a macro element compression for the moment method a new approach based on qr compression of the off diagonal blocks of the impedance matrix arising in integral equations solutions of electromagnetic em scattering problems previously implemented for finite and irregular gratings is extended here to an arbitrarily shaped obstacle the main tool consists in dividing the entire surface scatterer into subdomains and in assembling the global impedance matrix as a combination of the macromatrices associated to each interaction of any patch on the other the correlation between compression rate and accuracy of field computation is drawn up an experimental study is performed to optimize the method relatively to the two competing parameters accuracy and cost finally it is shown that this method works equally well in both the high and the low frequency domain and also with an inclusion of a small detail in a large object

spatially inhomogeneous linear inverse problems with possible singularities the objective of the present paper is to introduce the concept of a spatially inhomogeneous linear inverse problem which to the best of the author s knowledge has never been considered previously in statistical framework but is emerging due to a variety of practical applications the special feature of the problem is that the degree of ill posedness depends not only on the scale but also on location in this case the rates of convergence are determined by the interaction of four parameters the smoothness and spatial homogeneity of the unknown function f and degrees of ill posedness and spatial inhomogeneity of operator q an interesting property here is that if operator q is weakly inhomogeneous then the rates of convergence are not influenced by spatial inhomogeneity of operator q and coincide with the rates which are usual for homogeneous linear inverse problems on the other hand if operator q is moderately or strongly inhomogeneous convergence rates are significantly affected by the degree of spatial inhomogeneity estimators obtained in the paper are based either on wavelet vaguelette decomposition if the norms of all vaguelettes are finite or on a hybrid of wavelet vaguelette decomposition and galerkin method if vaguelettes in the neighborhood of the singularity point have infinite norms the hybrid estimator is a combination of a linear part in the vicinity of the singularity point and the nonlinear block thresholding wavelet estimator elsewhere to attain adaptivity an optimal resolution level for the linear singularity affected portion of the estimator is obtained using lepskii 1990 1999 method subsequently this resolution level is used as the lowest resolution level for the nonlinear wavelet estimator we show that convergence rates of the hybrid estimator lie within a logarithmic factor of the optimal minimax convergence rates the theory presented in the paper is supplemented by examples of deconvolution based on irregularly spaced observations or in the presence of a heteroscedastic noise the latter problem is also examined via a limited simulation study which demonstrates advantages of the hybrid estimator when the degree of spatial inhomogeneity is high

functional deconvolution in a periodic setting uniform case we extend deconvolution in a periodic setting to deal with functional data the resulting functional deconvolution model can be viewed as a generalization of a multitude of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation in the case when it is observed at a finite number of distinct points the proposed functional deconvolution model can also be viewed as a multichannel deconvolution model we derive minimax lower bounds for the l 2 risk in the proposed functional deconvolution model when f is assumed to belong to a besov ball and the blurring function is assumed to possess some smoothness properties including both regular smooth and super smooth convolutions furthermore we propose an adaptive wavelet estimator of f that is asymptotically optimal in the minimax sense or near optimal within a logarithmic factor in a wide range of besov balls in addition we consider a discretization of the proposed functional deconvolution model and investigate when the availability of continuous data gives advantages over observations at the asymptotically large number of points as an illustration we discuss particular examples for both continuous and discrete settings

adaptive sup norm estimation of the wigner function in noisy quantum homodyne tomography in quantum optics the quantum state of a light beam is represented through the wigner function a density on r 2 which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics in the framework of noisy quantum homodyne tomography with efficiency parameter 1 2 1 we study the theoretical performance of a kernel estimator of the wigner function we prove that it is minimax efficient up to a logarithmic factor in the sample size for the l risk over a class of infinitely differentiable functions we also compute the lower bound for the l 2 risk we construct an adaptive estimator i e which does not depend on the smoothness parameters and prove that it attains the minimax rates for the corresponding smoothness of the class of functions up to a logarithmic factor in the sample size finite sample behaviour of our adaptive procedure is explored through numerical experiments

adaptive wavelet estimator for nonparametric density deconvolution 1999 institute of mathematical statistics the electronic version of this article is the complete one and can be found online at http projecteuclid org euclid aos 1017939249

numerically efficient solution of dense linear system of equations arising in a class of electromagnetic scattering problems in this paper we present an efficient technique for solving a dense complex symmetric linear system of equations arising in the method of moments mom formulation to illustrate the application of the method we consider a finite array of scatterers which gives rise to a large number of unknowns the solution procedure utilizes preconditioned transpose free qmr ptfqmr iterations and computes the matrix vector products by employing a compressed impedance matrix the compression is achieved by reduced rank representation of the off diagonal blocks based on a partial qr decomposition which is followed by an iterative refinement both the preconditioning and the compression steps take advantage of the block structure of the matrix the convergence of the iterative procedure is investigated and the performance of the proposed algorithm is compared to that achieved by other schemes the effectiveness of the preconditioner and the degree of matrix compression are quantified finite arrays of variable shape and sizes are considered and it is demonstrated that the ability to solve large problems using this technique enables one to evaluate the edge effects in the finite array such array is basically flat and periodic but the algorithm is still efficient when variation with strict periodicity or flatness exists

efficient solution of efie via low rank compression of multilevel predetermined interactions this paper describes the predetermined interaction list oct tree pilot algorithm and its application in expediting the solution of full wave electric field integral equation efie based scattering problems for three dimensional arbitrarily shaped conductors pilot combines features of the fast multipole method fmm and qr decomposition based matrix compression techniques to optimize setup times solve times and memory requirements the method is kernel independent and stable for electrically small structures unlike traditional fmm the novel features of the algorithm namely the mixed potential compression scheme and the hierarchical multilevel predetermined matrix structure are explained in detail a complexity estimate is presented to demonstrate the scaling in time and memory requirements examples exhibiting the accuracy and the time and memory performances are also presented finally a quantitative study is included to address the expected but gradual degradation of qr based compression techniques for electrically large structures

forward fourier wavelet regularized deconvolution for ill conditioned systems we propose an efficient hybrid fourier wavelet regularized deconvolution forward algorithm that performs noise regularization via scalar shrinkage in both the fourier and wavelet domains the fourier shrinkage exploits the fourier transform s economical representation of the colored noise inherent in deconvolution whereas the wavelet shrinkage exploits the wavelet domain s economical representation of piecewise smooth signals and images we derive the optimal balance between the amount of fourier and wavelet regularization by optimizing an approximate mean squared error mse metric and find that signals with more economical wavelet representations require less fourier shrinkage forward is applicable to all ill conditioned deconvolution problems unlike the purely wavelet based wavelet vaguelette deconvolution wvd moreover its estimate features minimal ringing unlike the purely fourier based wiener deconvolution even in problems for which the wvd was designed we prove that forward s mse decays with the optimal wvd rate as the number of samples increases further we demonstrate that over a wide range of practical sample lengths forward improves on wvd s performance

dicussion on the meeting on statistical approaches to inverse problems johnstone kerkyacharian picard and raimondo johnstone kerkyacharian picard and raimondo are interested in the inverse problem of estimating f where f has been convolved with g and then contaminated with white noise this popular problem has been tackled by a wide variety of procedures and wavelet methods have recently generated considerable interest donoho s 1995 seminal wavelet vaguelette paper introduced the notion that wavelets would be a good choice for the representation of f since real life objects such as images are more likely to be efficiently represented using wavelets when compared with for example fourier representations johnstone and his colleagues have moved the field on significantly in particular their procedure is more direct than wavelet vaguelette or abramovich and silverman s 1998 vaguelette wavelet method it can handle boxcar blur theoretically and practically they have rates of convergence forp 2 p defines the type of loss and the paper innovates through use of the new maxiset approach for me the most appealing of these innovations is that of enabling the treatment of boxcar blur which is one of the most common types of inverse problem however is it really really the case that for rational a nothing can be done formula 4 compels us to say no nothing can but naively it still feels wrong formula 19 is the popular signal plus noise model but here it is a little different from what normally appears in the literature because the quantities are complex valued random variables more specifically the zl are zero mean gaussian variables which are complex valued and satisfy e zlzk lk one question is why threshold the k and not the yl directly the covariance of the k is given by cov k l n m km l m

fast directional multilevel algorithm for analyzing wave scattering a new method named fast directional multilevel algorithm is proposed for analyzing wave scattering similar to a fast multipole method an oct tree structure is used in the new method unlike the conventional mlfma the kernel is evaluated by a new method instead of approximating the green s function by multipole expansion the fast directional multilevel algorithm is first used to solve the wave scattering problem via combined field integral equation cfie using rao wilton glisson rwg basis functions the low rank representations are extended to apply in the low frequency regime which is proved efficient and stable the numerical results demonstrate that the computational complexity of this new multilevel algorithm can achieve the complexity of o n log n this algorithm is robust and performs well on other oscillatory kernels because of its not depending on the explicit form of kernels

wavelet regression in random design with heteroscedastic dependent errors we investigate function estimation in nonparametric regression models with random design and heteroscedastic correlated noise adaptive properties of warped wavelet nonlinear approximations are studied over a wide range of besov scales f b s r and for a variety of l p error measures we consider error distributions with long range dependence parameter 0 2 it is seen that there are three rate phases namely the dense sparse and long range dependence phase depending on the relative values of s p and furthermore we show that long range dependence does not come into play for shape estimation f f the theory is illustrated with some numerical examples

a multilevel matrix decomposition algorithm for analyzing scattering from large structures a multilevel algorithm is presented for analyzing scattering from electrically large surfaces the algorithm accelerates the iterative solution of integral equations that arise in computational electromagnetics the algorithm permits a fast matrix vector multiplication by decomposing the traditional method of moment matrix into a large number of blocks with each describing the interaction between distant scatterers the multiplication of each block by a trial solution vector is executed using a multilevel scheme that resembles a fast fourier transform fft and that only relies on well known algebraic techniques the computational complexity and the memory requirements of the proposed algorithm are o n log sup 2 n

a multilevel implementation of the qr compression for method of moments we present a multilevel version of a previously reported compression technique for the method of moments poirier j r et al ibid vol 46 p 1175 6 1998 this method is based on qr compression of the impedance matrix off diagonal blocks the block structure of the matrix is built up to optimize the compression rate it results from the study that the compression threshold is the unique adjustment parameter which makes this method very easy to handle some results are presented which stress the improvement in memory storage and computation time obtained through this approach with respect to the mono level algorithm

periodic boxcar deconvolution and diophantine approximation we consider the nonparametric estimation of a periodic function that is observed in additive gaussian white noise after convolution with a boxcar the indicator function of an interval this is an idealized model for the problem of recovery of noisy signals and images observed with motion blur if the length of the boxcar is rational then certain frequencies are irretreviably lost in the periodic model we consider the rate of convergence of estimators when the length of the boxcar is irrational using classical results on approximation of irrationals by continued fractions a basic question of interest is whether the minimax rate of convergence is slower than for nonperiodic problems with 1 f like convolution filters the answer turns out to depend on the type and smoothness of functions being estimated in a manner not seen with homogeneous filters

the principle of penalized empirical risk in severely ill posed problems we study a standard method of regularization by projections of the linear inverse problem y ax e where e is a white gaussian noise and a is a known compact operator it is assumed that the eigenvalues of aa converge to zero with exponential decay such behavior of the spectrum is typical for inverse problems related to elliptic differential operators as model example we consider recovering of unknown boundary conditions in the dirichlet problem for the laplace equation on the unit disk by using the singular value decomposition of a we construct a projection estimator of x the bandwidth of this estimator is chosen by a data driven procedure based on the principle of minimization of penalized empirical risk we provide non asymptotic upper bounds for the mean square risk of this method and we show in particular that this approach gives asymptotically minimax estimators in our model example

on convergence rates equivalency and sampling strategies in functional deconvolution models using the asymptotical minimax framework we examine convergence rates equivalency between a continuous functional deconvolution model and its real life discrete counterpart over a wide range of besov balls and for thel 2 risk for this purpose all possible models are divided into three groups for the models in the

mobius deconvolution on the hyperbolic plane with application to impedance density estimation in this paper we consider a novel statistical inverse problem on the poincare or lobachevsky upper complex half plane here the riemannian structure is hyperbolic and a transitive group action comes from the space of 2 x 2 real matrices of determinant one via mobius transformations our approach is based on a deconvolution technique which relies on the helgason fourier calculus adapted to this hyperbolic space this gives a minimax nonparametric density estimator of a hyperbolic density that is corrupted by a random mobius transform a motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the poincare plane exactly describes the physical system that is of statistical interest

on the use of the singular value decomposition in the fast multipole method the authors present a new matrix compression algorithm to improve the efficiency of the fast multipole method fmm the method is based on the application of the singular value decomposition svd to the plane wave fmm aggregation matrices these matrices are low ranked which is exploited to provide alternative sets of orthonormal singular basis functions obtained as linear combinations of the original basis by choosing only the most relevant singular functions a much more compact representation is obtained to accurately handle the interactions between the fmm groups the new formulation provides a reduction close to one order of magnitude both in computational cost and memory requirements with a moderate impact on the accuracy of the solution

on the optimal rates of convergence for nonparametric deconvolution problems deconvolution problems arise in a variety of situations in statistics an interesting problem is to estimate the density f of a random variable x based on n i i d observations from y x varepsilon where varepsilon is a measurement error with a known distribution in this paper the effect of errors in variables of nonparametric deconvolution is examined insights are gained by showing that the difficulty of deconvolution depends on the smoothness of error distributions the smoother the harder in fact there are two types of optimal rates of convergence according to whether the error distribution is ordinary smooth or supersmooth it is shown that optimal rates of convergence can be achieved by deconvolution kernel density estimators

wavelet deconvolution this paper studies the issue of optimal deconvolution density estimation using wavelets the approach taken here can be considered as orthogonal series estimation in the more general context of the density estimation we explore the asymptotic properties of estimators based on thresholding of estimated wavelet coefficients minimax rates of convergence under the integrated square loss are studied over besov classes b sub spl sigma pq of functions for both ordinary smooth and supersmooth convolution kernels the minimax rates of convergence depend on the smoothness of functions to be deconvolved and the decay rate of the characteristic function of convolution kernels it is shown that no linear deconvolution estimators can achieve the optimal rates of convergence in the besov spaces with p 2 when the convolution kernel is ordinary smooth and super smooth if the convolution kernel is ordinary smooth then linear estimators can be improved by using thresholding wavelet deconvolution estimators which are asymptotically minimax within logarithmic terms adaptive minimax properties of thresholding wavelet deconvolution estimators are also discussed

wavelet decomposition approaches to statistical inverse problems summary a wide variety of scientific settings involve indirect noisy measurements where one faces a linear inverse problem in the presence of noise primary interest is in some function f t but data are accessible only about some linear transform corrupted by noise the usual linear methods for such inverse problems do not perform satisfactorily when f t is spatially inhomogeneous one existing nonlinear alternative is the wavelet vaguelette decomposition method based on the expansion of the unknown f t in wavelet series in the vaguelette wavelet decomposition method proposed here the observed data are expanded directly in wavelet series the performances of various methods are compared through exact risk calculations in the context of the estimation of the derivative of a function observed subject to noise a result is proved demonstrating that with a suitable universal threshold somewhat larger than that used for standard denoising problems both the wavelet based approaches have an ideal spatial adaptivity property