a new approach to analyticity of dirichlet neumann operators this paper outlines the theoretical background of a new approach towards an accurate and well conditioned perturbative calculation of dirichlet neumann operators dnos on domains that are perturbations of simple geometries previous work on the analyticity of dnos has produced formulae that as we have found are very ill conditioned we show how a simple change of variables can lead to recursions that satisfy analyticity estimates without relying on subtle cancellation properties at the heart of previous formulae

a consistent metric for performance evaluation of multi object filters the concept of a miss distance or error between a reference quantity and its estimated controlled value plays a fundamental role in any filtering control problem yet there is no satisfactory notion of a miss distance in the well established field of multi object filtering in this paper we outline the inconsistencies of existing metrics in the context of multi object miss distances for performance evaluation we then propose a new mathematically and intuitively consistent metric that addresses the drawbacks of current multi object performance evaluation metrics

procedures for estimating the error in pade approximation kronrod s procedure is a method for estimating the error in gaussian quadrature methods pad6 approximants are formal gaussian quadrature formulas in a previous paper kronrod s method was used to obtain estimates of the error in pad6 approximation using a new interpretation of this procedure and three different expressions of the error of pad6 approximants extensions of the method are obtained they provide new error estimates for pad6 approximants these estimates are compared

entropic and conformational study of isolated double tethered chains by three dimensional lattice simulations abstract double tethered polymers are a kind of linear polymer with a peculiar topological constraint that is both of its end points are attached to a plane which the polymer segments cannot penetrate the effects of the constraint on the polymer s configurational and entropic properties were investigated by means of a three dimensional lattice simulation that combined a previously proposed idea with a very efficient chain generation algorithm in particular the value of a topology dependent critical exponent was estimated for the double tethered configurations this data is the first report on isolated and double tethered chains also two optional types of tethered polymer were investigated as asymptotes of the double and single tethered configurations

generalized recursive track before detect with proposal partitioning for tracking varying number of multiple targets in low snr track before detect tbd algorithms incorporate unthresholded measurements to track targets under low signal to noise ratio snr conditions in this paper we generalize a single target recursive tbd tracking algorithm to track a time varying number of targets in low snr and cluttered environments this algorithm is important for example in tracking scenes involving multiple closely spaced targets moving along the same direction such as a convoy of low observable vehicles moving through a forest or multiple targets moving in a crisscross fashion the proposed multi mode multi target tbd mm mm tbd algorithm is based on estimating the probabilities of all possible combinations of target existence scenarios to obtain the joint multi target posterior probability density function in a recursive bayesian framework we implement the algorithm recursively using particle filtering to also incorporate nonlinear non gaussian tracking models as the number of target existence combinations dynamically increases with the number of targets we also propose an efficient proposal density function through partitioning of the multiple target space in order to decrease the number of particles and thus improve the approximation accuracy of the particle filter we employ a heuristic decision directed based approach to keep the computational complexity as a linear function of the maximum number of possible targets by exploiting the information obtained from the estimated mode probabilities

phase field computations of single needle crystals crystal growth and motion by mean curvature the phase field model for free boundaries consists of a system of parabolic differential equations in which the variables represent a phase or order parameter and temperature respectively the parameters in the equations are related directly to the physical observables including the interfacial width epsilon which we can regard as a free parameter in computation the phase field equations can be used to compute a wide range of sharp interface problems including the classical stefan model its modification to incorporate surface tension and or surface kinetic terms the cahn allen motion by mean curvature the hele shaw model etc also included is anisotropy in the equilibrium and dynamical forms generally considered by materials scientists by adjusting the parameters the computations can be varied continuously from single needle dendritic to faceted crystals the computational method consists of smoothing a sharp interface problem within the scaling of distinguished limits of the phase field e

considerations on computing real logarithms of matrices hamiltonian logarithms and skew symmetric logarithms abstract the issue of computing a real logarithm of a real matrix is addressed after a brief review of some known methods more attention is paid to three 1 pade approximation techniques 2 newton s method and 3 a series expansion method newton s method has not been previously treated in the literature we address commutativity issues and simplify the algorithmic formulation we also address general structure preserving issues for two applications in which we are interested finding the real hamiltonian logarithm of a symplectic matrix and finding the skew symmetric logarithm of an orthogonal matrix the diagonal pade approximants and the proposed series expansion technique are proven to be structure preserving some algorithmic issues are discussed

bayesian multi object filtering with amplitude feature likelihood for unknown object snr in many tracking scenarios the amplitude of target returns are stronger than those coming from false alarms this information can be used to improve the multiple target state estimation by obtaining more accurate target and false alarm likelihoods target amplitude feature is well known to improve data association in conventional tracking filters such as probabilistic data association and multiple hypothesis tracking and results in better tracking performance of low signal to noise ratio snr targets the advantage of using the target amplitude approach is that targets can be identified earlier through the enhanced discrimination between target and false alarms one of the limitations of this approach is that it is usually assumed that the snr of the target is known we show that the reliable estimation of the snr requires a significant number of measurements and so we propose an alternative approach for situations where the snr is unknown we illustrate this approach in the context of multiple targets for different snrs in the framework of finite set statistics fisst furthermore we illustrate how this can be incorporated into approximate multiple object filters derived from fisst including probability hypothesis density phd and cardinalized phd cphd filters we present simulation results for gaussian mixture implementations of the filters that demonstrate a significant improvement in performance over just using location measurements

bayesian filtering with random finite set observations this paper presents a novel and mathematically rigorous bayes recursion for tracking a target that generates multiple measurements with state dependent sensor field of view and clutter our bayesian formulation is mathematically well founded due to our use of a consistent likelihood function derived from random finite set theory it is established that under certain assumptions the proposed bayes recursion reduces to the cardinalized probability hypothesis density cphd recursion for a single target a particle implementation of the proposed recursion is given under linear gaussian and constant sensor field of view assumptions an exact closed form solution to the proposed recursion is derived and efficient implementations are given extensions of the closed form recursion to accommodate mild nonlinearities are also given using linearization and unscented transforms

stochastic inversion in acoustic scattering a novel approach is presented for the probabilistic characterization of the solution in inverse acoustic scattering problems the approach consists of representing the unknown parameters of the scatterer as stochastic processes with spatially random fluctuations the stochastic processes are represented by their polynomial chaos decomposition the deterministic coefficients in this parametrization are then computed by solving a deterministic optimization algorithm with an appropriate objective function that captures the probabilistic significance of these coefficients the general procedure is demonstrated by its application to a two dimensional problem with the hidden object a hard obstacle being immersed in a heterogeneous medium the formulation provides as well a measure of validity of a particular predictive model at matching experimental evidence

analytic implementations of the cardinalized probability hypothesis density filter the probability hypothesis density phd recursion propagates the posterior intensity of the random finite set rfs of targets in time the cardinalized phd cphd recursion is a generalization of the phd recursion which jointly propagates the posterior intensity and the posterior cardinality distribution in general the cphd recursion is computationally intractable this paper proposes a closed form solution to the cphd recursion under linear gaussian assumptions on the target dynamics and birth process based on this solution an effective multitarget tracking algorithm is developed extensions of the proposed closed form recursion to accommodate nonlinear models are also given using linearization and unscented transform techniques the proposed cphd implementations not only sidestep the need to perform data association found in traditional methods but also dramatically improve the accuracy of individual state estimates as well as the variance of the estimated number of targets when compared to the standard phd filter our implementations only have a cubic complexity but simulations suggest favorable performance compared to the standard joint probabilistic data association jpda filter which has a nonpolynomial complexity

rapidly convergent two dimensional quasi periodic green function throughout the spectrum including wood anomalies we introduce a new methodology based on new quasi periodic green functions which converge rapidly even at and around wood anomaly configurations for the numerical solution of problems of scattering by periodic rough surfaces in two dimensional space as is well known the classical quasi periodic green function ceases to exist at wood anomalies the approach introduced in this text produces fast green function convergence throughout the spectrum on the basis of a certain finite differencing approach and smooth windowing of the classical green function lattice sum the resulting green function convergence is super algebraically fast away from wood anomalies and it reduces to an arbitrarily high user prescribed algebraic order of convergence at wood anomalies

self avoiding walks interacting with a surface discusses two models of polymer adsorption in one model a self avoiding walk on a d dimensional lattice interacts with a d 1 dimensional hyperplane in the other model this walk must also lie in or on one side of this hyperplane both models exhibit non analytic behaviour corresponding to a phase transition but these phase transitions do not occur at the same point the authors give numerical estimates of the locations of these transitions

stability of high order perturbative methods for the computation of dirichlet neumann operators abstract in this paper we present results on the stability of perturbation methods for the evaluation of dirichlet neumann operators dno defined on domains that are viewed as complex deformations of a basic simple geometry in such cases geometric perturbation methods based on variations of the spatial domains of definition have long been recognized to constitute efficient and accurate means for the approximation of dno and in fact several numerical implementations have been previously proposed inspired by our recent analytical work here we demonstrate that the convergence of these algorithms is quite generally limited by numerical instability indeed we show that these standard perturbative methods for the calculation of dno suffer from significant ill conditioning which is manifest even for very smooth boundaries and whose severity increases with boundary roughness moreover and again motivated by our previous work we introduce an alternative perturbative approach that we show to be numerically stable this approach can be interpreted as a reformulation of classical perturbative algorithms in suitable independent variables and thus it allows for both direct comparison and the possibility of analytic continuation of the perturbation series it can also be related to classical preconditioned spectral approaches and as such it retains in finite arithmetic the spectral convergence properties of classical perturbative methods albeit at a higher computational cost as it does not take advantage of possible dimensional reductions still as we show an alternative approach such as the one we propose may be mandated in cases where substantial information is contained in high order harmonics and or perturbation coefficients of the solution

finite element methods for the time dependent ginzburg landau model of superconductivity abstract the initial boundary value problem for the time dependent ginzburg landau equations that model the macroscopic behavior of superconductors is considered the convergence of finite dimensional semidiscrete galerkin approximations is studied as is a fully discrete scheme the results of some computational experiments are presented

a numerical methodology for the painleve equations the six painleve transcendents p i p v i have both applications and analytic properties that make them stand out from most other classes of special functions although they have been the subject of extensive theoretical investigations for about a century they still have a reputation for being numerically challenging in particular their extensive pole fields in the complex plane have often been perceived as numerical mine fields in the present work we note that the painleve property in fact provides the opportunity for very fast and accurate numerical solutions throughout such fields when combining a taylor pade based ode initial value solver for the pole fields with a boundary value solver for smooth regions numerical solutions become available across the full complex plane we focus here on the numerical methodology and illustrate it for the p i equation in later studies we will concentrate on mathematical aspects of both the p i and the higher painleve transcendents

efficient enforcement of far field boundary conditions in the transformed field expansions method the method of transformed field expansions tfe has been demonstrated to be a robust and highly accurate numerical scheme for simulating solutions of boundary value and free boundary problems from the sciences and engineering as a boundary perturbation method it builds highly accurate solutions based upon exact solutions in a simple canonical geometry and corrects these via taylor series to fit the actual geometry at hand the tfe method has significantly enhanced stability properties when compared with other boundary perturbation approaches however this comes at the cost of requiring a full volumetric discretization as opposed the surface formulation that other methods can realize in this paper we outline two techniques for ameliorating this shortcoming first by employing a legendre spectral element method to implement efficient graded meshes and second by utilizing an artificial boundary with a transparent boundary condition placed quite close to the boundary of the domain in this contribution we focus on the specific problem of simulating the dirichlet neumann operator associated to laplace s equation on a periodic cell which arises in the water wave problem while the details of our results are specific to this problem the general conclusions are valid for the wider class of problems to which the tfe method can be applied for each technique we discuss implementation details and display numerical results which support the conclusion that each of these techniques can greatly reduce the computational cost of using the tfe method

maxwell s equations in a periodic structure consider a diffraction of a beam of particles in r3 when the dielectric coefficient is a constant cl above a surface s and a constant e2 below a surface s and the magnetic permeability is constant throughout rii s is assumed to be periodic in the xl direction and of the form x1 f1 s x3 f3 s x2 arbitrary we prove that there exists a unique solution to the time harmonic maxwell equations in r 3 having the form of refracted waves for x3 1 and of transmitted waves for x3 1 if and only if there exists a unique solution to a certain system of two coupled fredholm equations thus in particular for all the e s except for a discrete number there exists a unique solution to the maxwell equations introduction in this paper we consider the maxwell equations for time harmonic solutions in the entire space r3 with piecewise constant dielectric coefficient having jump across a periodic surface the magnetic permeability u is assumed to be constant whereas the dielectric coefficient e is given by e 8 above a surface s x3 f xl and e e2 below the surface s e and 82 are different constants if s is a half space x3 0 then the solution eo ho can be computed explicitly we assume in this paper that s is periodic i e f xl l f xl vx1 e r l 0 we wish to find a solution e h such that e eo and h ho are superpositions of transmitted waves 0 1 in x3 a where a maxlfl in 1 7 we assume that f e c2 and we reduce the solution of the maxwell equations to a fredholm system of four integral equations in 8 we reduce it further to a fredholm system of two integral equations thus for all but a discrete sequence of values of the physical parameters there exists a unique solution to the integral equations yielding a solution of the maxwell equations received by the editors january 25 1989 1980 mathematics subject classification 1985 revision primary 78a10 78a45 secondary 35p25 47a40

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

self interacting polymer chains terminally anchored to adsorbing surfaces of three dimensional fractal lattices abstract we have studied the adsorption problem of self attracting linear polymers modeled by self avoiding walks saws situated on three dimensional fractal structures exemplified by 3d sierpinski gasket sg family of fractals as containers of a poor solvent members of sg family are enumerated by an integer b b 2 and it is assumed that one side of each sg fractal is an impenetrable adsorbing surface we calculate the critical exponents 1 11 and s which are related to the numbers of all possible saws with one both and no ends anchored to the adsorbing boundary respectively by applying the exact renormalization group rg method for the first three members of the sg fractal family b 2 3 and 4 we have obtained specific values of these exponents for chain and globular polymer phase we discuss their mutual relations and relations with corresponding values pertinent to extended polymer chain phase

a field expansions method for scattering by periodic multilayered media the interaction of acoustic and electromagnetic waves with periodic structures plays an important role in a wide range of problems of scientific and technological interest this contribution focuses upon the robust and high order numerical simulation of a model for the interaction of pressure waves generated within the earth incident upon layers of sediment near the surface herein described is a boundary perturbation method for the numerical simulation of scattering returns from irregularly shaped periodic layered media the method requires only the discretization of the layer interfaces so that the number of unknowns is an order of magnitude smaller than finite difference and finite element simulations while it avoids not only the need for specialized quadrature rules but also the dense linear systems characteristic of boundary integral element methods the approach is a generalization to multiple layers of bruno and reitich s method of field expansions for dielectric structures with two layers by simply considering the entire structure simultaneously rather than solving in individual layers separately the full field can be recovered in time proportional to the number of interfaces as with the original field expansions method this approach is extremely efficient and spectrally accurate

solution of a boundary value problem for the helmholtz equation via variation of the boundary into the complex domain in this paper we deal with the problem of diffraction of electromagnetic waves by a periodic interface between two materials this corresponds to a two dimensional quasi periodic boundary value problem for the helmholtz equation we prove that solutions behave analytically with respect to variations of the interface the interest of this result is both theoretical the legitimacy of power series expansions in the parameters of the problem has indeed been questioned and perhaps more importantly practical we have found that the solution can be computed on the basis of this observation the simple algorithm that results from such boundary variations is described to establish the property of analyticity of the solution for the grating with respect to the height we present a holomorphic formulation of the problem using surface potentials we show that the densities entering into the potential theoretic formulation are analytic with respect to variations of the boundary or in other words that the integral operator that results from the transmission conditions at the interface is invertible in a space of holomorphic functions of the variables x y this permits us to conclude in particular that the partial derivatives of u with respect to at 0 satisfy certain boundary value problems for the helmholtz equation in regions with plane boundaries which can be solved in a closed form

approximation of analytic functions a method of enhanced convergence we deal with a method of enhanced convergence for the approximation of analytic functions this method introduces conformal transformations in the approximation problems in order to help extract the values of a given analytic function from its taylor expansion around a point an instance of this method based on the euler transform has long been known recently we introduced more general versions of it in connection with certain problems in wave scattering in 2 we present a general discussion of this approach as is known in the case of the euler transform conformal transformations can enlarge the region of convergence of power series and can enhance substantially the convergence rates inside the circles of convergence we show that conformal maps can also produce a rather dramatic improvement in the conditioning of pade approximation this improvement which we discuss theoretically for stieltjes type functions is most notorious in cases of very poorly conditioned pade problems in many instances an application of enhanced convergence in conjunction with pade approximation leads to results which are many orders of magnitude more accurate than those obtained by either classical pade approximants or the summation of a truncated enhanced series

a stable high order method for two dimensional bounded obstacle scattering a stable and high order method for solving the helmholtz equation on a two dimensional domain exterior to a bounded obstacle is developed in this paper the method is based on a boundary perturbation technique transformed field expansions coupled with a well conditioned high order spectral galerkin solver the method is further enhanced with numerical analytic continuation implemented via pade approximation ample numerical results are presented to show the accuracy stability and versatility of the proposed method

algebraic computations of scaled pade fractions two companion algorithms are developed for constructing pade fractions along an off diagonal path of the pade table for a function a z b z where a z and b z are formal power series over a field one of the algorithms computes the first n pade fractions along the off diagonal in time o n 2 when a z and b z are finite power series i e polynomials it is shown that the algorithm is equivalent to euclid s extended algorithm for computing greatest common divisors the other algorithm a generalization of the first proceeds along the off diagonal in quadratic steps and is of complexity o n log 2 n when a z and b z are polynomials the second algorithm becomes a fast euclid s extended algorithm for computing greatest common divisors the algorithm is of the same complexity as other fast greatest common divisor methods but its iterative nature provides a practical advantage during implementation the algorithms may also be used for computing pade fractions along an

exact enumeration results for self avoiding walks on the honeycomb lattice attached to a surface we consider self avoiding walks on the honeycomb lattice interacting with a surface with different energies associated between sites in contact with a linear boundary to the left of the origin and those in contact with the right of the boundary we numerically confirm recent exact results for the polymer adsorption transition and corresponding critical exponents with mixed ordinary and special boundary conditions the phase diagram is elucidated with the aid of some rigorous arguments

inverse neumann obstacle problem results are reported for direct and inverse scattering of plane acoustic waves from sound hard scatterers of arbitrary shapes in an infinite homogeneous ambience the direct problem is solved via a shape deformation technique which is valid for finite deformations the resulting solution requires solving only a set of algebraic recursion relations but neither integral equations nor green s function the methodology has three essential features all calculations refer exclusively to a known simple boundary s instead of the iteratively updated scatterer shapes the jacobian of the scattered field is evaluated by solving a series of helmholtz s equations with different boundary data in the region exterior to the simple shape s and no ambiguity in the solution arises due to the interior eigenvalues the inverse problem of recovering the scatterer boundary shape is solved by least square minimization using the levenberg marquardt algorithm inversions of several two dimensional shapes with varying degre

analytic continuation of dirichlet neumann operators the analytic dependence of dirichlet neumann operators dno with respect to variations of their domain of definition has been successfully used to devise diverse computational strategies for their estimation these strategies have historically proven very competitive when dealing with small deviations from exactly solvable geometries as in this case the perturbation series of the dno can be easily and recursively evaluated in this paper we introduce a scheme for the enhancement of the domain of applicability of these approaches that is based on techniques of analytic continuation we show that in fact dno depend analytically on variations of arbitrary smooth domains in particular this implies that they generally remain analytic beyond the disk of convergence of their power series representations about a canonical separable geometry and this in turn guarantees that alternative summation mechanisms such as pade approximation can be effectively used to numerically access this extended domain of analyticity our method of proof is motivated by our recent development of stable recursions for the coefficients of the perturbation series here we again utilize this recursion as we compare and contrast the performance of our new algorithms with that of previously advanced perturbative methods the numerical results clearly demonstrate the beneficial effect of incorporating analytic continuation procedures into boundary perturbation methods moreover the results also establish the superior accuracy and applicability of our new approach which as we show allows for precise calculations corresponding to very large perturbations of a basic geometry

the gaussian mixture probability hypothesis density filter a new recursive algorithm is proposed for jointly estimating the time varying number of targets and their states from a sequence of observation sets in the presence of data association uncertainty detection uncertainty noise and false alarms the approach involves modelling the respective collections of targets and measurements as random finite sets and applying the probability hypothesis density phd recursion to propagate the posterior intensity which is a first order statistic of the random finite set of targets in time at present there is no closed form solution to the phd recursion this paper shows that under linear gaussian assumptions on the target dynamics and birth process the posterior intensity at any time step is a gaussian mixture more importantly closed form recursions for propagating the means covariances and weights of the constituent gaussian components of the posterior intensity are derived the proposed algorithm combines these recursions with a strategy for managing the number of gaussian components to increase efficiency this algorithm is extended to accommodate mildly nonlinear target dynamics using approximation strategies from the extended and unscented kalman filters

superalgebraically convergent smoothly windowed lattice sums for doubly periodic green functions in three dimensional space this work part i in a two part series presents i a simple and highly efficient algorithm for evaluation of quasi periodic green functions as well as ii an associated boundary integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three dimensional space except for certain wood frequencies at which the quasi periodic green function ceases to exist the proposed approach which is based on smooth windowing functions gives rise to tapered lattice sums which converge superalgebraically fast to the green function that is faster than any power of the number of terms used this is in sharp contrast to the extremely slow convergence exhibited by the lattice sums in the absence of smooth windowing the wood frequency problem is treated in part ii this paper establishes rigorously the superalgebraic convergence of the windowed lattice sums a variety of numerical results demonstrate the practical efficiency of the proposed approach