a bayesian approach to adaptive detection in nonhomogeneous environments we consider the adaptive detection of a signal of interest embedded in colored noise when the environment is nonhomogeneous i e when the training samples used for adaptation do not share the same covariance matrix as the vector under test a bayesian framework is proposed where the covariance matrices of the primary and the secondary data are assumed to be random with some appropriate joint distribution the prior distributions of these matrices require a rough knowledge about the environment this provides a flexible yet simple knowledge aided model where the degree of nonhomogeneity can be tuned through some scalar variables within this framework an approximate generalized likelihood ratio test is formulated accordingly two bayesian versions of the adaptive matched filter are presented where the conventional maximum likelihood estimate of the primary data covariance matrix is replaced either by its minimum mean square error estimate or by its maximum a posteriori estimate two detectors require generating samples distributed according to the joint posterior distribution of primary and secondary data covariance matrices this is achieved through the use of a gibbs sampling strategy numerical simulations illustrate the performances of these detectors and compare them with those of the conventional adaptive matched filter

a superconvergent ldg hybridizable galerkin method for second order elliptic problems we identify and study an ldg hybridizable galerkin method which is not an ldg method for second order elliptic problems in several space dimensions with remarkable convergence properties unlike all other known discontinuous galerkin methods using polynomials of degree k 0 for both the potential as well as the flux the order of convergence in l 2 of both unknowns is k 1 moreover both the approximate potential as well as its numerical trace superconverge in l 2 like norms to suitably chosen projections of the potential with order k 2 this allows the application of element by element postprocessing of the approximate solution which provides an approximation of the potential converging with order k 2 in l 2 the method can be thought to be in between the hybridized version of the raviart thomas and that of the brezzi douglas marini mixed methods

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

estimation for the linear model with uncertain covariance matrices we derive a maximum a posteriori estimator for the linear observation model where the signal and noise covariance matrices are both uncertain the uncertainties are treated probabilistically by modeling the covariance matrices with prior inverse wishart distributions the nonconvex problem of jointly estimating the signal of interest and the covariance matrices is tackled by a computationally efficient fixed point iteration as well as an approximate variational bayes solution the statistical performance of estimators is compared numerically to state of the art estimators from the literature and shown to perform favorably

covariance matrix estimation with heterogeneous samples we consider the problem of estimating the covariance matrix mp of an observation vector using heterogeneous training samples i e samples whose covariance matrices are not exactly mp more precisely we assume that the training samples can be clustered into k groups each one containing lk snapshots sharing the same covariance matrix mk furthermore a bayesian approach is proposed in which the matrices mk are assumed to be random with some prior distribution we consider two different assumptions for mp in a fully bayesian framework mp is assumed to be random with a given prior distribution under this assumption we derive the minimum mean square error mmse estimator of mp which is implemented using a gibbs sampling strategy moreover a simpler scheme based on a weighted sample covariance matrix scm is also considered the weights minimizing the mean square error mse of the estimated covariance matrix are derived furthermore we consider estimators based on colored or diagonal loading of the weighted scm and we determine theoretically the optimal level of loading finally in order to relax the a priori assumptions about the covariance matrix mp the second part of the paper assumes that this matrix is deterministic and derives its maximum likelihood estimator numerical simulations are presented to illustrate the performance of the different estimation schemes

superconvergence in a dpg method for an ultra weak formulation abstract in this work we study a dpg method for an ultra weak variational formulation of a reaction diffusion problem we improve existing a priori convergence results by sharpening an approximation result for the numerical flux by duality arguments we show that higher convergence rates for the scalar field variable are obtained if the polynomial order of the corresponding approximation space is increased by one furthermore we introduce a simple elementwise postprocessing of the solution and prove superconvergence numerical experiments indicate that the obtained results are valid beyond the underlying model problem

restriction to symmetric subgroups of unitary representations of rank one semisimple lie groups we consider the spherical complementary series of rank one lie groups h n so0 n 1 f for f r c h we prove that there exist finitely many discrete components in its restriction under the subgroup hn 1 so0 n 1 1 f this is proved by imbedding the complementary series into analytic continuation of holomorphic discrete series of g n su n 1 su n 1 x su n 1 and su 2n 2 and by the branching of holomorphic representations under the corresponding subgroup g n 1

berezin transform on real bounded symmetric domains let d be a bounded symmetric domain in a complex vector space vc with a real form v and d d v g k be the real bounded symmetric domain in the real vector space v we construct the berezin kernel and consider the berezin transform on the l2 space on d the corresponding representation of g is then unitarily equivalent to the restriction to g of a scalar holomorphic discrete series of holomorphic functions on d and is also called the canonical representation we find the spectral symbol of the berezin transform under the irreducible decomposition of the l2 space introduction the main purpose of the present paper is to calculate the spectral symbol of the berezin transform on real bounded symmetric domains to explain our results and motivations we let d be the unit disk in the complex plane with the lebesgue measure dm z we consider the weighted bergman space h 1 of holomorphic functions on d square integrable with respect to the measure 1 z 2 2dm z it has up to some positive constant the reproducing kernel kw z k z w 1 zw moreover the group gc su 1 1 of fractional transformations of d acts on the space h via f z 7 f gz g z 2 and it forms an irreducible unitary projective representation consider the subgroup so 1 1 consisting of transformations of the form z 7 az b bz a with a b r and a b 1 thus it is of interest to study the irreducible decomposition of the weighted bergman space under the subgroup g so 1 1 for that purpose we consider the unit interval d d r 1 1 as a trivial symmetric space g k so 1 1 1 and the restriction of holomorphic functions in h to the interval d more precisely consider r h c d rf x f x 1 x 2 x d let l d d 0 be the l space on d with the so 1 1 invariant measure d 0 x dx 1 x2 whose decomposition under so 1 1 can be done via the mellin transform see below the restriction r is a bounded operator from h into the space l d d 0 with dense image and intertwines the respective actions of so 1 1 received by the editors january 16 2000 and in revised form october 10 2000 2000 mathematics subject classification primary 22e46 43a85 32m15 53c35

an analysis of the practical dpg method we give a complete error analysis of the discontinuous petrov galerkin dpg method accounting for all the approximations made in its practical implementation specifi cally we consider the dpg method that uses a trial space consisting of polynomials of degree p on each mesh element earlier works showed that there is a trial to test operator t which when applied to the trial space defines a test space that guarantees stability in dpg formulations this operator t is local it can be applied element by element however an infinite dimensional problem on each mesh element needed to be solved to apply t in prac tical computations t is approximated using polynomials of some degree r p on each mesh element we show that this approximation maintains optimal convergence rates provided that r 2 p n where n is the space dimension two or more for the laplace equation we also prove a similar result for the dpg method for linear elasticity remarks on the conditioning of the stiffness matrix in dpg methods are also included

statistics of adaptive nulling and use of the generalized eigenrelation ger for modeling inhomogeneities in adaptive processing this paper examines the integrity of the generalized eigenrelation ger which is an approach to assessing performance in an adaptive processing context involving covariance estimation when the adaptive processors are subject to undernulled interference the ger is a mathematical relation which if satisfied often facilitates closed form analysis of adaptive processors employing estimated covariances subject to inhomogeneities the goal of this paper is to determine what impact this constraint has on the integrity of the adaptive nulling process in order to examine the impact of the ger constraint on adaptive nulling we establish fundamental statistical convergence properties of an adaptive null for the sample covariance based scb minimum variance distortionless response mvdr beamformer novel exact expressions relating the mean and variance of an adaptive null of a homogeneously trained beamformer to the mean and variance of a nonhomogeneous trained beamformer are derived in addition it is shown that the reed et al 1974 result for required sample support can be highly inaccurate under nonhomogeneous conditions indeed the required sample support can at times depend directly on the power of the undernulled interference

the adaptive coherence estimator a uniformly most powerful invariant adaptive detection statistic we show that the adaptive coherence estimator ace is a uniformly most powerful ump invariant detection statistic this statistic is relevant to a scenario appearing in adaptive array processing in which there are auxiliary signal free training data vectors that can be used to form a sample covariance estimate for clutter and interference suppression the result is based on earlier work by bose and steinhardt who found a two dimensional 2 d maximal invariant when test and training data share the same noise covariance their 2 d maximal invariant is given by kelly s generalized likelihood ratio test glrt statistic and the adaptive matched filter amf we extend the maximal invariant framework to the problem for which the ace is a glrt the test data shares the same covariance structure as the training data but the relative power level is not constrained in this case the maximal invariant statistic collapses to a one dimensional 1 d scalar which is also the ace statistic furthermore we show that the probability density function for the ace possesses the property of total positivity which establishes that it has monotone likelihood ratio thus a threshold test on the ace is ump invariant this means that it has a claim to optimality having the largest detection probability out of the class of detectors that are also invariant to affine transformations on the data matrix that leave the hypotheses unchanged this requires an additional invariance not imposed by bose and steinhardt invariance to relative scaling of test and training data the ace is invariant and has a constant false alarm rate cfar with respect to such scaling whereas kelly s glrt and the amf are invariant and cfar only with respect to common scaling

the dpg star method abstract this article introduces the dpg star from now on denoted dpg finite element method it is a method that is in some sense dual to the discontinuous petrov galerkin dpg method the dpg methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem in the same vein the dpg methodology is a means to solve an underdetermined discretization these two viewpoints are developed by embedding the same operator equation into two different saddle point problems the analyses of the two problems have many common elements comparison to other methods in the literature round out the newly garnered perspective notably dpg and dpg methods can be seen as generalizations of l l and least squares methods respectively a priori error analysis and a posteriori error control for the dpg method are considered in detail reports of several numerical experiments are provided which demonstrate the essential features of the new method a notable difference between the results from the dpg and dpg analyses is that the convergence rates of the former are limited by the regularity of an extraneous lagrange multiplier variable

branching laws for minimal holomorphic representations abstract in this paper we study the branching law for the restriction from su n m to so n m of the minimal representation in the analytic continuation of the scalar holomorphic discrete series we identify the group decomposition with the spectral decomposition of the action of the casimir operator on the subspace of s o n o m invariants the plancherel measure of the decomposition defines an l 2 space of functions for which certain continuous dual hahn polynomials furnish an orthonormal basis it turns out that the measure has point masses precisely when n m 2 under these conditions we construct an irreducible representation of so n m identify it with a parabolically induced representation and construct a unitary embedding into the representation space for the minimal representation of su n m

a robust petrov galerkin discretisation of convection diffusion equations a petrov galerkin discretisation is studied of an ultra weak variational formulation of the convection diffusion equation in a mixed form to arrive at an implementable method the truly optimal test space has to be replaced by its projection onto a finite dimensional test search space to prevent that this latter space has to be taken increasingly large for vanishing diffusion a formulation is constructed that is well posed in the limit case of a pure transport problem numerical experiments show approximations that are very close to the best approximations to the solution from the trial space uniformly in the size of the diffusion term

joint estimation of inverse covariance matrices lying in an unknown subspace we consider the problem of joint estimation of inverse covariance matrices lying in an unknown subspace of the linear space of symmetric matrices we perform the estimation using groups of measurements with different covariances assuming the inverse covariances span a low dimensional subspace our aim is to determine this subspace and to exploit this knowledge in order to improve the estimation we develop a novel optimization algorithm discovering and exploiting the underlying low dimensional subspace we provide a computationally efficient algorithm and derive a tight upper performance bound numerical simulations on synthetic and real world data are presented to illustrate the performance benefits of the algorithm

berezin kernels of tube domains abstract we study the berezin kernel decomposition into positive definite spherical functions on real forms of tube domains in other words we obtain for all tube domains associated to causal pairs g h the decomposition of holomorphic discrete series representations of g when restricted to the fully restrictive subgroup h

knowledge aided parametric adaptive matched filter with automatic combining for covariance estimation in this paper a knowledge aided parametric adap tive matched filter ka pamf is proposed that utilizing both observations including the test and training signals and ap riori knowledge of the spatial covariance matrix unlike existing ka pamf methods the proposed ka pamf is able to automat ically adjust the combining weight of ap rioricovariance matrix thus gaining enhanced robustness against uncertainty in the prior knowledge meanwhile the proposed ka pamf is significantly more efficient than its ka nonparametric counterparts when the amount of training signals is limited one distinct feature of the proposed ka pamf is the inclusion of both the test and training signals for automatic determination of the combining weights for the prior spatial covariance matrix and observations numerical results are presented to demonstrate the effectiveness of the proposed ka pamf especially in the limited training scenarios

a class of discontinuous petrov galerkin methods part iv the optimal test norm and time harmonic wave propagation in 1d the phase error or the pollution effect in the finite element solution of wave propagation problems is a well known phenomenon that must be confronted when solving problems in the high frequency range this paper presents a new method with no phase errors for one dimensional 1d time harmonic wave propagation problems using new ideas that hold promise for the multidimensional case the method is constructed within the framework of the discontinuous petrov galerkin dpg method with optimal test functions we have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm in this paper we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm this is answered in the specific case of the helmholtz equation with l 2 norm as the energy norm we obtain uniform stability with respect to the wave number we illustrate the method with a number of 1d numerical experiments using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally a 1d theoretical stability analysis is also developed

hardy spaces for non compactly causal symmetric spaces and the most continuous spectrum let g h be a semisimple symmetric space then the space l2 g h can be decomposed into a finite sum of series of representations induced from parabolic subgroups of g the most continuous part of the spectrum of l2 g h is the part induced from the smallest possible parabolic subgroup in this paper we introduce hardy spaces canonically related to this part of the spectrum for a class of non compactly causal symmetric spaces the hardy space is a reproducing hilbert space of holomorphic functions on a bounded symmetric domain of tube type containing g h as a boundary component a boundary value map is constructed and we show that it induces a g isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part lmc2 g h of l2 g h we also relate our hardy space to the classical hardy space on the bounded symmetric domain

a projection based error analysis of hdg methods we introduce a new technique for the error analysis of hybridizable discontinuous galerkin hdg methods the technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods this renders the analysis of the projections of the discretization errors simple and concise by showing that these projections of the errors are bounded in terms of the distance between the solution and its projection our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection we illustrate the technique on a specific hdg method applied to a model second order elliptic problem

bayesian complex amplitude estimation and adaptive matched filter detection in low rank interference we propose a bayesian method for complex amplitude estimation in low rank interference we assume that the received signal follows the generalized multivariate analysis of variance gmanova patterned mean structure and is corrupted by low rank spatially correlated interference and white noise an iterated conditional modes icm algorithm is developed for estimating the unknown complex signal amplitudes and interference and noise parameters we also discuss initialization of the icm algorithm and propose a non bayesian adaptive matched filter amf signal detector that utilizes the icm estimation results numerical simulations demonstrate the performance of the proposed methods

analysis of the dpg method for the poisson equation we give an error analysis of the recently developed dpg method applied to solve the poisson equation and a convection diffusion problem we prove that the method is quasioptimal error estimates in terms of both the mesh size h and the polynomial degree p for various element shapes can be derived from our results results of extensive numerical experiments are also presented

the cfar adaptive subspace detector is a scale invariant glrt the constant false alarm rate cfar matched subspace detector cfar msd is the uniformly most powerful invariant test and the generalized likelihood ratio test glrt for detecting a target signal in noise whose covariance structure is known but whose level is unknown previously the cfar adaptive subspace detector cfar asd or adaptive coherence estimator ace was proposed for detecting a target signal in noise whose covariance structure and level are both unknown and whose covariance structure is estimated with a sample covariance matrix based on training data we show here that the cfar asd is glrt when the test measurement is not constrained to have the same noise level as the training data as a consequence this glrt is invariant to a more general scaling condition on the test and training data than the well known glrt of kelly 1986

knowledge aided covariance matrix estimation and adaptive detection in compound gaussian noise let x be a signal to be sparsely decomposed over a redundant dictionary a i e a sparse coefficient vector s has to be found such that x as it is known that this problem is inherently unstable against

branching laws for discrete wallach points we consider the projective representations of the group of holomorphic automorphisms of a symmetric tube domain v i that are obtained by analytic continuation of the holomorphic discrete series for a representation corresponding to a discrete point in the wallach set we find the decomposition under restriction to the identity component of gl using riesz distributions an explicit intertwining operator is constructed as an analytic continuation of an integral operator the density of the plancherel measure involves quotients of functions and the c function for a symmetric cone of smaller rank

superconvergent discontinuous galerkin methods for second order elliptic problems we identify discontinuous galerkin methods for second order elliptic problems in several space dimensions having superconvergence properties similar to those of the raviart thomas and the brezzi douglas marini mixed methods these methods use polynomials of degree k 0 for both the potential as well as the flux we show that the approximate flux converges in l2 with the optimal order of k 1 and that the approximate potential and its numerical trace superconverge in l2 like norms to suitably chosen projections of the potential with order k 2 we also apply element by element postprocessing of the approximate solution to obtain new approximations of the flux and the potential the new approximate flux is proven to have normal components continuous across inter element boundaries to converge in l2 with order k 1 and to have a divergence converging in l2 also with order k 1 the new approximate potential is proven to converge with order k 2 in l2 numerical experiments validating these theoretical results are presented

degenerate principal series representations and their holomorphic extensions let x h l be an irreducible real bounded symmetric domain realized as a real form in an hermitian symmetric domain d g k the intersection s of the shilov boundary of d with x defines a distinguished subset of the topological boundary of x and is invariant under h it can be realized as s h p for certain parabolic subgroup p of h we study the spherical representations of h induced from p we find formulas for the spherical functions in terms of the macdonald hypergeometric function this generalizes the earlier result of faraut koranyi for hermitian symmetric spaces d we consider a class of h invariant integral intertwining operators from the representations on l2 s to the holomorphic representations of g restricted to h we construct a new class of complementary series for the groups h so n m su n m with n m 2 and sp n m with n m 1 we realize them as discrete components in the branching rule of the analytic continuation of the holomorphic discrete series of g su n m su n m su n m and su 2n 2m respectively

joint covariance estimation with mutual linear structure we consider the problem of joint estimation of structured covariance matrices assuming the structure is unknown estimation is achieved using heterogeneous training sets namely given groups of measurements coming from centered populations with different covariances our aim is to determine the mutual structure of these covariance matrices and estimate them supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices we develop a new efficient algorithm discovering the structure and using it to improve the estimation our technique is based on the application of principal component analysis in the matrix space we also derive an upper performance bound of the proposed algorithm in the gaussian scenario and compare it with the cramer rao lower bound numerical simulations are presented to illustrate the performance benefits of the proposed method

robust dpg method for convection dominated diffusion problems we propose and analyze a discontinuous petrov galerkin dpg method for convection dominated diffusion problems that provides robust l 2 error estimates for the field variables which are quasi optimal in the energy norm a key feature of the method is to construct test functions defined by a variational formulation with bilinear form test norm specifically designed for the goal of robustness a main theoretical ingredient is a stability analysis of the adjoint problem numerical experiments underline our theoretical results and in particular confirm robustness of the dpg method for well chosen test norms

plancherel formula for berezin deformation of l 2 on riemannian symmetric space consider natural representations of the pseudounitary group u p q in the space of holomorphic functions on the cartan domain hermitian symmetric space u p q u p u q berezin representations of o p q are the restrictions of such representations to the subgroup o p q we obtain the explicit plancherel formula for the berezin representations the support of the plancherel measure is a union of many series of representations the density of the plancherel measure on each piece of the support is an explicit product of functions we also show that the berezin representations give an interpolation between l2 on noncompact symmetric space o p q o p o q and l2 on compact symmetric space o p q o p o q

laplace and segal bargmann transforms on hermitian symmetric spaces and orthogonal polynomials let d g k be a complex bounded symmetric domain of tube type in a complex jordan algebra v and let dr j d d be its real form in a formally real euclidean jordan algebra j v dr h l is a bounded realization of the symmetric cone in j we consider representations of h that are gotten by the generalized segal bargmann transform from a unitary g space of holomorphic functions on d to an l2 space on dr we prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the laplace transform we find the extension to d of the spherical functions on dr and find their expansion in terms of the l spherical polynomials on d which are jack symmetric polynomials we prove that the coefficients are orthogonal polynomials in an l2 space the measure being the harish chandra plancherel measure multiplied by the symbol of the berezin transform we prove the difference equation and recurrence relation for those polynomials by considering the action of the lie algebra and the cayley transform on the polynomials on d finally we use the laplace transform to study generalized laguerre functions on symmetric cones

statistical analysis of the nonhomogeneity detector for non gaussian interference backgrounds we derive the nonhomogeneity detector nhd for non gaussian interference scenarios and present a statistical analysis of the method the non gaussian interference scenario is assumed to be modeled by a spherically invariant random process sirp we present a method for selecting representative homogeneous training data based on our statistical analysis of the nhd for finite sample support used in covariance estimation in particular an exact theoretical expression for the nhd test statistic probability density function pdf is derived performance analysis of the nhd is presented using both simulated data and measured data from the multichannel airborne radar measurement mcarm program a performance comparison with existing nhd approaches is also included

branching coefficients of holomorphic representations and segal bargmann transform let d g k be a complex bounded symmetric domain of tube type in a jordan algebra vc and let d h l d v be its real form in a jordan algebra v vc the analytic continuation of the holomorphic discrete series on dopf forms a family of interesting representations of g we consider the restriction on d and the branching rule under h of the scalar holomorphic representations the unitary part of the restriction map gives then a generalization of the segal bargmann transform the group l is a spherical subgroup of k and we find a canonical basis of l invariant polynomials in the components of the schmid decomposition and we express them in terms of the jack symmetric polynomials we prove that the segal bargmann transform of those l invariant polynomials are under the spherical transform on d multi variable wilson type polynomials and we give a simple alternative proof of their orthogonality relation we find the expansion of the spherical functions on d when extended to a holomorphic function in a neighborhood of 0 d in terms of the l spherical holomorphic polynomials on d the coefficients being the wilson polynomials

robust opportunistic inference from non homogeneous distribution free measurements a class of inference problems arising in wireless sensor network contexts is addressed where nodes aim at estimating cooperatively a parameter based on opportunistically gathered measurements to allow for generality non homogeneities are taken into account in the formulation furthermore knowledge of the data distribution is not assumed a novel linear estimation approach is devised based on a hierarchical modeling able to cope with local conditions e g due to manufacturing differences diverse classes of nodes and or spatial variability and heterogeneity of the samples which cannot be controlled in opportunistic inference the total variance of the hierarchy is derived then used to optimize the weights of the linear estimator obtaining in closed form the minimum variance solution of the general non homogeneous inference problem the proposed approach induces in the optimal weights a scalar parameter as additive term to the local sample sizes this acts like a tikhonov regularization on the variance contributions brought by the cooperating nodes which reflect local conditions and are inversely proportional to the sample size as a result of this regularization effect robustness against heavy tailed variations of the number of measurements available at the different nodes is achieved a thorough theoretical analysis corroborated by numerical results reveals that the proposed distribution free approach is superior to both linear and weighted least squares and can perform closely to the oracle minimum variance estimator

estimation of a covariance matrix under stein s loss on decrit la technique generale de stein pour ameliorer l estimateur sans biais invariant le meilleur et l estimateur minimax de la matrice de covariance normale

discrete components in restriction of unitary representations of rank one semisimple lie groups we consider spherical principal series representations of the semisimple lie group of rank one g so n 1 k g so n 1 k k r c hk r c h there is a family of unitarizable representations of g for in an interval on rr the so called complementary series and subquotients or subrepresentations of g for being negative integers we consider the restriction of g g under the subgroup h so n 1 1 k h so n 1 1 k we prove the appearing of discrete components the corresponding results for the exceptional lie group f4 20 f4 20 and its subgroup spin0 8 1 spin0 8 1 are also obtained

convergence rates of the dpg method with reduced test space degree this paper presents a duality theorem of the aubin nitsche type for discontinuous petrov galerkin dpg methods this explains the numerically observed higher convergence rates in weaker norms considering the specific example of the mild weak or primal dpg method for the laplace equation two further results are obtained first the dpg method continues to be solvable even when the test space degree is reduced provided it is odd second a non conforming method of analysis is developed to explain the numerically observed convergence rates for a test space of reduced degree

the discontinuous petrov galerkin methodology for the mixed multiscale finite element method abstract we present the application of the discontinuous petrov galerkin dpg methodology for the mixed multiscale finite element method msfem the msfem upscaling technique relies on incorporating fine scale features through special in a sense optimized for approximability trial functions while the dpg methodology allows for the selection of the optimal test functions to provide stability of the fem approximation the special trial functions are computed online by the solution of local boundary value problems we improved this process using the static condensation that restricted the construction of the functions to the coarse mesh skeleton element interfaces only we have verified by numerical tests that the proposed improvement of msfem reduced both the approximation error and computational cost moreover it simplified the algorithm significantly the key component of this prolongation construction is our novel method for prolongation of both traction and displacement vectors on element edges of arbitrary shape the proposed prolongation operator may be also used in the multigrid solver for direct analysis of composites with varying material parameters using an arbitrary well posed functional setting with or without the dpg methodology

performance of a class of adaptive detection algorithms in nonhomogeneous environments a two dimensional 2 d adaptive sidelobe blanker asb detection algorithm was developed through experimentation as an extenuate for false alarms caused by undernulled interference encountered when applying the adaptive matched filter amf in nonhomogeneous environments the algorithm s utility has been demonstrated empirically considering theoretic performance analyses of the asb detection algorithm as well as the amf generalized likelihood ratio test glrt and the adaptive cosine estimator ace under nonideal conditions can become fairly intractable rather quickly especially in an adaptive processing context involving covariance estimation in this paper however we have developed and exploited a theoretic framework through which the performance of these algorithms under nonhomogeneous conditions can be examined theoretically it is demonstrated through theoretic analysis that in the presence of undernulled interference the asb is a pliable false alarm regulatory far detector that maintains good target sensitivity a viable method of asb threshold selection is also presented and demonstrated

a robust dpg method for singularly perturbed reaction diffusion problems we present and analyze a discontinuous petrov galerkin method with optimal test functions for a reaction dominated diffusion problem in two and three space dimensions we start with an ultraweak formulation that comprises parameters alpha beta to allow for general varepsilon dependent weightings of three field variables varepsilon being the small diffusion parameter specific values of alpha and beta imply robustness of the method that is a quasi optimal error estimate with a constant that is independent of varepsilon moreover these values lead to a norm for the field variables that is known to be balanced in varepsilon for model problems with typical boundary layers several numerical examples underline our theoretical estimates and reveal stability of approximations even for very small varepsilon

the berezin transform and invariant differential operators the berezin calculus is important to quantum mechanics creation annihilation operators and operator theory toeplitz operators we study the basic berezin transform linking the contravariant and covariant symbol for all bounded symmetric domains and express it in terms of invariant differential operators

a posteriori error control for dpg methods a combination of ideas in least squares finite element methods with those of hybridized methods recently led to discontinuous petrov galerkin dpg finite element methods they minimize a residual inherited from a piecewise ultraweak formulation in a nonstandard locally computable dual norm this paper establishes a general a posteriori error analysis for the natural norms of the dpg schemes under conditions equivalent to a priori stability estimates it is proven that the locally computable residual norm of any discrete function is a lower and an upper error bound up to explicit data approximation errors the presented abstract framework for a posteriori error analysis applies to known dpg discretizations of laplace and lame equations and to a novel dpg method for the stress velocity formulation of stokes flow with symmetric stress approximations since the error control does not rely on the discrete equations it applies to inexactly computed or otherwise perturbed solutions within the discrete space

ultraweak formulation of linear pdes in nondivergence form and dpg approximation abstract we develop and analyze an ultraweak formulation of linear pdes in nondivergence form where the coefficients satisfy the cordes condition based on the ultraweak formulation we propose discontinuous petrov galerkin dpg methods we investigate fortin operators for the fully discrete schemes and provide a posteriori estimators for the methods under consideration numerical experiments are presented in the case of uniform and adaptive mesh refinement

breaking spaces and forms for the dpg method and applications including maxwell equations discontinuous petrov galerkin dpg methods are made easily implementable using broken test spaces i e spaces of functions with no continuity constraints across mesh element interfaces broken spaces derivable from a standard exact sequence of first order unbroken sobolev spaces are of particular interest a characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces this technique is used to provide a complete error analysis of dpg methods for maxwell equations with perfect electric boundary conditions the technique also permits considerable simplifications of previous analyses of dpg methods for other equations reliability and efficiency estimates for an error indicator also follow finally the equivalence of stability for various formulations of the same maxwell problem is proved including the strong form the ultraweak form and various forms in between

a class of discontinuous petrov galerkin methods part i the transport equation considering a simple model transport problem we present a new finite element method while the new method fits in the class of discontinuous galerkin dg methods it differs from standard dg and streamline diffusion methods in that it uses a space of discontinuous trial functions tailored for stability the new method unlike the older approaches yields optimal estimates for the primal variable in both the element size h and polynomial degree p and outperforms the standard upwind dg method

toeplitz quantization and asymptotic expansions for real bounded symmetric domains an analogue of the star product familiar from deformation quantization is studied in the setting of real bounded symmetric domains the analogue turns out to be a certain invariant operator which one might call star restriction from functions on the hermitification of the domain into functions on the domain itself in particular we establish the usual i e semiclassical asymptotic expansion of this star restriction and further obtain a real variable analogue of a theorem of arazy and orsted concerning the analogous expansion for the berezin transform

analysis on the minimal representation of o p q i realization via conformal geometry abstract this is the first in a series of papers devoted to an analogue of the metaplectic representation namely the minimal unitary representation of an indefinite orthogonal group this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of kirillov kostant we begin by applying methods from conformal geometry of pseudo riemannian manifolds to a general construction of an infinite dimensional representation of the conformal group on the solution space of the yamabe equation by functoriality of the constructions we obtain different models of the unitary representation as well as giving new proofs of unitarity and irreducibility the results in this paper play a basic role in the subsequent papers where we give explicit branching formulae and prove unitarization in the various models

hybridization and postprocessing techniques for mixed eigenfunctions we introduce hybridization and postprocessing techniques for the raviart thomas approximation of second order elliptic eigenvalue problems hybridization reduces the raviart thomas approximation to a condensed eigenproblem the condensed eigenproblem is nonlinear but smaller than the original mixed approximation we derive multiple iterative algorithms for solving the condensed eigenproblem and examine their interrelationships and convergence rates an element by element postprocessing technique to improve accuracy of computed eigenfunctions is also presented we prove that a projection of the error in the eigenspace approximation by the mixed method of any order superconverges and that the postprocessed eigenfunction approximations converge faster for smooth eigenfunctions numerical experiments using a square and an l shaped domain illustrate the theoretical results

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

discrete decomposability of the restriction of a q λ with respect to reductive subgroups and its applications letg g be real reductive lie groups and q a stable parabolic subalgebra of lie g this paper offers a sufficient condition on g g q that the irreducible unitary representation mathop a q limits ofg with non zero continuous cohomology splits into a discrete sum of irreducible unitary representations of a subgroupg each of finite multiplicity as an application to purely analytic problems new results on discrete series are also obtained for some pseudo riemannian non symmetric spherical homogeneous spaces which fit nicely into this framework some explicit examples of a decomposition formula are also found in the cases wherea q is not necessarily a highest weight module

canonical representations related to hyperbolic spaces abstract we extend the notion of canonical representation introduced by a m vershik i m gel fand and m i graev 1982 in representation theory cambridge univ press cambridge u k to all classical rank one semisimple lie groups we determine the spectral decomposition of the canonical representations in detail

function spaces and reproducing kernels on bounded symmetric domains there are several natural hilbert spaces of holomorphic functions on a bounded symmetric domain such are the bergman type spaces on which the holomorphic discrete series operates and the hardy type spaces which are related to the analytic continuation of the holomorphic discrete series also closely related is the bergmann space of entire functions on the ambient which arises as the closure of the polynomials with respect to a natural inner product the space of holomorphic polynomials decomposes into irreducible subspaces under the action of the isotropy group k of the domain the main facts about this decomposition were proved by schmid 21 for another proof see 22 each irreducible subspace contains a unique normalized l invariant spherical polynomial where l is the isotropy group of the shilov boundary in k our first main result is the explicit computation of the norms of the spherical polynomials with respect to each of the hilbert spaces considered for the domains of classical type they were considered by hua 9 and for some of the hilbert spaces this was done before by upmeier 24 using different methods for certain others there are partial results in 22 we are able to do this in a fairly simple unified way by making strong use of gindikin s generalized gamma function s next we obtain a description of the reproducing kernels of the k irreducible subspaces in each of our hilbert spaces and an expansion in terms of these for

a primal dpg method without a first order reformulation abstract we show that it is possible to apply the dpg methodology without reformulating a second order boundary value problem into a first order system by considering the simple example of the poisson equation the result is a new weak formulation and a new dpg method for the poisson equation which has no numerical trace variable but has a numerical flux approximation on the element interfaces in addition to the primal interior variable

bounds for estimation of covariance matrices from heterogeneous samples this correspondence derives lower bounds on the mean square error mse for the estimation of a covariance matrix mbi mp using samples mbi zk k 1 k whose covariance matrices mbi mk are randomly distributed around mbi mp this framework can be encountered e g in a radar system operating in a nonhomogeneous environment when it is desired to estimate the covariance matrix of a range cell under test using training samples from adjacent cells and the noise is nonhomogeneous between the cells we consider two different assumptions for mbi mp first we assume that mbi mp is a deterministic and unknown matrix and we derive the cramer rao bound for its estimation in a second step we assume that mbi mp is a random matrix with some prior distribution and we derive the bayesian bound under this hypothesis

the completion of locally refined simplicial partitions created by bisection recently in found comput math 7 2 2007 245 269 we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations which is essentially the method from sinum 38 2000 466 488 by morin nochetto and siebert converges with the optimal rate the number of triangles n in the output partition of such a method is generally larger than the number m of triangles that in all intermediate partitions have been marked for bisection because additional bisections are needed to retain conforming meshes a key ingredient to our proof was a result from numer math 97 2004 219 268 by binev dahmen and devore saying that n n 0 cm for some absolute constant c where n 0 is the number of triangles from the initial partition that have never been bisected in this paper we extend this result to bisection algorithms of n simplices with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions

geometric analysis on small unitary representations of gl n r abstract the most degenerate unitary principal series representations i r z 2 z of g gl n r attain the minimum of the gelfand kirillov dimension among all irreducible unitary representations of g this article gives an explicit formula of the irreducible decomposition of the restriction i h branching law with respect to all symmetric pairs g h for n 2 n with n 2 the restriction i h remains irreducible for h sp n r if 0 and splits into two irreducible representations if 0 the branching law of the restriction i h is purely discrete for h gl n c consists only of continuous spectrum for h gl p r gl q r p q n and contains both discrete and continuous spectra for h o p q p q 1 our emphasis is laid on geometric analysis which arises from the restriction of small representations to various subgroups