large time behavior of solutions for a system of nonlinear damped wave equations abstract we consider the cauchy problem of the semilinear damped wave system t 2 u u t u f u t 0 x r n u j 0 x a j x t u j 0 x b j x x r n where u t x u 1 t x u m t x with m 2 and j 1 m we show the asymptotic behavior of solutions under the sharp condition on the nonlinear exponents which is a natural extension of the results for the single nonlinear damped wave equations nishihara 2003 21 hayashi et al 2004 9 hosono and ogawa 2004 10 the proof is based on the l p l q type decomposition of the fundamental solutions of the linear damped wave equations into the dissipative part and hyperbolic part hosono and ogawa 2004 10 nishihara 2003 21

scaling variables and asymptotic expansions in damped wave equations abstract we study the long time behavior of small solutions to the nonlinear damped wave equationeu u a u n u u u r 0 whereeis a positive not necessarily small parameter we assume that the diffusion coefficienta converges to positive limitsa as and that the nonlinearity n u u u vanishes sufficiently fast asu 0 introducing scaling variables and using various energy estimates we compute an asymptotic expansion of the solutionu in powers of 1 2as and we show that this expansion is entirely determined up to the second order by a linear parabolic equation which depends only on the limiting valuesa in particular this implies that the small solutions of the damped wave equation behave for large like those of the parabolic equation obtained by settinge 0

renormalization group and asymptotics of solutions of nonlinear parabolic equations we present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations the method does not rely on a priori estimates such as the maximum principle it applies to systems of coupled equations to boundary conditions at infinity creating a front and to higher possibly fractional differential linear terms we present in detail the analysis for nonlinear diffusion type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity in an accompanying paper 5 the method is applied to systems of equations where some variables are slaved such as the complex ginzburg landau equation c 1994 john wiley sons inc

convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping we consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time asymptotically the hyperbolic model may be viewed as isentropic euler equations with friction term added to the momentum equation to model gas flow through a porous media in this case our result justifies darcy s law time asymptotically our model may also be viewed as an elastic model with damping

large time behavior of global solutions of the semilinear damped beam equation with slowly decaying data abstract we discuss the existence of the global solution and its asymptotic profile for the cauchy problem for the nonlinear damped beam equation we show that the equation admits a unique global solution with small slowly decaying data e g 1 x 2 k 2 0 k 1 moreover we show that the solution can be approximated by the solution of the linear heat equation with suitable data the proof is based upon the estimate for the evolution operator of the linearized equation in the homogeneous sobolev space

upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation abstract for a smooth bounded domain r n n 3 and e a real parameter consider the hyperbolic equation eu tt u t u u g in with dirichlet boundary conditions under certain conditions on this equation has a compact attractor a e in h 0 1 l 2 for e 0 the parabolic equation also has a compact attractor which can be naturally embedded into a compact set a 0 in h 0 1 l 2 it is shown that for any neighborhood u of a 0 the set a e u for e small

scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients abstract we study the asymptotic behavior of solutions for the semilinear damped wave equation with variable coefficients we prove that if the damping is effective and the nonlinearity and other lower order terms can be regarded as perturbations then the solution is approximated by the scaled gaussian of the corresponding linear parabolic problem the proof is based on the scaling variables and energy estimates

a fast poisson solver for complex geometries robust fast solvers for the poisson equation have generally been limited to regular geometries where direct methods based on fourier analysis or cyclic reduction and multigrid methods can be used while multigrid methods can be applied in irregular domains and to a broader class of partial differential equations they are difficult to implement in a robust fashion since they require an appropriate hierarchy of coarse grids which are not provided in many practical situations in this paper we present a new fast poisson solver based on potential theory rather than on direct discretization of the partial differential equation our method combines fast algorithms for computing volume integrals and evaluating layer potentials on a grid with a fast multipole accelerated integral equation solver the amount of work required is o m log m n where m is the number of interior grid points and n is the number of points on the boundary asymptotically the cost of our method is just twice that of a standard poisson solver on a rectangular domain in which the problem domain can be embedded independent of the complexity of the geometry

the singular limit dynamics of semilinear damped wave equations on considere la dynamique qualitative d une equation d onde amortie semi lineaire a une dimension dans la limite singuliere correspondant a l annulation du coefficient de la derivee d ordre 2 en temps

long time asymptotic expansion for the damped semilinear wave equation the first initial boundary value problem is considered for the damped semilinear wave equation with the quadratic nonlinearity for small initial data and homogeneous boundary conditions its solution is constructed in the form of a series in the eigenfunctions of the laplace operator the long time asymptotic expansion is obtained which shows the nonlinear effects of amplitude and frequency multiplication the same results hold for the admissible initial data that are not small

a dynamical system approach to the self similar large time behavior in scalar convection diffusion equations abstract we present a dynamical system method that provides both the existence of self similar solutions and the self similar large time behavior for convection diffusion equations in r n this method avoids a direct study of the elliptic problem related to the self similar profiles we concentrate our attention on the model example u t u a u 1 n u g x t in r n 0 u 0 u 0 l 1 r n l r n where a r n and g x t t 1 n 2 1h x formula with h l2 r n exp x 2 4 and l r n such that h y dy 0 and h y y dy being small enough in the natural similarity variables the self similar profiles become stationary solutions of a new convection diffusion equation by using lyapunov type arguments that rely in an essential manner on the l1 contraction property of the system we prove that those stationary solutions exists and that any trajectory converges to one of them the limit of any trajectory is completely determined by its mass which is conserved along the time in order to ensure the relative compactness of trajectories we work in the functional framework of the weighted sobolev spaces introduced by escobedo and kavian

a finite difference domain decomposition method using local corrections for the solution of poisson s equation parallel algorithms have become increasingly important in the field of scientific computing since the fastest computers currently all have some form of parallel architecture numerical solutions of elliptic partial differential equations are especially difficult to parallelize because the solution at each point in the domain requires global information many methods are available to facilitate efficient propagation of global information but most are inherently iterative and still require a large amount of inter processor communication parallel machines are generally able to perform computation on each processor orders of magnitude more quickly than communication among processors as a result the performance of algorithms with large communication requirements is often limited by communication rather than computational speed we present here a domain decomposition method for computing finite difference solutions to the poisson equation with infinite domain boundary conditions our method is a finite difference analogue of anderson s method of local corrections the solution is computed in three steps first fine grid solutions are computed in parallel using infinite domain boundary conditions on each subdomain second information is transferred globally through a coarse grid representation of the charge and a global coarse grid solution is found third a fine grid solution is computed on each subdomain using boundary conditions set with the global coarse solution corrected locally with fine grid information from nearby subdomains there are three important features of our algorithm first our method requires only a single iteration between the local fine grid solutions and the global coarse representation second the computed solution is second order accurate independent of the number of subdomains third the accuracy of the method is only weakly dependent on the coarse grid spacing as a result of these features we are able to compute accurate solutions in parallel with minimal communication overhead we present results from implementations of our algorithm in two different parallel language environments tested on a network of workstations these results verify the overall accuracy confirm the small communication costs and demonstrate the parallel scalability of the method

a fast direct solver for elliptic partial differential equations on adaptively refined meshes we present a new class of fast direct solvers for elliptic partial differential equations on adaptively refined meshes these solvers rely on a combination of standard fast solvers for uniform grids and potential theory unlike standard iterative approaches they have a well defined operation count they also preserve the order of accuracy of the uniform grid solver despite the presence of coarse fine interfaces

denoising enhancing images on elastic manifolds the conflicting demands for simultaneous low pass and high pass processing required in image denoising and enhancement still present an outstanding challenge although a great deal of progress has been made by means of adaptive diffusion type algorithms to further advance such processing methods and algorithms we introduce a family of second order in time partial differential equations these equations describe the motion of a thin elastic sheet in a damping environment they are also derived by a variational approach in the context of image processing the new operator enables better edge preservation in denoising applications by offering an adaptive lowpass filter which preserves high frequency components in the pass band better than the adaptive diffusion filter while offering slower error propagation across edges we explore the action of this powerful operator in the context of image processing and exploit for this purpose the wealth of knowledge accumulated in physics and mathematics about the action and behavior of this operator the resulting methods are further generalized for color and or texture image processing by embedding images in multidimensional manifolds a specific application of the proposed new approach to superresolution is outlined

fast triangulated vortex methods for the 2d euler equations abstract vortex methods for inviscid incompressible two dimensional fluid flow are usually based on blob approximations this paper presents a vortex method in which the vorticity is approximated by a piecewise polynomial interpolant on a delaunay triangulation of the vortices an efficient reconstruction of the delaunay triangulation at each step makes the method accurate for long times the vertices of the triangulation move with the fluid velocity which is reconstructed from the vorticity via a simplified fast multipole method for the biot savart law with a continuous source distribution the initial distribution of vortices is constructed from the initial vorticity field by an adaptive approximation method which produces good accuracy even for discontinuous initial data numerical results show that the method is highly accurate over long time intervals experiments with single and multiple circular and elliptical rotating patches of both piecewise constant and smooth vorticity indicate that the method produces much smaller errors than blob methods with the same number of degrees of freedom at little additional cost generalizations to domains with boundaries viscous flow and three space dimensions are discussed

asymptotic properties of the balanced repeated replication method for sample quantiles inference including variance estimation can be made from stratified samples by selecting a balanced set of subsamples this balanced subsampling method is generically called the balanced repeated replication method in survey data analysis which includes mccarthy s balanced half samples method and its extensions for more general stratified designs we establish the asymptotic consistency of the balanced repeated replication variance estimators when the parameter of interest is the population quantile the consistency results also hold when balanced subsampling is replaced by random subsampling as a key technical prerequisite we prove a bahadurtype representation for sample quantiles in stratified random sampling

continuity of non autonomous attractors for hyperbolic perturbation of parabolic equations abstract in this work we prove the lower and upper semicontinuity of pullback uniform and cocycle attractors for the non autonomous dynamical system given by hyperbolic equation on a bounded domain r 3 u t t u t u f t u for each 0 this equation has uniform pullback and cocycle attractors in h 0 1 l 2 and for 0 the limit parabolic equation u t u f 0 u has a global attractor a 0 in h 0 1 which can be naturally embedded into a compact set a 0 in h 0 1 l 2 we prove that all three types of non autonomous attractors converge both upper and lower semicontinuously to a 0 the study of the detailed structure of the non autonomous attractors under perturbation plays the crucial role in the arguments

large time behavior for convection diffusion equations in rn abstract we describe the large time behavior of solutions of the convection diffusion equation u t u a u q 1 u in 0 r n with a r n and q 1 1 n n 1 when q 1 1 n we prove that the large time behavior of solutions with initial data in l 1 r n is given by a uniparametric family of self similar solutions the relevant parameter is the mass of the solution that is conserved for all t our result extends to dimensions n 1 well known results on the large time behavior of solutions for viscous burgers equations in one space dimension the proof is based on la salle s invariance principle applied to the equation written in its self similarity variables when q 1 1 n the convection term is too weak and the large time behavior is given by the heat kernel in this case the result is easily proved applying standard estimates of the heat kernel on the integral equation related to the problem

an unsplit cell centered godunov method for ideal mhd we present a second order godunov algorithm for multidimensional ideal mhd our algorithm is based on the unsplit formulation of colella j comput phys 87 1990 with all of the primary dependent variables centered at the same location to properly represent the divergence free condition of the magnetic fields we apply a discrete projection to the intermediate values of the field at cell faces and apply a filter to the primary dependent variables at the end of each time step we test the method against a suite of linear and nonlinear tests to ascertain accuracy and stability of the scheme under a variety of conditions the test suite includes rotated planar linear waves mhd shock tube problems low beta flux tubes and a magnetized rotor problem for all of these cases we observe that the algorithm is second order accurate for smooth solutions converges to the correct weak solution for problems involving shocks and exhibits no evidence of instability or loss of accuracy due to the possible presence of non solenoidal fields

a completely iterative method for the infinite domain electrostatic problem with nonlinear dielectric media we present an iterative method for the solution of the exterior all space electrostatic problem for nonlinear dielectric media the electric potential is specified on interior boundaries and the electric field decays at infinity our approach uses a natural variational formulation based on the total energy of the nonlinear dielectric medium subject to boundary conditions the problem is decomposed into an exterior calculation and an interior calculation with the boundary specified electric potentials imposed as constraints between them together these enable an iterative method that is based on the variational formulation in contrast to direct solution of the electrostatic problems we avoid the construction storage and solution of dense and large linear systems this provides important advantages for multiphysics problems that couple the linear electrostatic poisson problem to nonlinear physics the latter necessarily involves iterative approaches and our approach replaces a large number of direct solves for the electrostatics with an iterative algorithm that can be coupled to the iterations of the nonlinear problem we present examples applying the method to inhomogeneous anisotropic nonlinear dielectrics a key advantage of our variational formulation is that we require only the free space isotropic homogeneous greens function for all these settings

inference from stratified samples properties of the linearization jackknife and balanced repeated replication methods the asymptotic normality of both linear and nonlinear statistics and the consistency of the variance estimators obtained using the linearization jackknife and balanced repeated replication brr methods in stratified samples are established the results are obtained as l oa within the context of a sequence of finite populations jil with l strata in ji l and are valid for any stratified multistage design in which the primary sampling units psu s are selected with replacement and in which independent subsamples are taken within those psu s selected more than once in addition some exact analytical results on the bias and stability of these alternative variance estimators in the case of ratio estimation are obtained for small l under a general linear regression model

reaction cattaneo systems with fluctuating relaxation time a reaction cattaneo equation with fluctuating relaxation time of the diffusive flux has been explored it has been shown that depending on the strength of fluctuations the dynamical system exhibits new oscillatory solutions as a result of hopf and double hopf bifurcations leading to spatiotemporal patterns this analysis has been applied to two model nonlinear systems

universal dynamics for the defocusing logarithmic schrodinger equation we consider the nonlinear schrodinger equation with a logarithmic nonlinearity in a dispersive regime we show that the presence of the nonlinearity affects the large time behavior of the solution the dispersion is faster than usual by a logarithmic factor in time and the positive sobolev norms of the solution grow logarithmically in time moreover after rescaling in space by the dispersion rate the modulus of the solution converges to a universal gaussian profile these properties are suggested by explicit computations in the case of gaussian initial data and remain when an extra power like nonlinearity is present in the equation one of the key steps of the proof consists in using the madelung transform to reduce the equation to a variant of the isothermal compressible euler equation whose large time behavior turns out to be governed by a parabolic equation involving a fokker planck operator

a fast high order solver for problems of scattering by heterogeneous bodies a new high order integral algorithm for the solution of scattering problems by heterogeneous bodies is presented here a scatterer is described by a continuously or discontinuously varying refractive index n x within a two dimensional 2d bounded region solutions of the associated helmholtz equation under given incident fields are then obtained by high order inversion of the lippmann schwinger integral equation the algorithm runs in o nlog n operations where n is the number of discretization points a wide variety of numerical examples provided include applications to highly singular geometries high contrast configurations as well as acoustically electrically large problems for which supercomputing resources have been used recently our method provides highly accurate solutions for such problems on small desktop computers in cpu times of the order of seconds

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

an iterative boundary potential method for the infinite domain poisson problem with interior dirichlet boundaries an iterative method is developed for the solution of poisson s problem on an infinite domain in the presence of interior boundaries held at fixed potential in three dimensions the method combines pre existing fast multigrid based poisson solvers for data represented on cartesian grids with the fast multipole method interior boundaries are represented with the embedded boundary formalism the implementation is in parallel and uses adaptive mesh refinement examples are presented for a smooth interior boundary for which an analytical result is known and for an irregular interior boundary problem second order accuracy in l 1 with respect to the grid resolution is demonstrated for both problems

asymptotically self similar solutions of the damped wave equation abstract we consider the damped hyperbolic equation 1 e u u a u u p 1 u r r where e 0 a 1 as and 1 p 3 we prove in this article that the exact self similar solutions of the semi linear parabolic equation obtained by setting e 0 and a 1 in 1 are also asymptotically stable self similar solutions of the eq 1 the proof of our result relies on various energy estimates rewritten in the variables ln

an adaptive fast gauss transform in two dimensions a variety of problems in computational physics and engineering require the convolution of the heat kernel a gaussian with either discrete sources densities supported on boundaries or continuous

an adaptive fast multipole accelerated poisson solver for complex geometries abstract we present a fast direct and adaptive poisson solver for complex two dimensional geometries based on potential theory and fast multipole acceleration more precisely the solver relies on the standard decomposition of the solution as the sum of a volume integral to account for the source distribution and a layer potential to enforce the desired boundary condition the volume integral is computed by applying the fmm on a square box that encloses the domain of interest for the sake of efficiency and convergence acceleration we first extend the source distribution the right hand side in the poisson equation to the enclosing box as a c 0 function using a fast boundary integral based method we demonstrate on multiply connected domains with irregular boundaries that this continuous extension leads to high accuracy without excessive adaptive refinement near the boundary and as a result to an extremely efficient black box fast solver

global asymptotics of solutions to the cauchy problem for the damped wave equation with absorption abstract we consider the cauchy problem for the damped wave equation with absorption u t t u u t u 1 u 0 t x r r n the behavior of u as t is expected to be same as that for the corresponding heat equation t 1 0 t x r r n in the subcritical case 1 c n 1 2 n there exists a similarity solution w b t x with the form t 1 1 f x t depending on b lim x x 2 1 f x 0 our first aim is to show the decay rates u t l 2 u t l 1 u t l 2 o t 1 1 n 4 t 1 1 n 2 1 t 1 1 1 2 n 4 provided that the initial data without initial data size restriction spatially decays with reasonable polynomial order the decay rates are sharp in the sense that they are same as those of the similarity solution the second aim is to show that the gauss kernel is the asymptotic profile in the supercritical case which has been shown in case of one dimensional space by hayashi kaikina and naumkin n hayashi e i kaikina p i naumkin asymptotics for nonlinear damped wave equations with large initial data preprint 2004 we show this assertion in two and three dimensional space to prove our results both the weighted l 2 energy method and the explicit formula of solutions will be employed the weight is an improved one originally developed in y todorova b yordanov critical exponent for a nonlinear wave equation with damping j differential equations 174 2001 464 489

a fast vortex method for computing 2d viscous flow abstract we present a fast version of the random vortex method for computing incompressible viscous flow at large reynolds numbers the basis of this method is anderson s method of local corrections and similar ideas for handling the potential and boundary layer flows the goal of these ideas is to reduce the cost involved in computing the velocity field at each time step from being quadratic to linear as a function of the number of vortex elements we present the results of a numerical study of the flow in a closed box due to a vortex fixed at its center our results demonstrate that the addition of the viscous portions of the random vortex method to the method of local corrections does not add appreciably to the cost furthermore the cost of the resulting method is linear when o 104 vortex elements are used in spite of the fact that the majority of these elements lie in a thin band adjacent to the boundary

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

fast integral equation methods for the modified helmholtz equation we present integral equation methods for the solution to the two dimensional modified helmholtz equation u x email protected email protected x 0 in bounded or unbounded multiply connected domains we consider both dirichlet and neumann problems we derive well conditioned fredholm integral equations of the second kind which are discretized using high order hybrid gauss trapezoid rules our fast multipole based iterative solution procedure requires only o n operations where n is the number of nodes in the discretization of the boundary we demonstrate the performance of our methods on several numerical examples and we show that they have both the ability to handle highly complex geometry and the potential to solve large scale problems

invariant manifolds for parabolic partial differential equations on unbounded domains in this paper finite dimensional invariant manifolds for nonlinear parabolic partial differential equations of the form

l estimation for linear models abstract linear combinations of order statistics or l estimators have played an extremely important role in the development of robust methods for the one sample problem we suggest analogs of l estimators for the parameters of the linear model based on the p dimensional regression quantiles proposed by koenker and bassett 1978 a uniform bahadur type asymptotic representation of regression quantiles is established and this yields a general asymptotic theory of l estimators for the linear model a leading example of the proposed estimators is an analog of the trimmed mean trq which is asymptotically equivalent to the trimmed least squares estimator studied by ruppert and carroll 1980 but appears to be somewhat less sensitive to influential design observations this estimator is also asymptotically equivalent to the well known huber m estimator but offers the significant advantage that it is scale invariant we illustrate the methods by reconsidering a mid 18th century linear model analyzed b

a fast multi level boundary element method for the helmholtz equation abstract most recently we have developed a novel multi level boundary element method mlbem for the solution of the steady heat diffusion equation involving asymptotically decaying non oscillatory log singular and strongly singular kernels this hierarchical approach generalizes the pioneering work of brandt and lubrecht on multi level multi integration mlmi and c cycle multi grid to the broader class of mixed boundary value problems the result is a fast accurate and efficient boundary element method the present paper extends this new computational methodology to the solution of the helmholtz equation involving oscillatory log singular and strongly singular kernels for two dimensional problems we consider a direct boundary element formulation and due to the nature of the fundamental solutions split the corresponding boundary integral equation into real and imaginary parts then we introduce double noded corners to facilitate a patch by patch application of the mlmi algorithm for fast matrix vector and matrix transpose vector multiplications within bi conjugate gradient methods the performance of the proposed fast mlbem is investigated using a numerical example that possesses an exact solution for wave numbers 20 and below we demonstrate that the fast mlbem algorithm for the helmholtz equation is robust accurate and exceptionally efficient

inference from stratified samples second order analysis of three methods for nonlinear statistics abstract for stratified samples and nonlinear statistics that can be expressed as functions of estimated totals second order asymptotic expansions of the linearization jackknife and balanced repeated replication variance estimators are obtained based on these comparisons are made in terms of their biases some higher order asymptotic equivalence results are also established the special case of a combined ratio estimator is investigated in detail some results on bias reduction achieved by the jack knife and balanced repeated replication estimators of a nonlinear function of totals are also given

invariant manifolds and the stability of traveling waves in scalar viscous conservation laws abstract the stability of traveling wave solutions of scalar viscous conservation laws is investigated by decomposing perturbations into three components two far field components and one near field component the linear operators associated to the far field components are the constant coefficient operators determined by the asymptotic spatial limits of the original operator scaling variables can be applied to study the evolution of these components allowing for the construction of invariant manifolds and the determination of their temporal decay rate the large time evolution of the near field component is shown to be governed by that of the far field components thus giving it the same temporal decay rate we also give a discussion of the relationship between this geometric approach and previous results which demonstrate that the decay rate of perturbations can be increased by requiring that initial data lie in appropriate algebraically weighted spaces

quantile estimation with a complex survey design estimation of the finite population distribution function and related statistics such as the median and interquartile range is considered large sample properties of estimators constructed from stratified cluster samples and properties of large sample confidence intervals are established the results are obtained within the context of a sequence of finite populations generated from a superpopulation

a jackknife variance estimator for unistage stratified samples with unequal probabilities summary existing jackknife variance estimators used with sample surveys can seriously overestimate the true variance under unistage stratified sampling without replacement with unequal probabilities a novel jackknife variance estimator is proposed which is as numerically simple as existing jackknife variance estimators under certain regularity conditions the proposed variance estimator is consistent under stratified sampling without replacement with unequal probabilities the high entropy regularity condition necessary for consistency is shown to hold for the rao sampford design an empirical study of three unequal probability sampling designs supports our findings

fast potential theory ii layer potentials and discrete sums we present three new families of fast algorithms for classical potential theory based on ewald summation and fast transforms of gaussians and fourier series ewald summation separates the green function for a cube into a high frequency localized part and a rapidly converging fourier series each part can then be evaluated efficiently with appropriate fast transform algorithms our algorithms are naturally suited to the use of green functions with boundary conditions imposed on the boundary of a cube rather than free space green functions our first algorithm evaluates classical layer potentials on the boundary of a d dimensional domain with d equal to two or three the quadrature error is o hm e where h is the mesh size on the boundary and m is the order of quadrature used the algorithm evaluates the discretized potential using n elements at o n points in o n log n arithmetic operations the constant in o n log n depends logarithmically on the desired error tolerance our second scheme evaluates a layer potential on the domain itself with the same accuracy it produces md values using n boundary elements in o n md log m arithmetic operations our third method evaluates a discrete sum of values of the green function of the type which occur in particle methods it attains error at a cost o na log n where a 2 1 dd and d is the hausdorff dimension of the set where the sources concentrate in the limit n thus it is o n log n when the sources do not cluster too much and close to o n log n in the important practical case when the points are uniformly distributed over a hypersurface we also sketch an o n log n algorithm based on special functions two dimensional numerical results are presented for all three algorithms layer potentials are evaluated to second order accuracy in times which exhibit considerable speedups even over a reasonably sophisticated direct calculation discrete sum calculations are speeded up astronomically our algorithm reduces the cpu time required for a calculation with 40 000 points from six months to one hour

fast multipole method for multivariable integrals we give a fast numerical algorithm to evaluate a class of multivariable integrals a direct numerical evaluation of these integrals costs n m where m is the number of variables and n is the number of the quadrature points for each variable for m 2 and m 3 and for only one dimensional variables we present an algorithm which is able to reduce this cost from n m to a cost of the order of o log epsilon mu m n where epsilon is the desired accuracy and mu m is a constant that depends only on m then we make some comments about possible extensions of such algorithms to number of variables m geq 4 and to higher dimensions this recursive algorithm can be viewed as an extension of fast multipole methods to situations where the interactions between particles are more complex than the standard case of binary interactions numerical tests illustrating the efficiency and the limitation of this method are presented

damped wave equation with a critical nonlinearity in higher space dimensions abstract we study the cauchy problem for nonlinear damped wave equations with a critical defocusing power nonlinearity u 2 n u where n denotes the space dimension for n 1 2 3 global in time existence of small solutions was shown in 4 in this paper we generalize the results to any spatial dimension via the method of decomposition of the equation into the high and low frequency components under the assumption that the initial data are small and decay rapidly at infinity furthermore we present a sharp time decay estimate of solutions with a logarithmic correction

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

a direct adaptive poisson solver of arbitrary order accuracy we present a direct adaptive solver for the poisson equation which can achieve any prescribed order of accuracy it is based on a domain decomposition approach using local spectral approximation as well as potential theory and the fast multipole method in two space dimensions the algorithm requireso nk work wherenis the number of discretization points andkis the desired order of accuracy

balanced repeated replication for stratified multistage survey data under imputation abstract balanced repeated replication brr is a popular method for variance estimation in surveys the standard brr method works by first creating a set of balanced pseudoreplicated datasets from the original dataset for a survey estimator the brr variance estimator is the average of squared deviations r where r is the same as but based on the data in the rth pseudoreplicated dataset only but when there are a large number of imputed missing values nonrespondents treating the imputed values as observed data and applying the standard brr variance estimation formula does not produce valid variance estimators intuitively the variation due to imputation can be captured by the brr method if every pseudoreplicated dataset is imputed in exactly the same way as the original dataset is imputed assuming that the dataset contains flags for nonrespondents but when a random imputation method such as random hot deck imputation random ratio imputation or random regression imputation is u

asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping we consider the asymptotic behavior of the solution of quasilinear hyperbolic equation with linear dampingvtt a vx x vt 0 x t r 0 subsequent to k nishihara j differential equations131 1996 171 188 in that article the system with dampingvt ux 0 ut p v x u p v 0 was treated and the convergence rates to the diffusion wave by l hsiao and t p liu comm math phys 143 1992 599 605 and l hsiao and t p liu chinese ann math ser b14 1993 465 480 were improved for initial data v0 u0 x satisfying v0 u0 v0 0 and v0 x v0 dx 0 this system is reduced to the quasilinear hyperbolic equation with linear dampingvtt p v0 vx p v0 x vt 0 and the decay rates v t l o t 1 2 vx t l o t 1 vt t l o t 3 2 were obtained provided that v vt t 0 h3 h2 is small and inl1 l1 from these decay rates vis expected to behave as a solution to the parabolic equation the aims of this paper are to seek the asymptotic profile x t satisfying t l o t 1 2 x t l o t 1 t t l o t 3 2 and to show that the solution of satisfies v t l o t 1 v x t l o t 3 2 v t t l o t 2 the proof is based on the elementary energy method and the green function method for the parabolic equation

krylov deferred correction accelerated method of lines transpose for parabolic problems in this paper a new class of numerical methods for the accurate and efficient solutions of parabolic partial differential equations is presented unlike traditional method of lines mol the new krylov deferred correction kdc accelerated method of lines transpose mol t first discretizes the temporal direction using gaussian type nodes and spectral integration and symbolically applies low order time marching schemes to form a preconditioned elliptic system which is then solved iteratively using newton krylov techniques such as newton gmres or newton bicgstab method each function evaluation in the newton krylov method is simply one low order time stepping approximation of the error by solving a decoupled system using available fast elliptic equation solvers preliminary numerical experiments show that the kdc accelerated mol t technique is unconditionally stable can be spectrally accurate in both temporal and spatial directions and allows optimal time stepsizes in long time simulations

the chebyshev fast gauss and nonuniform fast fourier transforms and their application to the evaluation of distributed heat potentials we present a method for the fast and accurate computation of distributed heat potentials in two dimensions the distributed source is assumed to be given in terms of piecewise space time chebyshev polynomials we discretize uniformly in time whereas in space the polynomials are defined on the leaf nodes of a quadtree the quadtree can vary at each time step we combine a product integration rule with fast algorithms fast heat potentials nonuniform fft fast gauss transform to obtain a high order accurate method with optimal complexity if n is the number of time steps m is the maximum number of leaf nodes over all the time steps and the input contains a qth order polynomial representation of f then our method requires o q 3mnlogm work to evaluate the heat potential at arbitrary mn space time target locations the overall convergence rate of the method is of order q we present numerical experiments for q 4 8 and 16 and we verify the theoretical convergence rate of the method when the solution is sufficiently smooth the 16th order variant results in significant computational savings even in the case in which we require only a few digits of accuracy

a new fast multipole accelerated poisson solver in two dimensions we present an adaptive fast multipole method for solving the poisson equation in two dimensions the algorithm is direct assumes that the source distribution is discretized using an adaptive quad tree and allows for dirichlet neumann periodic and free space conditions to be imposed on the boundary of a square the amount of work per grid point is comparable to that of classical fast solvers even for highly nonuniform grids

existence and nonexistence of time global solutions to damped wave equation on half line on the half line r 0 infinity the initial boundary value problems with null dirichlet boundary for both the semilinear heat equation and damped wave equation are considered the critical exponent p c n k of semilinear term for the existence and nonexistence about the semilinear heat equation on the halved space d n d k r k rn k is given by p c n k 1 2 n k j appl math phys 39 1988 135 149 arch rational mech anal 109 1990 63 71 since the damped wave equation is expected to be close to the heat equation j differential equations 191 2003 445 469 math z 244 2003 631 649 the critical exponent for the semilinear damped wave equation is expected to be same as that of the semilinear heat equation however there is no blow up result on the halved space for the damped wave equation in this paper the exponent pc l 1 2 is shown to be critical for the existence and nonexistence of time global solution to both the semilinear heat equation and damped wave equation on the half line r together with the derivation of the blow up time for the proof the explicit formulas of solutions are used in a similar fashion to those in li and zhou discrete continuous dynamic systems 1 1995 503 520 c 2005 elsevier ltd all rights reserved

large time behavior and lp lq estimate of solutions of 2 dimensional nonlinear damped wave equations abstract we show the asymptotic behavior of the solution to the cauchy problem of the two dimensional damped wave equation it is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the lp lq type estimate this is a two dimensional generalization of the three dimensional result due to nishihara math z 244 2003 631 to show this we use the fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations by using the lp lq estimate we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity u u our result covers the whole super critical case 1 where the 1 is well known as the fujita exponent when n 2

exact representation of the asymptotic drift speed and diffusion matrix for a class of velocity jump processes abstract this paper examines a class of linear hyperbolic systems which generalizes the goldstein kac model to an arbitrary finite number of speeds v i with transition rates i j under the basic assumptions that the transition matrix is symmetric and irreducible and the differences v i v j generate all the space the system exhibits a large time behavior described by a parabolic advection diffusion equation the main contribution is to determine explicit formulas for the asymptotic drift speed and diffusion matrix in term of the kinetic parameters v i and i j establishing a complete connection between microscopic and macroscopic coefficients it is shown that the drift speed is the arithmetic mean of the velocities v i the diffusion matrix has a more complicate representation based on the graph with vertices the velocities v i and arcs weighted by the transition rates i j the approach is based on an exhaustive analysis of the dispersion relation and on the application of a variant of the kirchoff s matrix tree theorem from graph theory

lp lq estimates of solutions to the damped wave equation in 3 dimensional space and their application it has been asserted that the damped wave equation has the diffusive structure as t in this paper we consider the cauchy problem in 3 dimensional space for the linear damped wave equation and the corresponding parabolic equation and obtain the lp lq estimates of the difference of each solution which represent the assertion precisely explicit formulas of the solutions are analyzed for the proof the second aim is to apply the lp lq estimates to the semilinear damped wave equation with power nonlinearity if the power is larger than the fujita exponent then the time global existence of small weak solution is proved and its optimal decay order is obtained

a damped hyperbolic equation on thin domains for a damped hyperbolic equation in a thin domain in r3 over a bounded smooth domain in r2 it is proved that the global attractors are upper semicontinuous it is shown also that a global attractor exists in the case of the critical sobolev exponent

l statistics in complex survey problems we study linear combinations of order statistics l statistics in survey problems with a stratified multistage sampling design two general types of l statistics are considered smooth l statistics with weights generated by a smooth function and nonsmooth l statistics sample quantiles the trimmed sample mean the decile mean and variance the sample lorenz curve and the sample gini s family parameters are examples of smooth lstatistics or functions of smooth l statistics used in survey problems it is shown that under weak conditions the smooth l statistics are asymptotically normal and their asymptotic variances can be consistently estimated by jackknifing for the sample quantiles their asymptotic normality requires more conditions on the finite population distribution functions consistent estimators for the asymptotic variances of the sample quantiles are derived asymptotic validity of the woodruff s confidence intervals for population quantiles is also proved

asymptotic behaviour of nonlinear parabolic equations with critical exponents a dynamical systems approach we investigate the large time behaviour of solutions to the nonlinear heat conduction equation with absorption ut u 1 u in q rn 0 e with n 1 0 and critical absorption exponent 1 2n the initial function u x 0 0 is assumed to be integrable nonnegative and compactly supported we prove that u converges as t to a unique self similar function which is a contracted version of one of the asymptotic profiles of the nonabsorptive problem ut u 1 the same for any initial data the cornerstone of the proof is a result about limits of infinite dimensional asymptotical dynamical systems combining this result with an asymptotic evaluation of the mass function as well as typical pde estimates gives the behaviour of e for large times similar unusual asymptotic behaviour is obtained for the equation ut div du du u with same conditions on and u x 0 and critical value for 1 2 n

diffusive stability of spatial periodic solutions of the swift hohenberg equation we are interested in the nonlinear stability of the eckhaus stable equilibria of the swift hohenberg equation on the infinite line with respect to small integrable perturbations the difficulty in showing stability for these stationary solutions comes from the fact that their linearizations possess continuous spectrum up to zero the nonlinear stability problem is treated with renormalization theory which allows us to show that the nonlinear terms are irrelevant i e that the fully nonlinear problem behaves asymptotically as the linearized one which is under a diffusive regime

efficient estimation of non linear finite population parameters by using non parametrics summary currently high precision estimation of non linear parameters such as gini indices low income proportions or other measures of inequality is particularly crucial we propose a general class of estimators for such parameters that take into account univariate auxiliary information assumed to be known for every unit in the population through a non parametric model assisted approach we construct a unique system of survey weights that can be used to estimate any non linear parameter that is associated with any study variable of the survey using a plug in principle based on a rigorous functional approach and a linearization principle the asymptotic variance of the estimators proposed is derived and variance estimators are shown to be consistent under mild assumptions the theory is fully detailed for penalized b spline estimators together with suggestions for practical implementation and guidelines for choosing the smoothing parameters the validity of the method is demonstrated on data extracted from the french labour force survey point and confidence interval estimation for the gini index and the low income proportion are derived theoretical and empirical results highlight our interest in using a non parametric versus a parametric approach when estimating non linear parameters in the presence of auxiliary information

a fast adaptive vortex method in three dimensions the method of local corrections mlc developed by anderson for two spatial dimensions is a particle particle particle mesh method in which the calculation of the velocity field induced by a collection of vortices is split into two parts i a finite difference velocity field calculation using a fast poisson solver the results of which are used to represent the velocity field induced by vortices far from the evaluation point and ii an n body calculation to compute the velocity field at a vortex induced by nearby vortices we present a fast vortex method for incompressible flow in three dimensions based on the extension of the mlc algorithm from two to three spatial dimensions and the use of adaptive mesh refinement in the finite difference calculation of the mlc calculations with a vortex ring in three dimensions show that the break even point between the mlc with amr and the direct method is at n 3000 on a cray y mp for n 64 000 mlc with amr can be 12 times faster than the direct method results from calculations of two colliding inviscid vortex rings demonstrate the increased resolution which can be obtained using fast methods

dynamic refinement algorithms for spectral element methods abstract a numerical method is presented that combines high order domain decomposition and adaptive mesh refinement the spatial discretization is based on a nonconforming spectral element method meshes are generated by adaptive refinement of a quadtree using either local solution gradients or the local polynomial spectrum as a refinement criteria the method is applied to solve scalar test problems and to simulate the flow in a lid driven cavity at moderate reynolds numbers

decay properties of solutions to the cauchy problem for the damped wave equation with absorption abstract we consider the cauchy problem for the damped wave equation with absorption u t t u u t u p 1 u 0 t x r r n the behavior of u as t is expected to be the same as that for the corresponding heat equation t p 1 0 t x r r n which has the similarity solution w a t x with the form t 1 p 1 f x t depending on a lim x x 2 p 1 f x 0 provided that p is less than the fujita exponent p c n 1 2 n in this paper as a first step if 1 p p c n and the data u 0 u 1 x decays exponentially as x without smallness condition the solution is shown to decay with rates as t u t l 2 u t l p 1 u t l 2 o t 1 p 1 n 4 t 1 p 1 n 2 p 1 t 1 p 1 1 2 n 4 those of which seem to be reasonable because the similarity solution w a t x has the same decay rates as for the proof the weighted l 2 energy method will be employed with suitable weight similar to that in todorova and yordanov y todorova b yordanov critical exponent for a nonlinear wave equation with damping j differential equations 174 2001 464 489

geometric stability analysis for periodic solutions of the swift hohenberg equation in this paper we describe invariant geometrical structures in the phase space of the swift hohenberg equation in a neighborhood of its periodic stationary states we show that in spite of the fact that these states are only marginally stable i e the linearized problem about these states has continuous spectrum extending all the way up to zero there exist finite dimensional invariant manifolds in the phase space of this equation which determine the long time behavior of solutions near these stationary solutions in particular using this point of view we obtain a new demonstration of schneider s recent proof that these states are nonlinearly stable

existence and stability of traveling fronts in the extended fisher kolmogorov equation abstract we study traveling wave solutions to a general fourth order differential equation that is a singular perturbation of the fisher kolmogorov equation we apply the geometric method for singularly perturbed systems to show that for every positive wavespeed there exists a traveling wave also we find that there exists a critical wavespeed c which divides these solutions into monotonic c c and oscillatory c c solutions we show that the monotonic fronts are locally stable under perturbations in appropriate weighted sobolev spaces by using various energy functionals

the rapid evaluation of volume integrals of potential theory on general regions abstract we present a new method for the rapid high order accurate evaluation of certain volume integrals in potential theory on general irregular regions the kernels of the integrals are either a fundamental solution or a linear combination of the derivatives of a fundamental solution of a second order linear elliptic differential equation instead of using a standard quadrature formula or the exact evaluation of any integral the methods rely on rapid methods of solving the differential equation which the kernel is the solution of therefore the number of operations needed to evaluate the volume integral is essentially equal to the number of operations needed to solve the differential equation on a rectangular region with a regular grid and the method requires no evaluation of the kernel