singular values of fractional integral operators a unification of theorems of hille tamarkin and chang abstract we obtain upper bounds on the singular values of fractional integral operators of the form l 0 x x y x 1 dy under the constraint 0 these bounds are employed to extend various results obtained over the last half century on the rate of decrease of eigenvalues and singular values of much more general integral operators apart from one relatively difficult theorem of hardy and littlewood math z 27 1928 565 606 the devices used are quite simple they involve no complex variable arguments

adaptive estimation over anisotropic functional classes via oracle approach we address the problem of adaptive minimax estimation in white gaussian noise model under lp loss 1 p 1 on the anisotropic nikolskii classes we present the es timation procedure based on a new data driven selection scheme from the family of kernel estimators with varying bandwidths for proposed estimator we establish so called lp norm oracle inequality and use it for deriving minimax adaptive results we prove the existence of rate adaptive estimators and fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk in particular some new asymptotics of the minimax risk are discovered including necessary and sufficient conditions for existencea uniformly consistent estimator we provide also with detailed overview of existing methods and results and formulate open problems in adaptive minimax estimation

renormalization group and asymptotics of solutions of nonlinear parabolic equations we present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations the method does not rely on a priori estimates such as the maximum principle it applies to systems of coupled equations to boundary conditions at infinity creating a front and to higher possibly fractional differential linear terms we present in detail the analysis for nonlinear diffusion type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity in an accompanying paper 5 the method is applied to systems of equations where some variables are slaved such as the complex ginzburg landau equation c 1994 john wiley sons inc

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

statistical approach to some ill posed problems for linear partial differential equations for linear partial differential equations some inverse source problems are treated statistically based on nonparametric estimation ideas by observing the solution in a small gaussian white noise the kernel type of estimators is used to estimate the unknown source function and its partial derivatives it is proved that such estimators are consistent as the noise intensity tends to zero depending on the principal part of the differential operator the optimal asymptotic rate of convergence is ascertained within a wide class of risk functions in a minimax sense

renormalization methods for higher order differential equations we adapt methodology of statistical mechanics and quantum field theory to approximate solutions to an arbitrary order ordinary differential equation boundary value problem by a second order equation in particular we study equations involving the derivative of a double well potential such as u u3 or u 2u3 using momentum fourier space variables we average over short length scales and demonstrate that the higher order derivatives can be neglected within the first cumulant approximation once length is properly renormalized yielding an approximation to solutions of the higher order equation from the second order the results are confirmed using numerical computations additional numerics confirm that the main role of the higher order derivatives is in rescaling the length

symmetry properties of a system of euler type equations for magnetized plasmas we provide all lie point symmetries of a system of partial differential equations of great interest to plasma physics we deduce the corresponding invariant solutions and discuss their physical interpretation we also consider some of its conditional and partial symmetries and obtain other interesting solutions finally we point out that although the system can be cast in divergence form and admits conserved currents it does not admit potential symmetries

adaptive estimates of linear functionals given a collection of nested closed convex symmetric sets and a linear functional we find estimates which are within a logarithm term of being simultaneously asymptotically minimax moreover these estimates can be constructed so that the loss of this logarithm term only occurs on a small subset of functions these estimates are quasi optimal since there do not exist estimators which do not lose a logarithm term on some part of the parameter spaces

error estimates for regularization methods in hilbert scales in this paper we study regularization methods to reconstruct the solution x of the linear ill posed problem ax y a x to y from noisy data y delta in y the regularization methods are of the general form x alpha delta bar x g alpha b s a a b s a y delta a bar x where b denotes an unbounded self adjoint strictly positive definite operator in the hilbert space x assuming ax sim x a and x bar x p le e for some bar x in x a ge 0 and p ge 0 x r b r 2 x is the norm in a hilbert scale x r r in rnus we derive error estimates which show that the accuracy of the above regularization methods is order optimal provided that the function g alpha lambda the regularization parameter alpha and the parameter s are chosen properly

convergence rates for regularized solutions of integral equations from discrete noisy data given data yi kg u ei where the e s are random errors the u s are known g is an unknown function in a reproducing kernel space with kernel r and k is a known integral operator it is shown how to calculate convergence rates for the regularized solution of the equation as the evaluation points u become dense in the interval of interest these rates are shown to depend on the eigenvalue asymptotics of krk where r is the integral operator with kernel r the theory is applied to abel s equation and the estimation of particle size densities in stereology rates of convergence of regularized histogram estimates of the particle size density are given

optimal discretization of inverse problems in hilbert scales regularization and self regularization of projection methods we study the efficiency of the approximate solution of ill posed problems based on discretized noisy observations which we assume to be given beforehand a basic purpose of the paper is the consideration of stochastic noise but deterministic noise is also briefly discussed we restrict ourselves to problems which can be formulated in hilbert scales within this framework we shall quantify the degree of ill posedness provide general conditions on projection schemes to achieve the best possible order of accuracy we pay particular attention on the problem of self regularization vs tikhonov regularization moreover we study the information complexity asymptotically any method which achieves the best possible order of accuracy must use at least such amount of noisy observations

on the balancing principle for some problems of numerical analysis we discuss a choice of weight in penalization methods the motivation for the use of penalization in computational mathematics is to improve the conditioning of the numerical solution one example of such improvement is a regularization where a penalization substitutes an ill posed problem for a well posed one in modern numerical methods for pdes a penalization is used for example to enforce a continuity of an approximate solution on non matching grids a choice of penalty weight should provide a balance between error components related with convergence and stability which are usually unknown in this paper we propose and analyze a simple adaptive strategy for the choice of penalty weight which does not rely on a priori estimates of above mentioned components it is shown that under natural assumptions the accuracy provided by our adaptive strategy is worse only by a constant factor than one could achieve in the case of known stability and convergence rates finally we successfully apply our strategy for self regularization of volterra type severely ill posed problems such as the sideways heat equation and for the choice of a weight in interior penalty discontinuous approximation on non matching grids numerical experiments on a series of model problems support theoretical results

numerical differentiation from a viewpoint of regularization theory in this paper we discuss the classical ill posed problem of numerical differentiation assuming that the smoothness of the function to be differentiated is unknown using recent results on adaptive regularization of general ill posed problems we propose new rules for the choice of the stepsize in the finite difference methods and for the regularization parameter choice in numerical differentiation regularized by the iterated tikhonov method these methods are shown to be effective for the differentiation of noisy functions and the order optimal convergence results for them are proved

sharp adaptive estimation of quadratic functionals estimation of a quadratic functional of a function observed in the gaussian white noise model is considered a data dependent method for choosing the amount of smoothing is given the method is based on comparing certain quadratic estimators with each other it is shown that the method is asymptotically sharp or nearly sharp adaptive simultaneously for the regular and irregular region we consider l p bodies and construct bounds for the risk of the estimator which show that for p 4 the estimator is exactly optimal and for example when p 3 100 then the upper bound is at most 1 055 times larger than the lower bound we show the connection of the estimator to the theory of optimal recovery the estimator is a calibration of an estimator which is nearly minimax optimal among quadratic estimators

asymptotically autonomous semiflows chain recurrence and lyapunov functions from the work of c conley it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set here we improve a result of l markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow in the special case that there is a lyapunov function for the limiting semiflow sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the lyapunov function

nonparametric estimation of the variogram and its spectrum in the study of intrinsically stationary spatial processes a new nonparametric variogram estimator is proposed through its spectral representation the methodology is based on estimation of the variogram s spectrum by solving a regularized inverse problem through quadratic programming the estimated variogram is guaranteed to be conditionally negative definite simulation shows that our estimator is flexible and generally has smaller mean integrated squared error than the parametric estimator under model misspecification our methodology is applied to a spatial dataset of decadal temperature changes copyright 2011 oxford university press

sparse recovery by the standard tikhonov method it is a common belief that the tikhonov scheme with the cdot l 2 penalty fails to reconstruct a sparse structure with respect to a given system i however in this paper we present a procedure for the sparse recovery which is totally based on the standard tikhonov method this procedure consists of two steps at first the tikhonov scheme is used as a sieve to find the coefficients near i which are suspected to be non zero within this step the performance of the standard tikhonov method is controlled in some sparsity promoting space rather than in the original hilbert one in the second step of the proposed procedure the coefficients with indices selected in the previous step are estimated by means of the data functional strategy the choice of the regularization parameter is a crucial issue for both steps we show that a recently developed parameter choice rule called the balancing principle can be effectively used here we also present the results of computational experiments giving the evidence of the reliability of our approach

smoothing spline semiparametric density models density estimation plays a fundamental role in many areas of statistics and machine learning parametric nonparametric and semiparametric density estimation methods have been proposed in the lite

a new method of solving noisy abel type equations abstract a new approach to solving noisy integral equations of the first kind is applied to the family of abel equations such equations play a role in stereology wicksell s unfolding problem medicine engineering and astronomy the method is based on an expansion in an arbitrary orthonormal basis coupled with exact inversion of the integral operator the inverse appears in the fourier coefficients of the expansion where it can be carried over to the usually well behaved basis elements in the form of the adjoint this method is an alternative to tikhonov regularization regularization of the inverse of the operator itself or a wavelet vaguelette singular value decomposition the method is particularly interesting in irregularity of the kernel the input or both because knowledge of the spectral properties of the operator is not required the method is also of interest in regular cases where these spectral properties are not sufficiently known or are hard to deal with for smooth input functions the simple basis of trigonometric functions yields input estimators whose mean integrated squared error converges at the optimal rate for the entire family of abel operators this can be shown when smooth wavelets are used for abel operators with index smaller than 1 2 and when the haar wavelet is used for operators with index larger than 1 2

a new lower bound for multiple hypothesis testing the purpose of this correspondence is to give a new easily tractable and sharp lower bound for the maximal error in multiple hypothesis testing with an application to nonasymptotic lower bounds for the minimax risk of estimators

boundary blow up solutions to elliptic systems of competitive type abstract we consider the elliptic system u u p v q v u r v s in where p s 1 q r 0 and r n is a smooth bounded domain subject to different types of dirichlet boundary conditions f u v i u v and sf u v on where 0 under several hypotheses on the parameters p q r s we show existence and nonexistence of positive solutions uniqueness and nonuniqueness we further provide the exact asymptotic behaviour of the solutions and their normal derivatives near some more general related problems are also studied

sharp adaptation for inverse problems with random noise we consider a heteroscedastic sequence space setup with polynomially increasing variances of observations that allows to treat a number of inverse problems in particular multivariate ones we propose an adaptive estimator that attains simultaneously exact asymptotic minimax constants on every ellipsoid of functions within a wide scale that includes ellipoids with polynomially and exponentially decreasing axes and at the same time satisfies asymptotically exact oracle inequalities within any class of linear estimates having monotone non increasing weights the construction of the estimator is based on a properly penalized blockwise stein s rule with weakly geometically increasing blocks as an application we construct sharp adaptive estimators in the problems of deconvolution and tomography

a remark on uniqueness of large solutions for elliptic systems of competitive type abstract we prove that the semilinear system u a x u p v q v b x u r v s in a smooth bounded domain r n has a unique positive solution with the boundary condition u v on provided that p s 1 q r 0 and p 1 s 1 q r 0 the main novelty is imposing a growth on the possibly singular weights a x b x near rather than requiring them to have a precise asymptotic behavior

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

regularization of some linear ill posed problems with discretized random noisy data for linear statistical ill posed problems in hilbert spaces we introduce an adaptive procedure to recover the unknown so lution from indirect discrete and noisy data this procedure is shown to be order optimal for a large class of problems smooth ness of the solution is measured in terms of general source condi tions the concept of operator monotone functions turns out to be an important tool for the analysis

cross validated spline methods for the estimation of three dimensional tumor size distributions from observations on two dimensional cross sections abstract we study the problem of estimating the distribution of the three dimensional radiuses of a collection of spheres given measurements of the two dimensional radiuses of a sample of planar cross sections this problem arises in the estimation of the tumor size distribution of spherical microtumors induced in mouse livers following injection of a carcinogen we first convert this problem to a form suitable for the application of cross validated spline methods for the solution of ill posed integral equations given noisy data then we develop special numerical techniques that will allow the spline methods to be accurately applied to integral equations like those associated with the present problem we apply the resulting method to some mouse liver data the subject mouse liver has been completely dissected allowing a rare comparison of the estimate with the truth the statistical properties of the estimate are explored via monte carlo methods the interplay between statistical and numerical analyti

poisson inverse problems in this paper we focus on nonparametric estimators in inverse problems for poisson processes involving the use of wavelet decompositions adopting an adaptive wavelet galerkin discretization we find that our method combines the well known theoretical advantages of wavelet vaguelette decompositions for inverse problems in terms of optimally adapting to the unknown smoothness of the solution together with the remarkably simple closed form expressions of galerkin inversion methods adapting the results of barron and sheu ann statist 19 1991 1347 1369 to the context of log intensity functions approximated by wavelet series with the use of the kullback leibler distance between two point processes we also present an asymptotic analysis of convergence rates that justifies our approach in order to shed some light on the theoretical results obtained and to examine the accuracy of our estimates in finite samples we illustrate our method by the analysis of some simulated examples

lie equations for asymptotic solutions of perturbation problems of ordinary differential equations lie theory is applied to perturbation problems of ordinary differential equations to construct approximate solutions and invariant manifolds according to the renormalization group approach of iwasa and nozaki a method to construct asymptotic solutions invariant under the renormalization group prog theor phys 116 605 2006 it is proved that asymptotic behavior of solutions is obtained from the lie equations even if original equations have no symmetries normal forms of the lie equations are introduced to investigate the existence of invariant manifolds

self regularization by projection for noisy pseudodifferential equations of negative order it is well known that pseudodifferential equations of negative order considered in sobolev spaces with small smoothness indices are ill posed on the other hand it is known that efficient discretization schemes with properly chosen discretization parameters allow to obtain a regularization effect for such equations the main accomplishment of the present paper is the principle for the adaptive choice of the discretization parameters directly from noisy discrete data we argue that the combination of this principle with wavelet based matrix compression techniques leads to algorithms which are order optimal in the sense of complexity

convergence rates of general regularization methods for statistical inverse problems and applications previously the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut off and tikhonov type estimators spectral cut off estimators achieve minimax rates for a broad range of smoothness classes and operators but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator tikhonov estimators are simpler to compute but still involve the inversion of an operator and achieve minimax rates only in restricted smoothness classes in this paper we introduce a unifying technique to study the mean square error of a large class of regularization methods spectral methods including the aforementioned estimators as well as many iterative methods such as nu methods and the landweber iteration the latter estimators converge at the same rate as spectral cut off but require only matrix vector products our results are applied to various problems in particular we obtain precise convergence rates for satellite gradiometry l 2 boosting and errors in variable problems

statistical inverse estimation in hilbert scales the recovery of signals from indirect measurements blurred by random noise is considered under the assumption that prior knowledge regarding the smoothness of the signal is avialable for greater flexibility the general problem is embedded in an abstract hilbert scale in the applications sobolev scales are used for the construction of estimators we employ preconditioning along with regularized operator inversion in the appropriate inner product where the operator is bounded but not necessarily compact a lower bound to certain minimax rates is included and it is shown that in generic examples the proposed estimators attain the asymptotic minimax rate examples include errors in variables deconvolution and indirect nonparametric regression special instances of the latter are estimation of the source term in a differential equation and the estimation of the initial state in the heat equation

renormalization group and the ginzburg landau equation we use renormalization group methods to prove detailed long time asymptotics for the solutions of the ginzburg landau equations with initial data approaching asx different spiraling stationary solutions a universal pattern is formed depending only on this asymptotics at spatial infinity

asymptotic equivalence theory for nonparametric regression with random design this paper establishes the global asymptotic equivalence between the nonparametric regression with random design and the white noise under sharp smoothness conditions on an unknown regression or drift function the asymptotic equivalence is established by constructing explicit equivalence mappings between the nonparametric regression and the whitenoise experiments which provide synthetic observations and synthetic asymptotic solutions from any one of the two experiments with asymptotic properties identical to the true observations and given asymptotic solutions from the other the impact of such asymptotic equivalence results is that an investigation in one nonparametric problem automatically yields asymptotically analogous results in all other asymptotically equivalent nonparametric problems 1 introduction the purpose of this paper is to establish the global asymptotic equivalence between the nonparametric regression with random design and the white noise under sharp smoothness conditions on an unknown regression or drift function we establish this asymptotic equivalence by constructing explicit equivalence mappings between the nonparametric regression and the white noise problems as in brown and low 1996 for their asymptotic equivalence results the equivalence mapping from the nonparametric regression to the white noise provides synthetic observations of the white noise from the nonparametric regression such that the distributions of the synthetic observations are asymptotically equivalent to those of the true observations of the white noise for any asymptotic solution to a white noise problem the application of the solution to the synthetic observations provides an asymptotic solution to the corresponding nonparametric regression problem with identical asymptotic properties likewise the equivalence mapping from the white noise produces synthetic observations of the nonparametric regression problem and synthetic asymptotic solutions to white noise problems based on those of the corresponding nonparametric regression problems the impact of such asymptotic equivalence results is that an investigation in one nonparametric problem automatically yields asymptotically analogous results in

log density estimation in linear inverse problems we estimate a probability density function p which is related by a linear operator k to a density function q in sequences of regular exponential families based on a random sample from q in this paper deconvolution and positron emission tomography are considered the logarithm of the density function is approximated by basis functions consisting of singular functions of k while direct maximum likelihood or minimum kullback leibler density estimation in exponential families selects the parameters to match the moments of the basis functions to the sample moments in the inverse problem the moment of each singular function is related to a corresponding moment of the direct problem by a factor given by a singular value a of k thus an appropriate analogue of the maximum likelihood estimate is obtained by matching moments with respect to p to 1 a times the empirical moments associated with the sample from q bounds on the kullback leibler distance between the true density and the estimators are obtained and rates of convergence are established for log density functions having a measure of smoothness the density estimator converges to the unknown density in the kullback leibler sense and in the l2 sense at a rate determined not only by the order of smoothness of the log density and the dimension of data but also by the decay rate of the singular values of the operator a minimax lower bound for deconvolution is provided under certain conditions numerical examples using simulated data are provided to illustrate the finite sample performance of the proposed method for deconvolution and positron emission tomography

a hermite pade perspective on the renormalization group with an application to the correlation function of lieb liniger gas renormalization group rg methods are based on symmetry principles that are peculiar to quantum field theory and certain related settings however in cases when the function depends only on the coupling the rg technology does include a part which is related to the equation for the running coupling and which we will call the single variable rg svrg that turns out to be a purely formal method for improving the convergence properties of series expansions in fact we show that svrg is an integral hermite pade approximation preceded by a series reversion a complementary viewpoint on what svrg does to an expansion is that it performs a minimal upgrade to a self similar form the minimality being crucial in applications svrg is especially useful in problems in which one knows not only the first few terms of an expansion valid for small values of a variable but also the exponent but not the prefactor of the scaling law for large values of the variable as a practical example we extract the scaling law prefactor for the one body density matrix of the lieb liniger gas using a new result for the fourth order term in the short distance expansion we find remarkable agreement with known quantum diffusion monte carlo numerical results

a study of least squares and maximum likelihood for image reconstruction in positron emission tomography positron emission tomography pet is a radiologic tool offering a unique capability for measuring tissue metabolism in vivo a number of biological and physical factors limit the resolution of pet so often the statistical aspects of image reconstruction have an appreciable effect on the quality of information obtained from a study to a first approximation reconstruction involves the solution of a linear inverse problem with a line integral radon type transform standard filtered back projection reconstruction is based on the method of least squares although computationally efficient the method does not enforce positivity constraints leading to undesirable negative artifacts in the results maximum likelihood based approaches to reconstruction do not suffer from this problem but their computational complexity has limited the ability to determine quantitatively the improvements in image quality in this paper asymptotic approximations and numerical simulations are used to examine the least squares and maximum likelihood methods in some detail the studies are carried out for idealized representations of conventional and time of flight tomographs the asymptotic analysis indicates that for a range of sobolev norms the rates of estimation of least squares and maximum likelihood reconstructions are of the same order this is borne out by numerical studies however in these studies maximum likelihood is found to be more efficient than least squares on a conventional distance angle tomograph the root mean square error is on the order of 10 20 smaller for maximum likelihood reconstructions the corresponding reduction in the root mean square error on a time of flight tomograph is in the 5 range similar results are found using more complex region of interest based measures of reconstruction accuracy in addition it is found that much of the effect of maximum likelihood can apparently be realized by postprocessing least squares solutions in an ad hoc manner to reduce negativity artifacts since the postprocessing scheme is computationally fast this suggests that in pet it may be possible to realize the benefits of maximum likelihood without the substantial computational overhead of the em algorithm

renormalization group and singular perturbations multiple scales boundary layers and reductive perturbation theory perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis with numerous examples we illustrate its application to ordinary differential equation problems involving multiple scales boundary layers with technically difficult asymptotic matching and wkb analysis in contrast to conventional methods the renormalization group approach requires neither ad hoc assumptions about the structure of perturbation series nor the use of asymptotic matching our renormalization group approach provides approximate solutions which are practically superior to those obtained conventionally although the latter can be reproduced if desired by appropriate expansion of the renormalization group approximant we show that the renormalization group equation may be interpreted as an amplitude equation and from this point of view develop reductive perturbation theory for partial differential equations describing spatially extended systems near bifurcation points deriving both amplitude equations and the center manifold s1063651x 96 00506 5

asymptotically exact nonparametric hypothesis testing in sup norm and at a fixed point for the signal in gaussian white noise model we consider the problem of testing the hypothesis h 0 f 0 the signal f is zero against the nonparametric alternative h 1 f where is a set of functions on r 1 of the form f f f c here is a holder or sobolev class of functions f is either the sup norm of f or the value of f at a fixed point c 0 is a constant is the minimax rate of testing and 0 is the asymptotic parameter of the model we find exact separation constants c 0 such that a test with the given summarized asymptotic errors of first and second type is possible for c c and is not possible for c c we propose asymptotically minimax test statistics

statistical estimation and optimal recovery new formulas are given for the minimax linear risk in estimating a linear functional of an unknown object from indirect data contaminated with random gaussian noise the formulas cover a variety of loss functions and do not require the symmetry of the convex a priori class it is shown that affine minimax rules are within a few percent of minimax even among nonlinear rules for a variety of loss functions it is also shown that difficulty of estimation is measured by the modulus of continuity of the functional to be estimated the method of proof exposes a correspondence between minimax affine estimates in the statistical estimation problem and optimal algorithms in the theory of optimal recovery

risk bounds for model selection via penalization performance bounds for criteria for model selection are devel oped using recent theory for sieves the model selection criteria are based on an empirical loss or contrast function with an added penalty term moti vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob servations most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function we show that the quadratic risk of the minimum penal ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve this accuracy index quanties the trade off among the candidate models between the approximation error and parameter dimension relative to sample size if we choose a list of models which exhibit good approximation prop erties with respect to different classes of smoothness the estimator can be simultaneously minimax rate optimal in each of those classes this is what is usually called adaptation the type of classes of smoothness in which one gets adaptation depends heavily on the list of models if too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously it may happen that the estimator is only approx

adaptive estimation of linear functionals in hilbert scales from indirect white noise observations we consider adaptive estimating the value of a linear functional from indirect white noise observations for a flexible approach the problem is embedded in an abstract hilbert scale we develop an adaptive estimator that is rate optimal within a logarithmic factor simultaneously over a wide collection of balls in the hilbert scale it is shown that the proposed estimator has the best possible adaptive properties for a wide range of linear functionals the case of discretized indirect white noise observations is studied and the adaptive estimator in this setting is developed

exact constants for pointwise adaptive estimation under the riesz transform we consider nonparametric estimation of a multivariate function and its partial derivatives at a fixed point when the riesz transform of the function is observed in gaussian white noise we assume that the unknown function belongs to some sobolev class and construct an estimation procedure which achieves the best asymptotic minimax risk when the smoothness of the function is unknown

classification of radial solutions for semilinear elliptic systems with nonlinear gradient terms abstract we are concerned with the classification of positive radial solutions for the system u v p v f u where p 0 and f c 1 0 is a nondecreasing function such that f t 0 for all t 0 we show that in the case where the system is posed in the whole space r n such solutions exist if and only if 1 0 s f t d t p 2 p 1 d s this is the counterpart of the keller osserman condition for the case of single semilinear equation similar optimal conditions are derived in case where the system is posed in a ball of r n if f t t q q 1 using dynamical system techniques we are able to describe the behaviour of solutions at infinity in case where the system is posed in the whole r n or around the boundary in case of a ball

asymptotic theory of filtering for linear operator equations with discrete noisy data on considere les equations integrales de fredholm de premiere espece a noyaux continus avec echantillonnage des donnees et alteration par le bruit blanc on etablit une condition suffisante de convergence des methodes de filtrage

oracle inequalities for inverse problems we consider a sequence space model of statistical linear inverse problems where we need to estimate a function f from indirect noisy observations let a finite set a of linear estimators be given our aim is to mimic the estimator in a that has the smallest risk on the true f under general conditions we show that this can be achieved by simple minimization of an unbiased risk estimator provided the singular values of the operator of the inverse problem decrease as a power law the main result is a nonasymptotic oracle inequality that is shown to be asymptotically exact this inequality can also be used to obtain sharp minimax adaptive results in particular we apply it to show that minimax adaptation on ellipsoids in the multivariate anisotropic case is realized by minimization of unbiased risk estimator without any loss of efficiency with respect to optimal nonadaptive procedures

group analysis and renormgroup symmetries an original regular approach to constructing special type symmetries for boundary value problems namely renormgroup symmetries is presented different methods of calculating these symmetries based on modern group analysis are described an application of the approach to boundary value problems is demonstrated with the help of a simple mathematical model

on the adaptive selection of the parameter in regularization of ill posed problems we study the possibility of using the structure of the regularization error for a posteriori choice of the regularization parameter as a result a rather general form of a selection criterion is proposed and its relation to the heuristical quasi optimality principle of tikhonov and glasko z vychisl mat mat fiz 4 1964 pp 564 571 and to an adaptation scheme proposed in a statistical context by lepskii theory probab appl 36 1990 pp 454 466 is discussed the advantages of the proposed criterion are illustrated by using such examples as self regularization of the trapezoidal rule for noisy abel type integral equations lavrentiev regularization for nonlinear ill posed problems and an inverse problem of the two dimensional profile reconstruction

global large time self similarity of a thermal diffusive combustion system with critical nonlinearity we study the initial value problem of the thermal diffusive combustion system u 1 t u 1 x x u 1 u 2 2 u 2 t du 2 xx u 1 u 2 2 x is an element of r 1 for non negative spatially decaying initial data of arbitrary size and for any positive constant d we show that if the initial data decay to zero sufficiently fast at infinity then the solution u 1 u 2 converges to a self similar solution of the reduced system u 1 t u 1 xx u 1 u 2 2 u 2 t du 2 xx in the large time limit in particular u 1 decays to zero like o t 1 2 delta where delta 0 is an anomalous exponent depending on the initial data and u 2 decays to zero with normal rate o t 1 2 the idea of the proof is to combine the a priori estimates for the decay of global solutions with the renormalization group method for establishing the self similarity of the solutions in the large time limit c 1996 academic press inc

expanding the asymptotic explosive boundary behavior of large solutions to a semilinear elliptic equation the main goal of this paper is to study the asymptotic expansion near the boundary of the large solutions of the equation delta u lambda u m f in omega where lambda 0 m 1 f is an element of c omega f 0 and omega is an open bounded set of r n n 1 with boundary smooth enough roughly speaking we show that the number of explosive terms in the asymptotic boundary expansion of the solution is finite but it goes to infinity as in goes to 1 we prove that the expansion consists in two eventual geometrical and non geometrical parts separated by a term independent on the geometry of partial derivative omega but dependent on the diffusion for low explosive sources the non geometrical part does not exist all coefficients depend on the diffusion and the geometry of the domain by means of well known properties of the distance function dist x partial derivative omega for high explosive sources the preliminary coefficients relative to the non geometrical part are independent on omega and the diffusion finally the geometrical part does not exist for very high explosive sources consists in two eventual geometrical and non geometrical parts separated by a term independent on the geometry of partial omega but dependent on the diffusion for low explosive sources the non geometrical part does not exist all coefficients depend on the diffusion and the geometry of the domain by means of well known properties of the distance function rm dist x partial omega dist x for high explosive sources the preliminary coefficients relative to the non geometrical part are independent on omega and the diffusion finally the geometrical part does not exist for very high explosive sources

minimax and minimax adaptive estimation in multiplicative regression locally bayesian approach this paper deals with the non parametric estimation in the regression with the multiplicative noise using the local polynomial fitting and the bayesian approach we construct the minimax on isotropic holder class estimator next applying lepski s method we propose an estimator which is optimally adaptive over the collection of isotropic holder classes to prove the optimality of the proposed procedure we establish in particular the exponential inequality for the deviation of locally bayesian estimator since the parameter estimated these theoretical results are illustrated by simulation study

variability assessment in positron emission tomography and related generalized deconvolution models abstract the problem of variance assessment for positron emission tomography pet image reconstructions is considered in the context of generalized deconvolution a refinement of an approximate technique proposed by carson and colleagues is examined computational implications of representing the reconstruction kernel in terms of a weighted sum of gaussian densities are developed bias and variance characteristics of the resulting variance estimators are examined by numerical simulation for typical regions the error in estimated standard deviations is found to be on the order of 10 the use of smoothing to obtain more reliable pointwise variance estimators is described and some theoretical analysis of this technique is carried out for the pet application simulations suggest that the percent improvement in the root mean squared error accuracy of pointwise variance estimators obtained by smoothing can be on the order of 30 a practical application of the methodology to a pet study is presented

renormalization group and lienard systems of differential equations autonomous lienard systems which constitute a huge family of periodic motions exhibit limit cycle behaviour in certain cases and centres in others in the literature the signature for the existence of these two different facets of periodic behaviour has been studied from different geometrical perspectives and not from a general viewpoint starting out from general considerations we show in this work that a certain renormalization scheme is capable of unifying these two different aspects of periodic motion we show that the renormalization group allows a unified analysis of the limit cycle and centre in a lienard system of differential equations while the approach is perturbative it is possible to make a stronger statement in this regard two different classes of lienard systems have been considered the analysis provides clear insight into how the frequency gets corrected at different orders of perturbation as one flips the parity of the damping term

oscillatory radial solutions for subcritical biharmonic equations it is well known that the biharmonic equation 2u u u p 1 with p 1 has positive solutions on rn if and only if the growth of the nonlinearity is critical or supercritical we close a gap in the existing literature by proving the existence and uniqueness up to scaling and symmetry of oscillatory radial solutions on rn in the subcritical case analyzing the nodal properties of these solutions we also obtain precise information about sign changing large radial solutions and radial solutions of the dirichlet problem on a ball

radial solutions of a polyharmonic equation with power nonlinearity abstract we classify the regular radial solutions of the equation m u u u p 1 with m a positive integer and p 1 any such solution u is uniquely determined by the center values of u and its iterated laplacians up to order m 1 that is by the center point u 0 u 0 m 1 u 0 most solutions have finite exit radius that is they are unbounded and defined on open balls of finite radius are eventually monotonic with respect to the radial variable and diverge to or the center points of these solutions form two open half spaces in r m separated by an m 1 dimensional manifold m solutions with center points on m are either global or they have finite exit radius and oscillate about 0 with unbounded amplitude our results yield precise information regarding the existence multiplicity and nature of large radial solutions of m u u u p 1 on a ball of given radius

sharp adaptive estimation of linear functionals we consider estimation of a linear functional t f where f is an unknown function observed in gaussian white noise we find asymptotically sharp adaptive estimators on various scales of smoothness classes in multidimensional situations the results allow evaluating explicitly the effect of dimension and treating general scales of classes furthermore we establish a connection between sharp adaptation and optimal recovery namely we propose a scheme that reduces the construction of sharp adaptive estimators on a scale of functional classes to a solution of the corresponding optimization problem

bandwidth selection for indirect density estimation based on corrupted histogram data abstract motivated by a number of practical applications we consider a class of indirect nonparametric density estimation problems in which the observed data consist of a histogram of binned empirically corrected counts due to variability in the process of correction the histogram cannot be modeled in terms of simple scaled poisson statistics this departure necessitates the development of a new methodology for bandwidth selection a variant of the method of unbiased risk estimation is proposed the methodology is studied using numerical simulations and asymptotic analysis tools in the context of a class of idealized density deconvolution problems the methodology is adapted for application to the practical reconstruction problem of positron emission tomography pet realistic numerical simulations and physical phantom data are presented to validate the approach in this setting some illustrations with cerebral glucose utilization and myocardial blood flow studies from some actual patient data sets

asymptotic behavior of large radial solutions of a polyharmonic equation with superlinear growth this paper concerns the blow up behavior of large radial solutions of polyharmonic equations with power nonlinearities and positive radial weights specifically we consider radially symmetric solutions of mu c x u p on an annulus x rn x with 0 and 0 that diverge to infinity as x here n m n p 1 and c is a positive continuous function on the interval letting r qc r q for r with q 2m p 1 q q q 1 q 2m 1 1 p 1 and c c 1 p 1 we show that as x the ratio u x x remains between positive constants that depend only on m and p extending well known results for the second order problem we prove in the fourth order case that u x x 1 as x and obtain precise asymptotic expansions if c is sufficiently smooth at in certain higher order cases we find solutions for which the ratio u x x does not converge but oscillates about 1 with non vanishing amplitude

speed of estimation in positron emission tomography and related inverse problems we study a continuous idealization of the image reconstruction problem in positron emission tomography considered as an example of bivariate density estimation based on indirect observations given a large sample of indirect observations we consider the size of the equivalent sample of observations whose original exact positions would allow equally accurate estimation of the image of interest both for indirect and for direct observations we establish exact minimax rates of convergence of estimation for all possible estimators over suitable smoothness classes of functions the approach of the paper is applicable to a wide class of linear inverse problems

camouflaged deconvolution with application to blood curve modeling in fdg pet studies abstract camouflaged deconvolution arises when the kernel in a simple convolution model is not completely specified we consider a situation in which the same fixed signal is repeatedly measured by separate convolutions with imprecisely known kernels we develop a regularization methodology for application to these problems the method involves simultaneous estimation of the target signal and the unknown parameters of the convolution kernels cross validation is used to determine the degree of smoothness of the solution we use simulation studies matched to the application to evaluate statistical performance these simulations find that the convergence of the regularization estimator is largely unaffected by the lack of information about the convolution kernels we illustrate the methodology by application to a blood curve modeling problem arising in the context of positron emission tomography pet studies with flurodeoxyglucose fdg a commonly used glucose tracer the results show promise towards the

large solutions for an elliptic system of competitive type existence uniqueness and asymptotic behavior abstract this paper deals with the existence uniqueness and asymptotic behavior of boundary blow up solutions for an elliptic system of competitive type where the weight functions a x b x vanish on the boundary of the underlying domain at different rates according to the point of the boundary the proof relies on subsolutions and supersolutions and the localization method

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

on the regularization of projection methods for solving iii posed problems in this paper we consider a class of regularization methods for a discretized version of an operator equation which includes the case that the problem is ill posed with approximately given right hand side we propose an a priori as well as an a posteriori parameter choice method which is similar to the discrepancy principle of ivanov morozov from results on fractional powers of selfadjoint operators we obtain convergence rates which are in many cases the same for both parameter choices

pointwise and sup norm sharp adaptive estimation of functions on the sobolev classes the problem of nonparametric function estimation in the gaussian white noise model is considered it is assumed that the unknown function belongs to one of the sobolev classes with an unknown regularity parameter asymptotically exact adaptive estimators of functions are proposed on the scale of sobolev classes with respect to pointwise and sup norm risks it is shown that unlike the case of l 2 risk a loss of efficiency under adaptation is inevitable here bounds on the value of the loss of efficiency are obtained

minimax estimation of functionals of discrete distributions we propose a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters and elaborate on the case of discrete distributions where the support size s is unknown and may be comparable with or even much larger than the number of observations n we treat the respective regions where the functional is nonsmooth and smooth separately in the nonsmooth regime we apply an unbiased estimator for the best polynomial approximation of the functional whereas in the smooth regime we apply a bias corrected version of the maximum likelihood estimator mle we illustrate the merit of this approach by thoroughly analyzing the performance of the resulting schemes for estimating two important information measures 1 the entropy h p sum i 1 s p i ln p i and 2 f alpha p sum i 1 s p i alpha alpha 0 we obtain the minimax l 2 rates for estimating these functionals in particular we demonstrate that our estimator achieves the optimal sample complexity n asymp s ln s for entropy estimation we also demonstrate that the sample complexity for estimating f alpha p 0 is n asymp s 1 alpha ln s which can be achieved by our estimator but not the mle for 1 we show the minimax l 2 rate for estimating f alpha p is n ln n 2 alpha 1 for infinite support size while the maximum l 2 rate for the mle is n 2 alpha 1 for all the above cases the behavior of the minimax rate optimal estimators with n samples is essentially that of the mle plug in rule with n ln n samples which we term effective sample size enlargement we highlight the practical advantages of our schemes for the estimation of entropy and mutual information we compare our performance with various existing approaches and demonstrate that our approach reduces running time and boosts the accuracy moreover we show that the minimax rate optimal mutual information estimator yielded by our framework leads to significant performance boosts over the chow liu algorithm in learning graphical models the wide use of information measure estimation suggests that the insights and estimators obtained in this paper could be broadly applicable