the underlying structure of nonnested hypothesis tests summary a comparison between fitted and observed log likelihoods is used to study likelihood ratio statistics for both nested and nonnested hypothesis tests a study of the geometry involved in the nonnested case clarifies the relationship between cox s likelihood ratio test and the complete parametric wald encompassing test of mizon and richard further it is shown why the asymptotic distribution of these test statistics is sometimes degenerate also a new test statistic is proposed which combines the cox and mizon richard tests

a level set based eulerian method for paraxial multivalued traveltimes we apply the level set methodology to compute multivalued solutions of the paraxial eikonal equation in both isotropic and anisotropic metrics this paraxial equation is obtained from 2d stationary eikonal equations by using one of the spatial directions as the artificial evolution direction the advection velocity field used to move level sets is obtained by the method of characteristics therefore the motion of level sets is defined in a phase space and the zero level set yields the location of bicharacteristic strips in the reduced phase space the multivalued traveltime is obtained from solving another advection equation with a source term the complexity of the algorithm is o n3 logn in the worst case and o n3 in the average case where n is the number of the sampling points along one of the spatial directions numerical experiments including the well known marmousi synthetic model illustrate the accuracy and the efficiency of the eulerian method

a local level set method for paraxial geometrical optics we propose a local level set method for constructing the geometrical optics term in the paraxial formulation for the high frequency asymptotics of two dimensional 2 d acoustic wave equations the geometrical optics term consists of two multivalued functions a travel time function satisfying the eikonal equation locally and an amplitude function solving a transport equation locally the multivalued travel times are obtained by solving a level set equation and a travel time equation with a forcing term the multivalued amplitudes are computed by a neweulerian formula based on the gradients of travel times and takeoff angles as a byproduct the method is also able to capture the caustic locations the proposed eulerian method has complexity of o n 2 rm log n rather than o n 4 as typically seen in the lagrangian ray tracing method several examples including the well known marmousi synthetic model illustrate the accuracy and efficiency of the eulerian method

analysis of subspace fitting and ml techniques for parameter estimation from sensor array data it is shown that the multidimensional signal subspace method termed weighted subspace fitting wsf is asymptotically efficient this results in a novel compact matrix expression for the cramer rao bound crb on the estimation error variance the asymptotic analysis of the maximum likelihood ml and wsf methods is extended to deterministic emitter signals the asymptotic properties of the estimates for this case are shown to be identical to the gaussian emitter signal case i e independent of the actual signal waveforms conclusions concerning the modeling aspect of the sensor array problem are drawn

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice i homogeneous problems we present a computational approach for the wkb approximation of the wave function of an electron moving in a periodic one dimensional crystal lattice we derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the bloch spectrum of the schrodinger operator relying on the framework of the multibranch entropy solutions introduced by brenier and corrias we compute efficiently multiphase solutions using well adapted and simple numerical schemes in this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method

upwind finite difference calculation of traveltimes seismic traveltimes can be computed efficiently on a regular grid by an upwind finite difference method the method solves a conservation law that describes changes in the gradient components of the traveltime field the traveltime field itself is easily obtained from the solution of the conservation law by numerical integration the conservation law derives from the eikonal equation and its solution depicts the first arrival time field the upwind finite difference scheme can be implemented in fully vectorized form in contrast to a similar scheme proposed recently by vidale the resulting traveltime field is useful both in kirchhoff migration and modeling and in seismic tomography many reliable methods exist for the numerical solution of conservation laws which appear in fluid mechanics as statements of the conservation of mass momentum etc a first order upwind finite difference scheme proves accurate enough for seismic applications upwind schemes are stable because they mimic the behavior of flu

a bloch band based level set method for computing the semiclassical limit of schrodinger equations a novel bloch band based level set method is proposed for computing the semiclassical limit of schrodinger equations in periodic media for the underlying equation subject to a highly oscillatory initial data a hybrid of the wkb approximation and homogenization leads to the bloch eigenvalue problem and an associated hamilton jacobi system for the phase in each bloch band with the bloch eigenvalue be part of the hamiltonian we formulate a level set description to capture multi valued solutions to the band wkb system and then evaluate total homogenized density over a sample set of bands a superposition of band densities is established over all bands and solution branches when away from caustic points the numerical approach splits the solution process into several parts i initialize the level set function from the band decomposition of the initial data ii solve the bloch eigenvalue problem to compute bloch waves iii evolve the band level set equation to compute multi valued velocity and density on each bloch band iv evaluate the total position density over a sample set of bands using bloch waves and band densities obtained in steps ii and iii respectively numerical examples with different number of bands are shown to demonstrate the capacity of the method

distributed source modeling and direction of arrival estimation techniques in direction of arrival doa estimation the direction of a signal is usually assumed to be a point if the direction of a signal is distributed due to some environmental phenomenon however doa estimation methods based on the point source assumption may result in poor performance we consider doa estimation when the signal sources are distributed parametric and nonparametric models are proposed and estimation methods are considered under these models in addition the asymptotic distribution of estimation errors is obtained to show the models statistical properties

a fast level set method for propagating interfaces a method is introduced to decrease the computational labor of the standard level set method for propagating interfaces the fast approach uses only points close to the curve at every time step we describe this new algorithm and compare its efficiency and accuracy with the standard level set approach

asymptotic evaluation of integrals related to time dependent fields near caustics integrals which arise in the theory of time dependent wave fields near caustics in nondispersive problems are much simpler than their time harmonic counterparts because they involve delta functions rather than exponentials fields near smooth caustics for instance involve solutions of cubic equations which can be expressed in elementary terms instead of the airy functions of the better known time harmonic theory the pearcey functions which arise in the theory of the cusped caustic are similarly replaced by expressions involving solutions of quartic equations etc as is well known each type of caustic is related to a particular type of catastrophe the theory presented here deals conveniently with fields reducible to single integrals since the delta function and the integral will annihilate each other leaving closed form expressions fields involving multiple integrals will be reduced somewhat in complexity but not in general to closed form and are not treated here

a new eulerian method for the computation of propagating short acoustic and electromagnetic pulses a new method is described to compute short acoustic or electromagnetic pulses that propagate according to geometrical optics the pulses are treated as zero thickness sheets that can propagate over long distances through inhomogeneous media with multiple reflections the method has many of the advantages of lagrangian ray tracing but is completely eulerian typically using a uniform cartesian grid accordingly it can treat arbitrary configurations of pulses that can reflect from surfaces and pass through each other without requiring special computational marker arrays for each pulse also information describing the pulses which are treated as continuous surfaces can be available throughout the computational grid rather than only at isolated individual markers the method uses a new type of representation which we call dynamic surface extension the basic idea is to propagate or broadcast defining fields from each pulse surface through a computational grid these fields carry information about a nearby pulse surface that is used at each node to compute the location of the pulse surfaces and other attributes such as amplitude thus the emphasis is on the dynamics of these propagating defining fields which obey only local eulerian equations at each node the dynamic surface extension representation can be thought of as dual to level set representation the defining fields involve single valued variables which are constant at each time along lines that are normal to the evolving surface whereas level set techniques involve a function which has constant values on the evolving surface and neighboring surfaces in this way the new method overcomes the inability of level set or eikonal methods to treat intersecting pulses that obey a wave equation and can pass through each other while still using only single valued variables propagating thin pulse surfaces in 1 d 2 d and 3 d that can reflect from boundaries and pass through each other are computed using the new method the method was first presented as a new general representation of surfaces filaments and particles by j steinhoff and m fan 1998 eulerian computation of evolving surfaces curves and discontinuous fields utsi preprint

a local level set method for 3d inversion of gravity gradient data abstractwe have developed a local level set method for inverting 3d gravity gradient data to alleviate the inherent nonuniqueness of the inverse gradiometry problem we assumed that a homogeneous density contrast distribution with the value of the density contrast specified a priori was supported on an unknown bounded domain d so that we may convert the original inverse problem into a domain inverse problem because the unknown domain d may take a variety of shapes we parametrized the domain d by a level set function implicitly so that the domain inverse problem was reduced to a nonlinear optimization problem for the level set function because the convergence of the level set algorithm relied heavily on initializing the level set function to enclose the gravity center of a source body we applied a weighted l1 regularization method to locate such a gravity center so that the level set function can be properly initialized to rapidly compute the gradient of the nonlinear functional arising in the level

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

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

geometric optics in a phase space based level set and eulerian framework we introduce a level set approach for ray tracing and the construction of wavefronts in geometric optics this is important in a wide variety of applications in wave propagation our approach automatically handles the multivalued solutions that appear and automatically resolves the wavefronts this is achieved through solving for the bicharacteristic strips whose projection to spatial space gives the wavefronts in a reduced phase space under a eulerian and partial differential equation based framework the bicharacteristic strips are represented using a level set approach for handling higher codimensional objects and the partial differential equations responsible for the evolution are reduced forms of the liouville equations results for the two dimensional case for constant and variable indices of refraction are shown and compared to those of other current methods in the field results are also introduced to show the ability to handle reflection and to extend the method to the three dimensional case

detection and localization of multiple sources via the stochastic signals model a method for the detection of coherent and noncoherent signals based on the application of j rissanen s 1978 minimum description length principle for model selection to the stochastic signals model is presented the detection and localization are done simultaneously with the location estimator coinciding with the maximum likelihood estimator derived by j f bohme 1986 the proposed method outperforms the method of m wax and i ziskind 1989 especially in the threshold region simulation results demonstrating the improved performance are included

towards a quantum analog of weak kam theory we discuss a quantum analogue of mather s minimization principle for lagrangian dynamics and provide some formal calculations suggesting the corresponding euler lagrange equation we then rigorously construct from the dual eigenfunctions of a certain non selfadjoint operator a candidate for a minimizer and recover aspects of weak kam theory in the limit as h 0 regarding our state as a quasimode we furthermore derive some error estimates although it remains an open problem to improve these bounds

invariant distributions and cohomology for geodesic flows and higher cohomology of higher rank anosov actions abstract we are motivated by a conjecture of a and s katok to study the smooth cohomologies of a family of weyl chamber flows the conjecture is a natural generalization of the livsic theorem to anosov actions by higher rank abelian groups it involves a description of top degree cohomology and a vanishing statement for lower degrees our main result proved in part ii verifies the conjecture in lower degrees for our systems and steps in the correct direction in top degree in part i we study our base case geodesic flows of finite volume hyperbolic manifolds we describe obstructions invariant distributions to solving the coboundary equation in unitary representations of the group of orientation preserving isometries of hyperbolic n space and we study sobolev regularity of solutions one byproduct is a smooth livsic theorem for geodesic flows of hyperbolic manifolds with cusps part i provides the tools needed in part ii for the main theorem

a simple level set method for solving stefan problems a simple level set method for solving stefan problems is presented this method can be applied to problems involving dendritic solidification our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance our method is accurate with respect to some exact solutions of the stefan problem results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

simple and efficient nonparametric method for estimating the number of signals without eigendecomposition inspired by the computational simplicity and numerical stability of qr decomposition a nonparametric method for estimating the number of signals without eigendecomposition mense is proposed for the coherent narrowband signals impinging on a uniform linear array ula by exploiting the array geometry and its shift invariance property to decorrelate the coherency of signals through subarray averaging the number of signals is revealed in the rank of the qr upper trapezoidal factor of the autoproduct of a combined hankel matrix formed from the cross correlations between some sensor data since the infection of additive noise is defused signal detection capability is improved a new detection criterion is then formulated in terms of the row elements of the qr upper triangular factor when finite array data are available and the number of signals is determined as a value of the running index for which this ratio criterion is maximized where the qr decomposition with column pivoting is also used to improve detection performance the statistical analysis clarifies that the mense detection criterion is asymptotically consistent furthermore the proposed mense algorithm is robust against the array uncertainties including sensor gain and phase errors and mutual coupling and against the deviations from the spatial homogeneity of noise model the effectiveness of the mense is verified through numerical examples and the simulation results show that the mense is superior in detecting closely spaced signals with a small number of snapshots and or at relatively low signal to noise ratio snr

robust properties of likelihood ratio tests summary the usual asymptotic chi squared distribution for the likelihood ratio test statistic is based on the assumptions that the data come from the parametric model under consideration and that the parameter satisfies the null hypothesis in this paper we examine the distribution of the likelihood ratio statistic when the data do not come from the parametric model but when the nearest member of the parametric family still satisfies the null hypothesis in general the likelihood ratio statistic no longer follows an asymptotic chi squared distribution and an alternative statistic based on the unionintersection approach is proposed

using k branch entropy solutions for multivalued geometric optics computations this paper is devoted to a numerical simulation of the classical wkb system arising in geometric optics expansions it contains the nonlinear eikonal equation and a linear conservation law whose coefficient can be discontinuous we address the problem of treating it in such a way that superimposed signals can be reproduced by means of the kinetic formulation of multibranch solutions originally due to brenier and corrias some existence and uniqueness results are given together with computational test cases of increasing difficulty displaying up to five multivaluations

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

eulerian gaussian beams for high frequency wave propagation we design an eulerian gaussian beam summation method for solving helmholtz equations in the high frequency regime the traditional gaussian beam summation method is based on lagrangian ray tracing and local ray centered coordinates we propose a new eulerian formulation of gaussian beam theory which adopts global cartesian coordinates level sets and liouville equations yielding uniformly distributed eulerian traveltimes and amplitudes in phase space simultaneously for multiple sources the time harmonic wavefield can be constructed by summing up gaussian beams with ingredients provided by the new eulerian formulation the conventional gaussian beam summation method can be derived from the proposed method there are three advantages of the new method 1 we have uniform resolution of ray distribution 2 we can obtain wavefields excited at different sources by varying only source locations in the summation formula 3 we can obtain wavefields excited at different frequencies by varying only frequencies in the summation formula numerical experiments indicate that the gaussian beam summation method yields accurate asymptotic wavefields even at caustics the new method may be used for seismic modeling and migration

traveltime and amplitude estimation using wavefront construction we have developed and implemented a new method for estimating traveltimes and amplitudes in a general smooth two dimensional 2 d model the basic idea of this wavefront wf construction approach is to use ray tracing to estimate a new wf from the old one the wf is defined as a curve in 2 d of constant traveltime from the source the ray direction and amplitude will then be a function of s the distance along the front to maintain a sufficiently small sampling distance along the wf it is scanned at every time step and new rays are interpolated whenever the distance between two rays becomes larger than a predefined limit as the wavefronts are constructed the data i e traveltimes amplitude coefficients etc are transferred to the receivers by interpolation within the ray cells advantages of the wf construction method are its flexibility robustness and accuracy first second and later arrivals may be found at any point in the model any shape of the initial wavefront is possible the drawb

on tests for global maximum of the log likelihood function given the location of a relative maximum of the log likelihood function how to assess whether it is the global maximum this paper investigates an existing statistical tool which based on asymptotic analysis answers this question by posing it as a hypothesis testing problem a general framework for constructing tests for global maximum is given the characteristics of the tests are investigated for two cases correctly specified model and model mismatch a finite sample approximation to the power is given which gives a tool for performance prediction and a measure for comparison between tests the sensitivity of the tests to model mismatch is analyzed in terms of the renyi divergence and the kullback leibler divergence between the true underlying distribution and the assumed parametric class and tests that are insensitive to small deviations from the model are derived thereby overcoming a fundamental weakness of existing tests the tests are illustrated for three applications passive localization or direction finding using an array of sensors estimating the parameters of a gaussian mixture model and estimation of superimposed exponentials in noise problems that are known to suffer from local maxima

livsic theorem for matrix cocycles we prove the liv sic theorem for arbitrary gl m r cocycles we con sider a hyperbolic dynamical system f x x and a holder continuous function a x gl m r we show that if a has trivial periodic data i e a f n 1 p a fp a p id for each periodic point p f n p then there exists a holder continuous function c x gl m r satisfying a x c fx c x 1 for all x2 x the main new ingredients in the proof are results of independent interest on relations between the periodic data lyapunov exponents and uniform estimates on growth of products along orbits for an arbitrary holder function a

optimal and suboptimal broad band source location estimation two maximum likelihood ml estimators are considered for direction of arrival doa estimation of broadband sources with unknown spectral parameters one is based on the assumption that the sources radiate stochastic gaussian signals and therefore is called the stochastic gaussian ml sgml estimator the other using estimates of the actual signals not their assumed distribution is called the conditional ml cml estimator neither is efficient if the source spectral parameters are completely arbitrary and unknown but the problem can be avoided for a version of the sgml estimation if the signal and noise spectra are known to satisfy certain smoothness conditions while this version of the sgml is formally superior to the cml it is demonstrated that the performance difference is small with underconditions not infrequently encountered in practice when these are satisfied the computationally simpler cml can be used without significant loss the required conditions become more stringent as the source separation decreases or correlation between sources increases a closed form analytic expression is obtained for the small error variance of the cml estimator of the doa of the nth source in the presence of n 1 other sources

music maximum likelihood and cramer rao bound further results and comparisons the problem of determining the direction of arrival of narrowband plane waves using sensor arrays and the related problem of estimating the parameters of superimposed signals from noisy measurements are studied a number of results have been recently presented by the authors on the statistical performance of the multiple signal characterization music and the maximum likelihood ml estimators for the above problems this work extends those results in several directions first it establishes that in the class of weighted music estimators the unweighted music achieves the best performance i e the minimum variance of estimation errors in large samples next it derives the covariance matrix of the ml estimator and presents detailed analytic studies of the statistical efficiency of music and ml estimators these studies include performance comparisons of music and mle with each other as well as with the ultimate performance corresponding to the cramer rao bound finally some numerical examples are given which provide a more quantitative study of performance for the problem of finding two directions with uniform linear sensor arrays

multi phase computations in geometrical optics in this work we propose a new set of partial differential equations pdes which can be seen as a generalization of the classical eikonal and transport equations to allow for solutions with multiple phases the traditional geometrical optics pair of equations suffer from the fact that the class of physically relevant solutions is limited in particular it does not include solutions with multiple phases corresponding to crossing waves our objective has been to generalize these equations to accommodate solutions containing more than one phase the new equations are based on the same high frequency approximation of the scalar wave equation as the eikonal and the transport equations however they also incorporate a finite superposition principle the maximum allowed number of intersecting waves in the solution can be chosen arbitrarily but a higher number means that a larger system of pdes must be solved the pdes form a hyperbolic system of conservation laws with source terms although the equations are only weakly hyperbolic and thus not well posed in the strong sense several examples show the viability of solving the equations numerically the technique we use to capture multivalued solutions is based on a closure assumption for a system of equations representing the moments

maximum likelihood array processing in non gaussian noise with gaussian mixtures many approaches have been studied for the array processing problem when the additive noise is modeled with a gaussian distribution but these schemes typically perform poorly when the noise is non gaussian and or impulsive this paper is concerned with maximum likelihood array processing in non gaussian noise we present the cramer rao bound on the variance of angle of arrival estimates for arbitrary additive independent identically distributed iid symmetric non gaussian noise then we focus on non gaussian noise modeling with a finite gaussian mixture distribution which is capable of representing a broad class of non gaussian distributions that include heavy tailed impulsive cases arising in wireless communications and other applications based on the gaussian mixture model we develop an expectation maximization em algorithm for estimating the source locations the signal waveforms and the noise distribution parameters the important problems of detecting the number of sources and obtaining initial parameter estimates for the iterative em algorithm are discussed in detail the initialization procedure by itself is an effective algorithm for array processing in impulsive noise novel features of the em algorithm and the associated maximum likelihood formulation include a nonlinear beamformer that separates multiple source signals in non gaussian noise and a robust covariance matrix estimate that suppresses impulsive noise while also performing a model based interpolation to restore the low rank signal subspace the em approach yields improvement over initial robust estimates and is valid for a wide snr range the results are also robust to pdf model mismatch and work well with infinite variance cases such as the symmetric stable distributions simulations confirm the optimality of the em estimation procedure in a variety of cases including a multiuser communications scenario we also compare with existing array processing algorithms for non gaussian noise

direct computation of multivalued phase space solutions for hamilton jacobi equations a method is presented for the exact and complete resolution of hamiltonian systems only using the corresponding hamilton jacobi equation we show that it possible to get the ode solution from the corresponding pde

the stochastic caustic the propagation of a high frequency initially plane wave through a homogeneous and isotropic random medium with small order o sigma fluctuations in index of refraction is investigated using geometrical optics it is shown that caustics occur along every ray in a distance scale of order o sigma 2 3 a where a is the correlation length of the medium on this scale the ray angle deviations are small of order o sigma 2 3 while the ray position deviates o 1 from its deterministic value furthermore it is shown that if t gamma 1 3 s where s is arclength along a ray gamma 2 int 0 infty dr left frac 1 r frac partial partial r right 2 r r and r r is the correlation function of the medium then as a function of t the probability density of distance to first caustic is as sigma to 0 a universal curve with no free parameters and thus does not depend on the detailed statistics of the random medium for small values of t this density is given by t

an eulerian approach for computing the finite time lyapunov exponent we propose efficient eulerian methods for approximating the finite time lyapunov exponent ftle the idea is to compute the related flow map using the level set method and the liouville equation there are several advantages of the proposed approach unlike the usual lagrangian type computations the resulting method requires the velocity field defined only at discrete locations no interpolation of the velocity field is needed also the method automatically stops a particle trajectory in the case when the ray hits the boundary of the computational domain the computational complexity of the algorithm is o dx d 1 with d the dimension of the physical space since there are the same number of mesh points in the x t space the computational complexity of the proposed eulerian approach is optimal in the sense that each grid point is visited for only o 1 time we also extend the algorithm to compute the ftle on a co dimension one manifold the resulting algorithm does not require computation on any local coordinate system and is simple to implement even for an evolving manifold

a maximal invariant framework for adaptive detection with structured and unstructured covariance matrices we introduce a framework for exploring array detection problems in a reduced dimensional space by exploiting the theory of invariance in hypothesis testing this involves calculating a low dimensional basis set of functions called the maximal invariant the statistics of which are often tractable to obtain thereby making analysis feasible and facilitating the search for tests with some optimality property using this approach we obtain a locally most powerful invariant test for the unstructured covariance case and show that all invariant tests can be expressed in terms of the previously published kelly s generalized likelihood ratio glrt and robey s adaptive matched filter amf test statistics applying this framework to structured covariance matrices corresponding to stochastic interferers in a known subspace for which the glrt is unavailable we obtain the maximal invariant and propose several new invariant detectors that are shown to perform as well or better than existing ad hoc detectors these invariant tests are unaffected by most nuisance parameters hence the variation in the level of performance is sharply reduced this framework facilitates the search for such tests even when the usual glrt is unavailable

marginal likelihood for estimation and detection theory this paper derives and analyzes the asymptotic performances of the maximum likelihood ml estimator and the generalized likelihood ratio test glrt derived under the assumption of independent identically distribution i i d samples where in the actual model the signal samples are m dependent the ml and glrt under such a modeling mismatch are based on the marginal likelihood function and they are referred to as marginal maximum likelihood mml and generalized sum marginal log likelihood ratio test gmlrt respectively under some regularity conditions the asymptotical distributions of the mml and gmlrt are derived the asymptotical distributions in some signal processing examples are analyzed simulation results support the theory via several examples

eigenvalues and eigenvectors of covariance matrices for signals closely spaced in frequency the eigenstructures of common covariance matrices are identified for the general case of m closely spaced signals it is shown that the largest signal space eigenvalue is relatively insensitive to signal separation by contrast the ith largest eigenvalue is proportional to delta omega sup 2 i 1 or delta omega sup 4 i 1 where delta omega is a measure of signal separation therefore matrix conditioning degrades rapidly as signal separation is reduced it is also shown that the limiting eigenvectors have remarkably simple structures the results are very general and apply to planar far field direction finding problems involving almost arbitrary scenarios and also to time series analysis of sinusoids exponentials and other signals

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

invariant distributions and time averages for horocycle flows there are infinitely many obstructions to existence of smooth solutions of the cohomological equation uu f where u is the vector field generating the horocycle flow on the unit tangent bundle sm of a riemann m surface of finite area and f is a given function on sm we study the sobolev regularity of these obstructions construct smooth solutions of the cohomological equation and derive asymptotics for the ergodic averages of horocycle flows

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

approximate maximum likelihood estimators for array processing in multiplicative noise environments we consider the problem of localizing a source by means of a sensor array when the received signal is corrupted by multiplicative noise this scenario is encountered for example in communications owing to the presence of local scatterers in the vicinity of the mobile or due to wavefronts that propagate through random inhomogeneous media since the exact maximum likelihood ml estimator is computationally intensive two approximate solutions are proposed originating from the analysis of the high and low signal to noise ratio snr cases respectively first starting with the no additive noise case a very simple approximate ml aml sub 1 estimator is derived the performance of the aml sub 1 estimator in the presence of additive noise is studied and a theoretical expression for its asymptotic variance is derived its performance is shown to be close to the cramer rao bound crb for moderate to high snr next the low snr case is considered and the corresponding aml sub 2 solution is derived it is shown that the approximate ml criterion can be concentrated with respect to both the multiplicative and additive noise powers leaving out a two dimensional 2 d minimization problem instead of a four dimensional 4 d problem required by the exact ml numerical results illustrate the performance of the estimators and confirm the validity of the theoretical analysis

general asymptotic analysis of the generalized likelihood ratio test for a gaussian point source under statistical or spatial mismodeling this paper investigates the robustness of the generalized likelihood ratio test glrt for a far field gaussian point source given measurements from an array of sensors the performance of the glrt under two types of common modeling errors is investigated the first type is spatial mismodeling which relates to errors due to multipath effects or errors in the assumed number of sources i e deviation from the single point source assumption the second type is statistical mismodeling which relates to errors due to non gaussianity in either the noise or the signal i e deviation from the gaussian assumption it is shown that for some types of modeling errors the detector s performance improves and general conditions for such an improvement are found moreover for both types of errors the change in performance is analyzed and quantified this analysis shows that for a distributed source with small spatial spreading the degradation in performance is significant whereas for a constant modulus point source the performance improves simulations of various cases are shown to verify the analytical results

fundamental limits of linear equalizers diversity capacity and complexity linear equalizers les have been widely adopted for practical systems due to their low computational complexity however it is also well known that les provide inferior performance relative to a maximum likelihood equalizer mle or other near mles because les usually cannot collect the diversity order enabled by the transmitter and at the same time they lose mutual information more importantly unlike mle or near mles the performance of les has not been well quantified this hinders more general applications of les in wireless systems in this paper we reveal a fundamental link between a channel parameter orthogonality deficiency od of the channel matrix and the diversity and capacity of les we identify that when the od of channel matrix has an upper bound strictly less than 1 the same diversity order as that of mles is collected by les and the outage capacity loss relative to mles is also a constant over signal to noise ratio snr these results can be applied to designing a framework for hybrid equalizers furthermore by studying the statistical properties of the od and comparing the complexity of different equalizers we show that hybrid equalizers can trade off between performance and complexity by tuning the channel matrix od the theoretical analysis is corroborated by computer simulations

high frequency wave propagation by the segment projection method geometrical optics is a standard technique used for the approximation of high frequency wave propagation computational methods based on partial differential equations instead of the traditional ray tracing have recently been applied to geometrical optics these new methods have a number of advantages but typically exhibit difficulties with linear superposition of waves in this paper we introduce a new partial differential technique based on the segment projection method in phase space the superposition problem is perfectly resolved and so is the problem of computing amplitudes in the neighborhood of caustics the computational complexity is of the same order as that of ray tracing the new algorithm is described and a number of computational examples are given including a simulation of waveguides

eulerian gaussian beams for schrodinger equations in the semi classical regime we propose gaussian beam based eulerian methods to compute semi classical solutions of the schrodinger equation traditional gaussian beam type methods for the schrodinger equation are based on the lagrangian ray tracing based on the first eulerian gaussian beam framework proposed in leung et al s leung j qian r burridge eulerian gaussian beams for high frequency wave propagation geophysics 72 2007 sm61 sm76 we develop a new eulerian gaussian beam method which uses global cartesian coordinates level set based implicit representation and liouville equations the resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously to obtain semi classical solutions to the schrodinger equation with different initial wave functions we only need to slightly modify the summation formula this yields a very efficient method for computing semi classical solutions to the schrodinger equation for instance in the one dimensional case the proposed algorithm requires only o snm2 operations to compute s different solutions with s different initial wave functions under the influence of the same potential where n o 1 is the planck constant and m n is the number of computed beams which depends on weakly numerical experiments indicate that this eulerian gaussian beam approach yields accurate semi classical solutions even at caustics

on posterior distributions for signals in gaussian noise with unknown covariance matrix a bayesian approach to estimate parameters of signals embedded in complex gaussian noise with unknown color is presented the study specifically focuses on a bayesian treatment of the unknown noise covariance matrix making up a nuisance parameter in such problems by integrating out uncertainties regarding the noise color an enhanced ability to estimate both the signal parameters as well as properties of the error is exploited several noninformative priors for the covariance matrix such as the reference prior the jeffreys prior and modifications to this are considered some of the priors result in analytical solutions whereas others demand numerical approximations in the linear signal model connections are made between the standard adaptive maximum likelihood aml estimate and a bayesian solution using the jeffreys prior with adjustments to the jeffreys prior correspondence to the regularized solution is also established this in turn enables a formal treatment of the regularization parameter simulations indicate that significant improvements compared to the aml estimator can be obtained by considering both the derived regularized solutions as well as the one obtained using the reference prior the simulations also indicate the possibility of enhancing the predictions of properties of the error as uncertainties in the noise color are acknowledged

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

detection of signals by information theoretic criteria general asymptotic performance analysis detecting the number of sources is a well known and a well investigated problem in this problem the number of sources impinging on an array of sensors is to be estimated the common approach for solving this problem is to use an information theoretic criterion like the minimum description length mdl or the akaike information criterion aic although it has been gaining much popularity and has been used in a variety of problems the performance of information theoretic criteria based estimators for the unknown number of sources has not been sufficiently studied yet in the context of array processing the performance of such estimators were analyzed only for the special case of gaussian sources where no prior knowledge of the array structure if given is used based on the theory of misspecified models this paper presents a general asymptotic analysis of the performance of any information theoretic criterion based estimator and especially of the mdl estimator in particular the performance of the mdl estimator which assumes gaussian sources and structured array when applied to gaussian sources is analyzed in addition it is shown that the performance of a certain mdl estimator is not very sensitive to the actual distribution of the source signals however appropriate use of prior knowledge about the array geometry can lead to significant improvement in the performance of the mdl estimator simulation results show a good fit between the empirical and the theoretical results

parametric localization of distributed sources most array processing algorithms are based on the assumption that the signals are generated by point sources this is a mathematical constraint that is not satisfied in many applications in this paper we consider situations where the sources are distributed in space with a parametric angular cross correlation kernel we propose an algorithm that estimates the parameters of this model using a generalization of the music algorithm the method involves maximizing a cost function that depends on a matrix array manifold and the noise eigenvectors we study two particular cases coherent and incoherent spatial source distributions the spatial correlation function for a uniformly distributed signal is derived from this we find the array gain and show that in contrast to point sources it does not increase linearly with the number of sources we compare our method to the conventional point source music algorithm the simulation studies show that the new method outperforms the music algorithm by reducing the estimation bias and the standard deviation for scenarios with distributed sources it is also shown that the threshold signal to noise ratio required for resolving two closely spaced distributed sources is considerably smaller for the new method

source number estimators using transformed gerschgorin radii we introduce effective uses of gerschgorin radii of the unitary transformed covariance matrix for source number estimation there are two approaches likelihood and heuristic used for developing the detection criteria the likelihood approach combines the gerschgorin radii to the well known source number detectors and improves their detection performances for gaussian and white noise processes it is verified that the gerschgorin likelihood estimators gle are consistent the gerschgorin aic yields a consistent estimate and the gerschgorin mdl criterion does not tend to underestimate for small or moderate data samples the heuristic approach applying the gerschgorin disk estimator gde developed from the projection concept overcomes the problem in cases of small data samples an unknown noise model and data dependency furthermore the detection performances of both approaches through the suggested rotations and averaging can be further improved finally the proposed and existing criteria are evaluated in various conditions by using simulated and measured experimental data

logarithm laws for flows on homogeneous spaces in this paper we generalize and sharpen d sullivan s logarithm law for geodesics by specifying conditions on a sequence of subsets at t n of a homogeneous space g g a semisimple lie group an irreducible lattice and a sequence of elements ft of g under which t n ftx at is infinite for a e x g the main tool is exponential decay of correlation coefficients of smooth functions ong besides the general higher rank version of sullivan s result as a consequence we obtain a new proof of the classical khinchin groshev theorem on simultaneous diophantine approximation and settle a conjecture recently made by m skriganov