and wave decomposition method a transparent interface condition is derived to couple these two regions together numerical examples show that the simulated field is close to the field obtained using the full helmholtz equation in the whole domain

the bishop phelps bollobas theorem for operators from l1 μ to banach spaces with the radon nikodým property abstract let y be a banach space and be a finite measure space where is an infinite algebra of measurable subsets of we show that if the couple l 1 y has the bishop phelps bollobas property for operators then y has the ahsp further for a banach space y with the radon nikod m property we prove that the couple l 1 y has the bishop phelps bollobas property for operators if and only if y has the ahsp

transparent boundary conditions for split step pade approximations of the one way helmholtz equation in this paper we generalize the nonlocal discrete transparent boundary condition introduced by f schmidt and p deuflhard 1995 comput math appl 29 53 76 and by f schmidt and d yevick 1997 j comput phys 134 96 107 to propagation methods based on arbitrary pade approximations of the two dimensional one way helmholtz equation our approach leads to a recursive formula for the coefficients appearing in the nonlocal condition which then yields an unconditionally stable propagation method

the bishop phelps bollobas theorem for operators on l1 μ in this paper we show that the bishop phelps bollobas theorem holds for l l1 l1 for all measures and and also holds for l l1 l for every arbitrary measure and every localizable measure finally we show that the bishop phelps bollobas theorem holds for two classes of bounded linear operators from a real l1 into a real c k if is a finite measure and k is a compact hausdorff space in particular one of the classes includes all bochner representable operators and all weakly compact operators

higher order paraxial wave equation approximations in heterogenous media a new family of paraxial wave equation approximations is derived these approximations are of higher order accuracy than the parabolic approximation and they can be applied to the same computationa

the bishop phelps bollobas property for lipschitz maps abstract in this paper we introduce and study a lipschitz version of the bishop phelps bollobas property lip bpb property this property deals with the possibility of making a uniformly simultaneous approximation of a lipschitz map f and a pair of points at which f almost attains its norm by a lipschitz map g and a pair of points such that g strongly attains its norm at the new pair of points we first show that if m is a finite pointed metric space and y is a finite dimensional banach space then the pair m y has the lip bpb property and that both finiteness assumptions are needed next we show that if m is a uniformly gromov concave pointed metric space i e the molecules of m form a set of uniformly strongly exposed points then m y has the lip bpb property for every banach space y we further prove that this is the case for finite concave metric spaces ultrametric spaces and holder metric spaces the extension of the lip bpb property from m r to some banach spaces y and some results for compact lipschitz maps are also discussed

an inclusion principle for hereditary systems abstract given two hereditary dynamic systems having different dimensions the conditions are provided under which a part of the motion of the larger system is reproduced by the smaller system that is the larger system includes the smaller one the conditions for inclusion are useful in applying the concept of vector liapunov functions to stability analysis of systems composed of overlapping subsystems by expanding the systems into a larger space the overlapping subsystems appear as disjoint and standard methods can be used to conclude stability of the expanded system under the inclusion conditions stability of the expansion implies stability of the original system an example is provided to show stability where the standard disjoint decompositions fail

parabolic and gaussian white noise approximation for wave propagation in random media the parabolic or forward scattering approximation has been used extensively in the study of wave propagation this approximation is combined with a gaussian white noise approximation for waves propagating in a random medium the validity of this approximation is proved for stratified weakly fluctuating random media in the high frequencies regime the limiting distribution of the wave field is characterized as the unique solution of a random schrodinger equation studied by dawson and papanicolaou appl math optim 12 1984 pp 97 114 the proofs are based on various generalizations of the perturbed test function method developed by kushner approximation and weak convergence methods for random processes mit press 1984

bishop phelps bollobas property for bilinear forms on spaces of continuous functions it is shown that the bishop phelps bollobas theorem holds for bilinear forms on the complex c 0 l 1 times c 0 l 2 for arbitrary locally compact topological hausdorff spaces l 1 and l 2

the bishop phelps bollobas property for operators between spaces of continuous functions abstract we show that the space of bounded linear operators between spaces of continuous functions on compact hausdorff topological spaces has the bishop phelps bollobas property a similar result is also proved for the class of compact operators from the space of continuous functions vanishing at infinity on a locally compact and hausdorff topological space into a uniformly convex space and for the class of compact operators from a banach space into a predual of an l 1 space

decentralized control using quasi block diagonal dominance of transfer function matrices the purpose of this paper is to present a generalization of the nyquist array method to blockwise decompositions which is based upon a new version of block diagonal dominance an additional flexibility of the proposed method is in partitioning of the system matrices into disjoint as well as overlapping submatrices which increases considerably the class of control systems which can be designed via block diagonal dominance within this framework the controllers for individual subsystems can be designed independently of each other so that their union represents a decentralized controller for the overall system