asymptotic equivalence of nonparametric regression and white noise the principal result is that under conditions to any nonparametric regression problem there corresponds an asymptotically equivalent sequence of white noise with drift problems and conversely this asymptotic equivalence is in a global and uniform sense any normalized risk function attainable in one problem is asymptotically attainable in the other with the difference in normalized risks converging to zero uniformly over the entire parameter space the results are constructive a recipe is provided for producing these asymptotically equivalent procedures some implications and generalizations of the principal result are also discussed

the intrinsic distribution and selection bias of long period cometary orbits a question that arises in the study of cometary orbits is whether or not the directed normals to the orbits are uniformly distributed on the celestial sphere previous studies by statisticians have not taken selection effects into account and have tended to reject uniformity here a plausible selection mechanism is proposed that gives rise to a one parameter family of distributions on the sphere data on long period comets are analyzed using this one parameter family a nonzero selection effect is detected and its size is estimated subject to this selection effect uniformity of the directed normals can no longer be ruled out

projective fourier analysis for patterns abstract identifying psl 2 c as a projective group for patterns in the conformal camera model the projective harmonic analysis on its double covering group sl 2 c is presented in the noncompact and compact pictures the pictures used to study different aspects of irreducible unitary representations of semisimple lie groups bypassing technicalities of representation theory but stressing the motivation and similarities with euclidean fourier analysis each constructed picture of the projective fourier analysis includes the fourier transform plancherel s theorem and convolution property projectively covariant characteristics of the analysis in the noncompact picture allow rendering any of image projective transformations of a pattern after removing conformal distortions by using only one projective fourier transform of the original pattern what is demonstrated in a computer simulation the convolution properties in both pictures must by used to develop algorithms for projectively invariant matching of patterns work in progress on fast algorithms for computing with projective fourier transforms and for rendering image projective transformations is discussed efficient computations of the convolutions would follow from the both fast projective fourier transforms and their inverses

3d magnetotelluric modeling by using finite difference method comparison study of different forward modeling approaches we have investigated different approaches to solve the magnetotelluric forward modeling problems beside classical direct electromagnetic formulation ungauged and gauged lorenz coulomb and axial vector and scalar potential formulations are discretized using finite difference numerical solution technique linear matrix equations obtained from finite difference discretization for each electromagnetic field formulations are solved by using the conjugate gradient iterative solution method we compared finite different solutions of each approaches with respect to accuracy and speed we show that all approaches generate accurate results however ungauged approach gave solutions with less conjugate gradient iteration and accordingly less computer time all stiffness matrices arising from each formulation are examined in terms of their sizes and type finite difference solution of the coulomb gauge and ungauged approaches generates bigger size stiffness matrices than the other approaches direct electromagne

bayesian aspects of some nonparametric problems we study the bayesian approach to nonparametric function estimation problems such as nonparametric regression and signal estimation we consider the asymptotic properties of bayes procedures for conjugate gaussian priors we show that so long as the prior puts nonzero measure on the very large parameter set of interest then the bayes estimators are not satisfactory more specifically we show that these estimators do not achieve the correct minimax rate over norm bounded sets in the parameter space thus all bayes estimators for proper gaussian priors have zero asymptotic efficiency in this minimax sense we then present a class of priors whose bayes procedures attain the optimal minimax rate of convergence these priors may be viewed as compound or hierarchical mixtures of suitable gaussian distributions

time variant channel estimation using discrete prolate spheroidal sequences we propose and analyze a low complexity channel estimator for a multiuser multicarrier code division multiple access mc cdma downlink in a time variant frequency selective channel mc cdma is based on orthogonal frequency division multiplexing ofdm the time variant channel is estimated individually for every flat fading subcarrier assuming small intercarrier interference the temporal variation of every subcarrier over the duration of a data block is upper bounded by the doppler bandwidth determined by the maximum velocity of the users slepian showed that time limited snapshots of bandlimited sequences span a low dimensional subspace this subspace is also spanned by discrete prolate spheroidal dps sequences we expand the time variant subcarrier coefficients in terms of orthogonal dps sequences we call slepian basis expansion this enables a time variant channel description that avoids the frequency leakage effect of the fourier basis expansion the square bias of the slepian basis expansion per subcarrier is three magnitudes smaller than the square bias of the fourier basis expansion we show simulation results for a fully loaded mc cdma downlink with classic linear minimum mean square error multiuser detection the users are moving with 19 4 m s using the slepian basis expansion channel estimator and a pilot ratio of only 2 we achieve a bit error rate performance as with perfect channel knowledge

computationally attractive reconstruction of bandlimited images from irregular samples an efficient method for the reconstruction of bandlimited images and the approximation of arbitrary images from nonuniform sampling values is developed the novel method is based on the observation that the reconstruction problem can be formulated as linear system of equations using two dimensional 2 d trigonometric polynomials where the matrix is of block toeplitz type with toeplitz blocks this system is solved iteratively by the conjugate gradient cg method we show that the use of so called adaptive weights in the establishment of the block toeplitz matrix can be seen as efficient preconditioning the superiority of the new method over conventional approaches is demonstrated by numerical experiments

multivariate smoothing spline functions given data z i g t i varepsilon i 1 leqq i leqq n where g is the unknown function the t i are known d dimensional variables in a domain omega and the varepsilon i are i

nonuniform sampling of periodic bandlimited signals digital processing techniques are based on representing a continuous time signal by a discrete set of samples this paper treats the problem of reconstructing a periodic bandlimited signal from a finite number of its nonuniform samples in practical applications only a finite number of values are given extending the samples periodically across the boundaries and assuming that the underlying continuous time signal is bandlimited provides a simple way to deal with reconstruction from finitely many samples two algorithms for reconstructing a periodic bandlimited signal from an even and an odd number of nonuniform samples are developed in the first the reconstruction functions constitute a basis while in the second they form a frame so that there are more samples than needed for perfect reconstruction the advantages and disadvantages of each method are analyzed specifically it is shown that the first algorithm provides consistent reconstruction of the signal while the second is shown to be more stable in noisy environments next we use the theory of finite dimensional frames to characterize the stability of our algorithms we then consider two special distributions of sampling points uniform and recurrent nonuniform and show that for these cases the reconstruction formulas as well as the stability analysis are simplified significantly the advantage of our methods over conventional approaches is demonstrated by numerical experiments

an empirical bayes approach to directional data and efficient computation on the sphere this paper proposes a consistent nonparametric empirical bayes estimator of the prior density for directional data the methodology is to use fourier analysis on s 2 to adapt euclidean techniques to this non euclidean environment general consistency results are obtained in addition a discussion of efficient numerical computation of fourier transforms on s 2 is given and their applications to the methods suggested in this paper are sketched

three dimensional induction logging problems part i an integral equation solution and model comparisons a 3 d frequency domain solution based on a volume integral equation approach has been implemented to simulate induction log responses in our treatment of the problem we assume that the electrical properties of the bedding as well as the borehole and invasion zones can exhibit transverse anisotropy the solution process uses a krylov subspace iteration to solve the scattering equation which is based on the modified iterative dissipative method internal consistency checks and comparisons with mode matching and finite difference solutions for vertical borehole models demonstrate the accuracy of the solution there are no known analytical solutions for induction log responses arising from deviated boreholes intersecting horizontal bed boundaries to simulate such responses requires the numerical solution of maxwell s equations in three dimensions along with independent tests to validate the solution approach and its accuracy in this paper we compare two independent 3 d frequency domain solutions for the

deconvolution density estimation on so n this paper develops nonparametric deconvolution density estimation over so n the group of n x n orthogonal matrices of determinant 1 the methodology is to use the group and manifold structures to adapt the euclidean deconvolution techniques to this lie group environment this is achieved by employing the theory of group representations explicit to so n general consistency results are obtained with specific rates of convergence achieved under sufficient smoothness conditions application to empirical bayes prior estimation and inference is also discussed

three dimensional magnetotelluric modeling using difference equations theory and comparisons to integral equation solutions we have developed an algorithm for computing the magnetotelluric response of three dimensional 3 d earth models it is a difference equation algorithm that is based on the integral forms of maxwell s equations rather than the differential forms this formulation does not require approximating derivatives of earth properties or electromagnetic fields as happens when using the second order vector diffusion equation rather one must determine how averages are to be computed side boundary values for the h fields are obtained from putting two dimensional 2 d slices of the model into a larger scale 2 d model and solving for the fields at the 3 d boundary positions to solve the 3 d system of equations we propagate an impedance matrix which relates all the horizontal e fields in a layer to all the horizontal h fields in that same layer up through the earth model applying a plane wave source condition and the side boundary h field values allows us to solve for the unknown fields within the model the r

performance of linear field reconstruction techniques with noise and uncertain sensor locations we consider a wireless sensor network sampling a bandlimited field described by a limited number of harmonics sensor nodes are irregularly deployed over the area of interest or subject to random displacement in addition sensors measurements are affected by noise our goal is to obtain a high quality reconstruction of the field with the mean square error mse of the estimate as performance metric in particular we analytically derive the performance of several reconstruction estimation techniques based on linear filtering for each technique we obtain the mse as well as its asymptotic expression in the case where the number of field harmonics and the number of sensors grow to infinity while their ratio is kept constant through numerical simulations we show the validity of the asymptotic analysis even for a small number of sensors we provide some novel guidelines for the design of sensor networks when many parameters such as field bandwidth number of sensors reconstruction quality and sensor displacement characteristics to be traded off

spline interpolation and smoothing on hyperspheres the authors generalize wahba s theory siam j sci statist comput 2 1981 pp 5 16 on spline interpolation and smoothing on the surface of the two dimensional unit sphere to arbitrary dimensional hyperspheres as a consequence practical solutions to minimum norm interpolation and smoothing problems on hyperspheres are provided in terms of certain hyperspherical splines in addition wahba s results for powers of the laplace beltrami operator are extended to more general operators and wahba s hilbert space of constant functions is expanded to allow more than just constant functions extensive curve fitting calculations are made for some two and higher dimensional test problems using hyperspherical harmonics and hyperspherical splines it is found that hyperspherical splines yield better fits than hyperspherical harmonics for test functions that possess no symmetry and are not infinitely differentiable tests have been run using several continuous functions satisfactory absolute errors can be obt

approximation of method of regularization estimators la methode de tikhonov est une methode generale d estimation du parametre de regression non parametrique dans un modele lineaire abstrait avec un bruit discret on donne une analyse asymptotique dans laquelle le probleme d estimation discrete est approche par une estimation continue

electromagnetic modeling of three dimensional bodies in layered earths using integral equations we have developed an algorithm based on the method of integral equations to simulate the electromagnetic responses of three dimensional bodies in layered earths the inhomogeneities are replaced by an equivalent current distribution which is approximated by pulse basis functions a matrix equation is constructed using the electric tensor green s function appropriate to a layered earth and it is solved for the vector current in each cell subsequently scattered fields are found by integrating electric and magnetic tensor green s functions over the scattering currents efficient evaluation of the tensor green s functions is a major consideration in reducing computation time we find that tabulation and interpolation of the six electric and five magnetic hankel transforms defining the secondary green s functions is preferable to any direct hankel transform calculation using linear filters a comparison of responses over elongate three dimensional 3 d bodies with responses over two dimensional 2 d bodie

channel estimation in wireless ofdm systems with irregular pilot distribution this paper addresses pilot assisted channel estimation for wireless orthogonal frequency division multiplexing ofdm systems with irregular pilot arrangements using nonuniform sampling techniques we propose a novel channel estimator that is based on conjugate gradient iterations and features very low computational complexity we investigate the impact of the pilot arrangement on the channel estimation mean square error and we provide a heuristic stopping criterion for early termination of the conjugate gradient iterations one dimensional and two dimensional channel estimator implementations and applications of our method to multiple antenna and multiuser ofdm systems are discussed extensive numerical simulations corroborate that the proposed method performs similarly to computationally much more expensive minimum mean square error estimators

estimation of time limited channel spectra from nonuniform samples this paper deals with the estimation of a time invariant channel spectrum from its own nonuniform samples assuming there is a bound on the channel s delay spread except for this last assumption this is the basic estimation problem in systems providing channel spectral samples however as shown in the paper the delay spread bound leads us to view the spectrum as a band limited signal rather than the fourier transform of a tapped delay line tdl using this alternative model a linear estimator is presented that approximately minimizes the expected root mean square rms error for a deterministic channel its main advantage over the tdl is that it takes into account the spectrum s smoothness time width thus providing a performance improvement the proposed estimator is compared numerically with the maximum likelihood ml estimator based on a tdl model in pilot assisted channel estimation pace for ofdm

3d magnetotelluric inversion using a limited memory quasi newton optimization the limited memory quasi newton method with simple bounds is used to develop a novel fully 3d magnetotelluric mt inversion technique this nonlinear inversion is based oniterativeminimizationofaclassicaltikhonovregularized penalty function however instead of the usual model space of log resistivities the approach iterates in a model space with simple bounds imposed on the conductivities of the 3d target the method requires storage proportional to 2 ncp n wherenisthenumberofconductivitiestoberecovered and ncp is the number of correction pairs practically only a few these requirements are much less than those imposed byothernewtonmethods whichusuallyrequirestorageproportional ton m orn n where m is the number of data tobeinverted thederivativesofthepenaltyfunctionarecalculated using an adjoint method based on electromagnetic fieldreciprocity theinversioninvolvesallfourentriesofthe mt impedance matrix the x3d integral equation forwardmodeling code is used as an engine for this inversion convergence performance and accuracy of the inversion are demonstrated on synthetic numerical examples after investigating erratic resistivities in the upper part of the model obtainedforoneoftheexamples weconcludethatthestandard tikhonov regularization is not enough to provide consistently smooth underground structures an additional regularizationhelpstoovercometheproblem

advances in three dimensional magnetotelluric modeling using integral equations recent progress in integral equation modeling of three dimensional magnetotelluric responses includes the ability to simulate 3 d structures which outcrop excluding topography which transect layer interfaces and which extend indefinitely in one or more directions the most important factor in achieving this capability is an accurate treatment of the electric surface charge in particular a previous integro difference formulation for evaluating charges has been abandoned in favor of true surface integrations over the source cells with potential differencing across the field cells in the 3 d body the new procedure constitutes a good approximation to galerkin s method while preserving internal consistency in terms of pulse basis functions to verify outcropping structures juxtaposed conductive and resistive prisms at the surface are simulated and compared to 2 d results an elongate version of the 3 d model shows good agreement with both transverse electric and transverse magnetic modes of the 2 d res

asymptotics for spherical needlets we investigate invariant random fields on the sphere using a new type of spherical wavelets called needlets these are compactly supported in frequency and enjoy excellent localization properties in real space with quasi exponentially decaying tails we show that for random fields on the sphere the needlet coefficients are asymptotically uncorrelated for any fixed angular distance this property is used to derive clt and functional clt convergence results for polynomial functionals of the needlet coefficients here the asymptotictheory is considered in the high frequency sense our proposals emerge from strong empirical motivations especially in connection with the analysis of cosmological data sets 1 introduction over the last two decades wavelets have emerged as one of the most interesting tools of statistical investigation in this paper we give an application to the statistical analysis of data sets indexed by the unit sphere s 2 this is motivated mostly by the analysis of the cosmic

spline smoothing and optimal rates of convergence in nonparametric regression models linear estimation is considered in nonparametric regression models of the form yi f xi ei xi e a b where the zero mean errors are uncorrelated with common variance a2 and the response function f is assumed only to have a bounded square integrable qth derivative the linear estimator which minimizes the maximum mean squared error summed over the observation points is derived and the exact minimax rate of convergence is obtained for practical problems where bounds on 11 f2q 11 and a2 may be unknown generalized cross validation is shown to give an adaptive estimator which achieves the minimax optimal rate under the additional assumption of normality 1 the model consider the nonparametric regression model

3d marine magnetotelluric modeling and inversion with the finite difference time domain method abstractfrequency domain methods which are typically applied to 3d magnetotelluric mt modeling require solving a system of linear equations for every frequency of interest this is memory and computationally intensive we developed a finite difference time domain algorithm to perform 3d mt modeling in a marine environment in which maxwell s equations are solved in a so called fictitious wave domain boundary conditions are efficiently treated via convolutional perfectly matched layers for which we evaluated optimized parameter values obtained by testing over a large number of models in comparison to the typically applied frequency domain methods two advantages of the finite difference time domain method are 1 that it is an explicit low memory method that entirely avoids the solution of systems of linear equations and 2 that it allows the computation of the electromagnetic field unknowns at all frequencies of interest in a single simulation we derive a design criterion for vertical node spacin

backfitting in smoothing spline anova a computational scheme for fitting smoothing spline anova models to large data sets with a near tensor product design is proposed such data sets are common in spatial temporal analyses the proposed scheme uses the backfitting algorithm to take advantage of the tensor product design to save both computational memory and time several ways to further speed up the backfitting algorithm such as collapsing component functions and successive over relaxation are discussed an iterative imputation procedure is used to handle the cases of near tensor product designs an application to a global historical surface air temperature data set which motivated this work is used to illustrate the scheme proposed

closest point search in lattices in this semitutorial paper a comprehensive survey of closest point search methods for lattices without a regular structure is presented the existing search strategies are described in a unified framework and differences between them are elucidated an efficient closest point search algorithm based on the schnorr euchner 1995 variation of the pohst 1981 method is implemented given an arbitrary point x spl isin spl ropf sup m and a generator matrix for a lattice spl lambda the algorithm computes the point of spl lambda that is closest to x the algorithm is shown to be substantially faster than other known methods by means of a theoretical comparison with the kannan 1983 1987 algorithm and an experimental comparison with the pohst 1981 algorithm and its variants such as the viterbo boutros see ibid vol 45 p 1639 42 1999 decoder modifications of the algorithm are developed to solve a number of related search problems for lattices such as finding a shortest vector determining the kissing number computing the voronoi 1908 relevant vectors and finding a korkine zolotareff 1873 reduced basis

simulation models with correct statistical properties for rayleigh fading channels in this paper new sum of sinusoids statistical simulation models are proposed for rayleigh fading channels these new models employ random path gain random initial phase and conditional random doppler frequency for all individual sinusoids it is shown that the autocorrelations and cross correlations of the quadrature components and the autocorrelation of the complex envelope of the new simulators match the desired ones exactly even if the number of sinusoids is as small as a single digit integer moreover the probability density functions of the envelope and phase the level crossing rate the average fade duration and the autocorrelation of the squared fading envelope which contains fourth order statistics of the new simulators asymptotically approach the correct ones as the number of sinusoids approaches infinity while good convergence is achieved even when the number of sinusoids is as small as eight the new simulators can be directly used to generate multiple uncorrelated fading waveforms for frequency selective fading channels multiple input multiple output channels and diversity combining scenarios statistical properties of one of the new simulators are evaluated by numerical results finding good agreements

intrinsic regression models for medial representation of subcortical structures the aim of this article is to develop a semiparametric model to describe the variability of the medial representation of subcortical structures which belongs to a riemannian manifold and establish its association with covariates of interest such as diagnostic status age and gender we develop a two stage estimation procedure to calculate the parameter estimates the first stage is to calculate an intrinsic least squares estimator of the parameter vector using the annealing evolutionary stochastic approximation monte carlo algorithm and then the second stage is to construct a set of estimating equations to obtain a more efficient estimate with the intrinsic least squares estimate as the starting point we use wald statistics to test linear hypotheses of unknown parameters and establish their limiting distributions simulation studies are used to evaluate the accuracy of our parameter estimates and the finite sample performance of the wald statistics we apply our methods to the detection of the diffe

electromagnetic modeling of 3 d structures by the method of system iteration using integral equations a new approach for electromagnetic modeling of three dimensional 3 d earth conductivity structures using integral equations is introduced a conductivity structure is divided into many substructures and the integral equation governing the scattering currents within a substructure is solved by a direct matrix inversion the influence of all other substructures are treated as external excitations and the solution for the whole structure is then found iteratively this is mathematically equivalent to partitioning the scattering matrix into many block submatrices and solving the whole system by a block iterative method this method reduces computer memory requirements since only one submatrix at a time needs to be stored the diagonal submatrices that require direct inversion are defined by local scatterers only and thus are generally better conditioned than the matrix for the whole structure the block iterative solution requires much less computation time than direct matrix inversion or conventional point

thomas bayes walk on manifolds convergence of the bayes posterior measure is considered in canonical statistical settings where observations sit on a geometrical object such as a compact manifold or more generally on a compact metric space verifying some conditions a natural geometric prior based on randomly rescaled solutions of the heat equation is considered upper and lower bound posterior contraction rates are derived

fft based preconditioners for toeplitz block least squares problems discretized two dimensional deconvolution problems arising e g in image restoration and seismic tomography can be formulated as least squares computations min b tx 2 where t is often a large scale rectangular toeplitz block matrix the authors consider solving such block least squares problems by the preconditioned conjugate gradient algorithm using square nonsingular circulant block and related preconditioners constructed from the blocks of the rectangular matrix t preconditioning with such matrices allows efficient implementation using the one dimensional or two dimensional fast fourier transform fft two block preconditioners related to those proposed by t chan and j olkin for square nonsingular toeplitz block systems are derived and analyzed it is shown that for important classes of t the singular values of the preconditioned matrix are clustered around one this extends the authors earlier work on preconditioners for toeplitz least squares iterations for one dimensiona

multivariate bayesian function estimation bayesian methods are developed for the multivariate nonparametric regression problem where the domain is taken to be a compact riemannian manifold in terms of the latter the underlying geometry of the manifold induces certain symmetries on the multivariate nonparametric regression function the bayesian approach then allows one to incorporate hierarchical bayesian methods directly into the spectral structure thus providing a symmetry adaptive multivariate bayesian function estimator one can also diffuse away some prior information in which the limiting case is a smoothing spline on the manifold this together with the result that the smoothing spline solution obtains the minimax rate of convergence in the multivariate nonparametric regression problem provides good frequentist properties for the bayes estimators an application to astronomy is included

3d finite difference frequency domain modeling of controlled source electromagnetic data direct solution and optimization for high accuracy three dimensional modeling of marine controlled source electromagnetic csem data is vital to improve the understanding of electromagnetic em responses collected in increasingly complex geologic settings a modeling tool for simulating 3d marine csem surveys based on a finite difference discretization of the helmholtz equation for the electric fields has been developed optimizations for csem simulations include the use of a frequency domain technique a staggering scheme that reduces inaccuracies especially for horizontal electric dipole sources located near the seafloor and a new interpolation technique that provides highly accurate em field values for receivers located in the immediate vicinity of the seafloor source singularities are eliminated through a secondary field approach in which the primary fields are computed analytically for a homogeneous or a 1d layered background the secondary fields are computed using the finite difference technique exploiting recent advances in computer technology and algorithmic developments the system of finite difference equations is solved using the mumps direct matrix solver in combination with the other optimizations this allows accurate em field computations for moderately sized models on small computer clusters the explicit availability of matrix factorizations is advantageous for multisource modeling and makes the algorithm well suited for future use within an inversion scheme comparisons of simulated data for 1 1d models to data generated using a 1d reflectivity technique and 2 3d models to published 3d data demonstrate the accuracy and benefits of the approach

fft interpolation from nonuniform samples lying in a regular grid this paper presents a method to interpolate a periodic band limited signal from its samples lying at nonuniform positions in a regular grid which is based on the fft and has the same complexity order as this last algorithm this kind of interpolation is usually termed the missing samples problem in the literature and there exists a wide variety of iterative and direct methods for its solution the one presented in this paper is a direct method that exploits the properties of the so called erasure polynomial and provides a significant improvement on the most efficient method in the literature which seems to be the burst error recovery ber technique of marvasti s the paper includes numerical assessments of the method s stability and complexity

efficient numerical methods in non uniform sampling theory summary we present a new second generation reconstruction algorithm for irregular sampling i e for the problem of recovering a band limited function from its non uniformly sampled values the efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems the choice of adaptive weights can be seen as a simple but very efficient method of preconditioning further substantial acceleration is achieved by utilizing the toeplitztype structure of the system matrix this new algorithm can handle problems of much larger dimension and condition number than have been accessible so far furthermore if some gaps between samples are large then the algorithm can still be used as a very efficient extrapolation method across the gaps

reconstruction of multidimensional signals from irregular noisy samples we focus on a multidimensional field with uncorrelated spectrum and study the quality of the reconstructed signal when the field samples are irregularly spaced and affected by independent and identically distributed noise more specifically we apply linear reconstruction techniques and take the mean square error mse of the field estimate as a metric to evaluate the signal reconstruction quality we find that the mse analysis could be carried out by using the closed form expression of the eigenvalue distribution of the matrix representing the sampling system unfortunately such distribution is still unknown thus we first derive a closed form expression of the distribution moments and we find that the eigenvalue distribution tends to the marcenko pastur distribution as the field dimension goes to infinity finally by using our approach we derive a tight approximation to the mse of the reconstructed field

nonparametric estimation of a probability density on a riemannian manifold using fourier expansions supposing a given collection y 1 cdots y n of i i d random points on a riemannian manifold we discuss how to estimate the underlying distribution from a differential geometric viewpoint the main hypothesis is that the manifold is closed and that the distribution is sufficiently smooth under such a hypothesis a convergence arbitrarily close to the n 1 2 rate is possible both in the l 2 and the l infty senses

conjugate direction relaxation solutions for 3 d magnetotelluric modeling in recent years there has been a tremendous amount of progress made in three dimensional 3 d magnetotelluric modeling algorithms much of this work has been devoted to the integral equation technique e g hohmann 1975 weidelt 1975 wannamaker et al 1984 wannamaker 1991 this method has contributed significantly to our understanding of electromagnetic field behavior in 3 d models however some of the very earliest work in 3 d modeling concentrated on differential methods e g jones and pascoe 1972 reddy et al 1977 it is generally recognized that differential methods are better suited than integral equation methods to model arbitrarily complex geometries and consequently this area has recently been receiving a great deal of attention e g madden and mackie 1989 xinghua et al 1991 mackie et al 1993 smith 1992 personnal communication differential methods lead to large sparse systems of equations to be solved for the unknown field values it is possible to use relaxation al

tensor product space anova models to deal with the curse of dimensionality in high dimensional nonparametric problems we consider using tensor product space anova models which extend the popular additive models and are able to capture interactions of any order the multivariate function is given an anova decomposition that is it is expressed as a constant plus the sum of functions of one variable main effects plus the sum of functions of two variables two factor interactions and so on we assume the interactions to be in tensor product spaces we show in both regression and white noise settings the optimal rate of convergence for the tps anova model is within a log factor of the one dimensional optimal rate and that the penalized likelihood estimator in tps anova achieves this rate of convergence the quick optimal rate of the tps anova model makes it very preferable in high dimensional function estimation many properties of the tensor product space of sobolev hilbert spaces are also given

semiparametric analysis of variance with tensor product thin plate splines we consider fitting multivariate models to response data based on analysis of variance anova like decompositions of functions of several variables f t 1 t d c f t f t t a theory for fitting some components of these models with polynomial smoothing splines exists when each t is in a subset of the real line in this case the various estimated components turn out to be certain tensor sums and products of polynomial splines this approach may not be natural when one or more of the variables are geographic in particular where nature does not know north from east

occam s inversion to generate smooth two dimensional models from magnetotelluric data magnetotelluric mt data are inverted for smooth 2 d models using an extension of the existing 1 d algorithm occam s inversion since an mt data set consists of a finite number of imprecise data an infinity of solutions to the inverse problem exists fitting field or synthetic electromagnetic data as closely as possible results in theoretical models with a maximum amount of roughness or structure however by relaxing the misfit criterion only a small amount models which are maximally smooth may be generated smooth models are less likely to result in overinterpretation of the data and reflect the true resolving power of the mt method the models are composed of a large number of rectangular prisms each having a constant conductivity apriori information in the form of boundary locations only or both boundary locations and conductivity may be included providing a powerful tool for improving the resolving power of the data joint inversion of te and tm synthetic data generated from known models al

integral equation solution for the transient electromagnetic response of a three dimensional body in a conductive half space the time domain integral equation for the three dimensional vector electric field is formulated as a convolution of the scattering current with the tensor green s function the convolution integral is divided into a sum of integrals over successive time steps so that a numerical scheme can be formulated with a time stepping approximation of the convolution of past values of the solution with the system impulse response this together with spatial discretization leads to a matrix equation in which previous solution vectors are multiplied by a series of matrices and fed back into the system by adding to the primary field source vector the spatial discretization based on a modification of the usual pulse basis formulation in the frequency domain includes an additional subset of divergence free basis functions generated by integrating the green s function around concentric closed rectangular paths the inductive response of the body is more accurately modeled with these additional basis functions and a