the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

an improved domain decomposition method for the 3d helmholtz equation the efficiency of a non overlapping domain decomposition method is strictly related to the transmission operator acting at the interfaces between subdomains a new operator arising from the non reflecting boundary conditions theory and a new iteration scheme lead to a dramatic improvement of the performances of the scheme which was originally proposed by bruno despres and which is recalled in the first part of this paper the convergence of the new method is proved and some numerical results show the comparison of the performances

empirical processes and random projections in this note we establish some bounds on the supremum of certain empirical processes indexed by sets of functions with the same l2 norm we present several geometric applications of this result the most important of which is a sharpening of the johnson lindenstrauss embedding lemma our results apply to a large class of random matrices as we only require that the matrix entries have a subgaussian tail

a new kernel based approach for linear system identification this paper describes a new kernel based approach for linear system identification of stable systems we model the impulse response as the realization of a gaussian process whose statistics differently from previously adopted priors include information not only on smoothness but also on bibo stability the associated autocovariance defines what we call a stable spline kernel the corresponding minimum variance estimate belongs to a reproducing kernel hilbert space which is spectrally characterized compared to parametric identification techniques the impulse response of the system is searched for within an infinite dimensional space dense in the space of continuous functions overparametrization is avoided by tuning few hyperparameters via marginal likelihood maximization the proposed approach may prove particularly useful in the context of robust identification in order to obtain reduced order models by exploiting a two step procedure that projects the nonparametric estimate onto the space of nominal models the continuous time derivation immediately extends to the discrete time case on several continuous and discrete time benchmarks taken from the literature the proposed approach compares very favorably with the existing parametric and nonparametric techniques

characterizing the function space for bayesian kernel models kernel methods have been very popular in the machine learning literature in the last ten years mainly in the context of tikhonov regularization algorithms in this paper we study a coherent bayesian kernel model based on an integral operator defined as the convolution of a kernel with a signed measure priors on the random signed measures correspond to prior distributions on the functions mapped by the integral operator we study several classes of signed measures and their image mapped by the integral operator in particular we identify a general class of measures whose image is dense in the reproducing kernel hilbert space rkhs induced by the kernel a consequence of this result is a function theoretic foundation for using non parametric prior specifications in bayesian modeling such as gaussian process and dirichlet process prior distributions we discuss the construction of priors on spaces of signed measures using gaussian and levy processes with the dirichlet processes being a special case the latter computational issues involved with sampling from the posterior distribution are outlined for a univariate regression and a high dimensional classification problem

random martingales and localization of maximal inequalities abstract let x d be a metric measure space for r 0 consider the hardy littlewood maximal operator m r f x def sup r r 1 b x r b x r f d we show that if there is an n 1 such that one has the microdoubling condition b x 1 1 n r b x r for all x x and r 0 then the weak 1 1 norm of m r has the following localization property m r l 1 x l 1 x sup r 0 m r r n r l 1 x l 1 x an immediate consequence is that if x d is ahlfors david n regular then the weak 1 1 norm of m r is n log n generalizing a result of stein and stromberg 1983 47 we show that this bound is sharp by constructing a metric measure space x d that is ahlfors david n regular for which the weak 1 1 norm of m 0 is n log n the localization property of m r is proved by assigning to each f l 1 x a distribution over random martingales for which the associated random doob maximal inequality controls the weak 1 1 inequality for m r

a non overlapping domain decomposition method with non matching grids for modeling large finite antenna arrays a non overlapping domain decomposition method ddm is proposed herein to solve maxwell equations in r 3 in this work the maxwell equations are discretized using a vector finite element method with hierarchical h curl vector basis functions there are two major ingredients in the proposed non overlapping ddm a a proper 1st order transmission condition to enforce field continuity across domain boundaries and b a cement technique to allow non matching grids for neighboring domains moreover a detail fourier analysis of the transmission condition for a canonical half space example is presented the analysis provides significant insights into the convergence behavior of the proposed non overlapping ddm for solving electromagnetic radiation problems such as the large finite antenna arrays particularly for the antenna arrays the proposed non overlapping ddm is extremely efficient since the formulation can easily incorporate geometrical repetitions exponentially tapered notch vivaldi antenna arrays with size up to 100x100 elements are solved on a common pc to validate the proposed non overlapping ddm

the connection between bayesian estimation of a gaussian random field and rkhs reconstruction of a function from noisy data is key in machine learning and is often formulated as a regularized optimization problem over an infinite dimensional reproducing kernel hilbert space rkhs the solution suitably balances adherence to the observed data and the corresponding rkhs norm when the data fit is measured using a quadratic loss this estimator has a known statistical interpretation given the noisy measurements the rkhs estimate represents the posterior mean minimum variance estimate of a gaussian random field with covariance proportional to the kernel associated with the rkhs in this brief we provide a statistical interpretation when more general losses are used such as absolute value vapnik or huber specifically for any finite set of sampling locations that includes where the data were collected the maximum a posteriori estimate for the signal samples is given by the rkhs estimate evaluated at the sampling locations this connection establishes a firm statistical foundation for several stochastic approaches used to estimate unknown regularization parameters to illustrate this we develop a numerical scheme that implements a bayesian estimator with an absolute value loss this estimator is used to learn a function from measurements contaminated by outliers

donsker classes of sets we study the central limit theorem clt and the law of large numbers lln for empirical processes indexed by a countable class of sets c the main result of purely measure theoretical nature relates different ways to measure the size of c it relies on a new rearrangement inequality that has been inspired by techniques used in the local theory of banach spaces as an application we give sharp necessary conditions for the clt that are in some sense the best possible we also obtain a way to compute the rate of convergence in the lln

on generic chaining and the smallest singular value of random matrices with heavy tails we present a very general chaining method which allows one to control the supremum of the empirical process suph h n 1 i 1nh2 xi eh2 in rather general situations we use this method to establish two main results first a quantitative non asymptotic version of the celebrated bai yin theorem on the singular values of a random matrix with i i d entries that have heavy tails and second a sharp estimate on the quadratic empirical process when h t t t t rn and is an isotropic unconditional log concave measure

the supremum of some canonical processes 1 introduction consider a random variable r v h such that eh2 l distributed like h to each t tk k 1 of 2 one can associate a r v xi ek l tkhk this defines a stochastic process xt t e2 by stochastic process we simply mean a collection of r v this process could be called the canonical process associated to h a long range and rather ambitious program would be to understand given h for which subsets t of 2 the process xt tct is bounded let us first discuss the case where h is centered gaussian the special importance of that case stems from the fact that for the question of boundedness discussed here gaussian processes of the type xt tct are the most general gaussian processes this fact is closely related to the rotational invariance of gaussian measures for a stochastic process xt tct where the set t is possibly uncountable we set

selector processes on classes of sets consider the random subset x of obtained by selecting independently each integer with a probability consider a finite class open image in new window of finite sets we describe a combinatorial quantity that is of the same order as open image in new window we then give a related result allowing to compute the supremum of the empirical process on a class of sets

variational inference of the fiber orientation density using diffusion mr imaging diffusion mr imaging has enabled the in vivo exploration of the connectional architecture in human brain this method particularly reveals the complex system of long range nerve fibers that integrate the functionally distinct areas of the cerebral cortex since the fibers are not directly observed but the diffusion process of water molecules in the underlying material a forward model is established that maps the microgeometry of nervous tissue onto the diffusion weighted signals this article proposes the spherical deconvolution of the fiber orientation density in a reproducing kernel hilbert space thereby generalizing previous approaches that perform a truncated fourier analysis on the sphere the specified inverse problem is solved within a smoothing spline framework which preserves the characteristic properties of a density function namely its normalization and non negativity a gaussian process model allows the specification of confidence bands for the estimated fiber orientation density and the rigorous selection of the hyperparameters here the high frequency content in the density function and the noise variance of the mr observations in addition we weaken the constant diffusivity assumption frequently made in the spherical convolution methodology the novel approach which uncovers the fiber orientation field of white matter is demonstrated with diffusion weighted data sets featuring high angular resolution

on the boundedness of bernoulli processes we present a positive solution to the so called bernoulli conjecture concerning the characterization of sample boundedness of bernoulli processes we also discuss some applications and related open problems

a chain rule for the expected suprema of gaussian processes the expected supremum of a gaussian process indexed by the image of an index set under a function class is bounded in terms of separate properties of the index set and the function class the bound is relevant to the estimation of nonlinear transformations or the analysis of learning algorithms whenever hypotheses are chosen from composite classes as is the case for multi layer models

a new kernel based approach for nonlinearsystem identification we present a novel nonparametric approach for identification of nonlinear systems exploiting the framework of gaussian regression the unknown nonlinear system is seen as a realization from a gaussian random field its covariance encodes the idea of fading memory in the predictor and consists of a mixture of gaussian kernels parametrized by few hyperparameters describing the interactions among past inputs and outputs the kernel structure and the unknown hyperparameters are estimated maximizing their marginal likelihood so that the user is not required to define any part of the algorithmic architecture e g the regressors and the model order once the kernel is estimated the nonlinear model is obtained solving a tikhonov type variational problem the hilbert space the estimator belongs to is characterized benchmarks problems taken from the literature show the effectiveness of the new approach also comparing its performance with a recently proposed algorithm based on direct weight optimization and with parametric approaches with model order estimated by aic or bic

robust kernel principal component analysis this letter discusses the robustness issue of kernel principal component analysis a class of new robust procedures is proposed based on eigenvalue decomposition of weighted covariance the proposed procedures will place less weight on deviant patterns and thus be more resistant to data contamination and model deviation theoretical influence functions are derived and numerical examples are presented as well both theoretical and numerical results indicate that the proposed robust method outperforms the conventional approach in the sense of being less sensitive to outliers our robust method and results also apply to functional principal component analysis

homoclinic solution and chaos in xf x 0 is followed by a move in the opposite direction i t 0 however there are equations 1 2 which are closely related to applications and where the negative feedback condition 3 is only true in a certain neighbourhood of x 0 furumochi 3 studied equation

the central limit theorem for empirical processes under local conditions the case of radon infinitely divisible limits without gaussian component on obtient des resultats de convergence faible pour des processus empiriques indices par des classes f de fonctions dans le cas de limites de radon purement de poisson infiniment divisibles

quantitative geometry rather than describing a traditional field of mathematics quantitative geometry refers to a modern way of thinking about geometry and its applications that occur in a wide range of mathematical disciplines the overarching theme here is that certain geometric problems can only be formulated after the introduction of auxiliary quantitative parameters and asking for their asymptotic or approximate behavior this leads to meaningful questions that do not necessarily have counterparts in classical qualitative geometric investigations a viewpoint that allows one to uncover deep and useful structural information that appears only in the quantitative regime quantitative reasoning allows one to tame complicated objects and phenomena that are unwieldy if one insists on exact measurements it is often the case that a rich yet structured picture emerges only if one asks the correct questions by allowing for controlled errors

regularity of gaussian processes on donne la validite de la conjecture de fernique 1974 sur l existence de mesures de majoration pouvant caracteriser la limitation des processus gaussiens

continuity of l 2 valued ornstein uhlenbeck processes a stationary l 2 valued ornstein uhlenbeck process is considered which is given formally by dx t ax t dt 2a db t where a is a positive self adjoint operator on l 2 b t is a cylindrical brownian motion on l 2 and a is a positive diagonal operator on l 2 a simple criterion is given for the almost sure continuity of x t in l 2 which is shown to be quite sharp furthermore in certain special cases we obtain simple necessary and sufficient conditions for this almost sure continuity

an iterative method for elliptic problems on regions partitioned into substructures some new preconditioners for discretizations of elliptic boundary problems are studied with these preconditioners the domain under consideration is broken into subdomains and preconditioners are defined which only require the solution of matrix problems on the subdomains analytic estimates are given which guarantee that under appropriate hypotheses the preconditioned iterative procedure converges to the solution of the discrete equations with a rate per iteration that is independent of the number of unknowns numerical examples are presented which illustrate the theoretically predicted iterative convergence rates

characterizations of almost surely continuous p stable random fourier series and strongly stationary processes on etend le probleme de dudley fernique aux processus p stables fortement stationnaires 1 p 2

discrepancy chaining and subgaussian processes the research leading to the results presented here has received funding from the european research council under the european community s seventh frame work programme fp7 2007 2013 erc grant agreement 203134 from the israel science foundation grant 666 06 and from the australian research council grant dp0986563

majorizing measures the generic chaining majorizing measures provide bounds for the supremum of stochastic processes they represent the most general possible form of the chaining argument going back to kolmogorov majorizing measures arose from the theory of gaussian processes but they now have applications far beyond this setting the fundamental question is the construction of these measures this paper focuses on the tools that have been developed for this purpose and in particular the use of geometric ideas applications are given to several natural problems where entropy methods are powerless

functional mixed effects spectral analysis in many experiments time series data can be collected from multiple units and multiple time series segments can be collected from the same unit this article introduces a mixed effects cramer spectral representation which can be used to model the effects of design covariates on the second order power spectrum while accounting for potential correlations among the time series segments collected from the same unit the transfer function is composed of a deterministic component to account for the population average effects and a random component to account for the unit specific deviations the resulting log spectrum has a functional mixed effects representation where both the fixed effects and random effects are functions in the frequency domain it is shown that when the replicate specific spectra are smooth the log periodograms converge to a functional mixed effects model a data driven iterative estimation procedure is offered for the periodic smoothing spline estimation of the fixed effects penalized estimation of the functional covariance of the random effects and unit specific random effects prediction via the best linear unbiased predictor

absorbing boundary conditions for diffusion equations summary we construct and analyse a family of absorbing boundary conditions for diffusion equations with variable coefficients curved artifical boundary and arbitrary convection it relies on the geometric identification of the dirichlet to neumann map and rational interpolation of z 1 2 in the complex plane the boundary conditions are stable accurate and practical for computations

low rank tensor recovery via iterative hard thresholding abstract we study extensions of compressive sensing and low rank matrix recovery matrix completion to the recovery of low rank tensors of higher order from a small number of linear measurements while the theoretical understanding of low rank matrix recovery is already well developed only few contributions on the low rank tensor recovery problem are available so far in this paper we introduce versions of the iterative hard thresholding algorithm for several tensor decompositions namely the higher order singular value decomposition hosvd the tensor train format tt and the general hierarchical tucker decomposition ht we provide a partial convergence result for these algorithms which is based on a variant of the restricted isometry property of the measurement operator adapted to the tensor decomposition at hand that induces a corresponding notion of tensor rank we show that subgaussian measurement ensembles satisfy the tensor restricted isometry property with high probability under a certain almost optimal bound on the number of measurements which depends on the corresponding tensor format these bounds are extended to partial fourier maps combined with random sign flips of the tensor entries finally we illustrate the performance of iterative hard thresholding methods for tensor recovery via numerical experiments where we consider recovery from gaussian random measurements tensor completion recovery of missing entries and fourier measurements for third order tensors

donsker classes and random geometry on donne des conditions geometriques aleatoires pour que le processus empirique indice par la classe des fonctions de carre integrable satisfasse le theoreme central limite

the central limit theorem and the law of iterated logarithm for empirical processes under local conditions a clt and a lil are proved under weak l2 gaussian bracketing conditions weaker than the usual ones these results have wide applicability and in particular provide an improvement of the jain marcus central limit theorem for c s valued random variables

some global bifurcations of two dimensional endomorphisms by use of critical lines many studies have been devoted to discrete dynamical systems represented by homeomorphisms and diffeomorphisms since the works by smale l and gavrilov and shilnikov 2 31 but comparatively few works have been devoted to endomorphisms that is to maps with a nonunique inverse however the mathematical models derived from the description of evolutive phenomena in several applicative fields are often represented by discrete dynamical systems or maps with a nonunique inverse this is particularly true in the socio economic field 4 61 but examples can also be found in ecology 7 81 biology 9 l l and control systems 121 this work concerns the study of the global dynamics of recurrences represented by twodimensional endomorphisms of applicative interest in the economic context useful notions for the local and global analysis of nonlinear endomorphisms have been introduced by gumowski and mira ll 13 141 for two dimensional endomorphisms in particular this is the notion of critical lines or critical curves representing the twodimensional extension of the critical points in one dimensional maps and the notions of absorbing area and chaotic area bounded by elements of critical lines which determine phase plane domains where chaotic solutions are located these notions which will be recalled in the next section together with some bifurcations involving the chaotic areas are the basic concepts and properties that we shall use in the analysis of the subsequent sections the aim of this paper is to illustrate by an example how powerful the critical lines may be in studying the global dynamics and bifurcations of plane nonlinear endomorphisms particularly when they are used in combination with more traditional techniques it is known that in diffeomorphisms global bifurcations which qualitatively change the structure of a chaotic attractor called interior crises or catastrophes by some authors 15 181 are characterized by the stable manifold of a saddle approaching and touching the chaotic attractor we shall find that this phenomenology also occurs in the endomorphisms of our model with a particular role being played by the critical lines in section 3 we shall describe our mathematical model a two dimensional piecewise linear map which we call map f together with some of its elementary properties and the definition of three classes of endomorphisms into which the analysis can be conveniently subdivided endomorphisms of the first class have been the object of a previous work 191 here we shall show in section 4 how several bifurcations involving chaotic areas can be explained by use of critical lines the main result is the analytical determination of the bifurcation which discriminates between the global attractivity of an absorbing area and the nonglobal attractivity

periodic solutions and chaos in a delay differential equation modelling haematopoesis simple mathematical models have been devised to describe the process of haematopoesis the formation and differentiation of blood cells within the human body l 21 haematopoesis is an example of a biological control system with an inherent delay in it this is due to the time required for primitive blood cells to mature such a delayed response is known to cause oscillatory behaviour in control systems and it has been found that similar oscillatory instabilities in blood cell counts are associated with diseases such as aplastic anaemia 2 this suggest that the study of such equations may be of biological importance mathematically they are of interest because they can exhibit complex aperiodic behaviour a model put forward for consideration by mackey glass 3 is

variable selection for the prediction of c 0 1 valued autoregressive processes using reproducing kernel hilbert spaces a model for the prediction of functional time series is introduced where observations are assumed to be continuous random functions we model the dependence of the data with a nonstandard autoregr

a fem domain decomposition method for photonic and electromagnetic band gap structures this paper introduces an efficient domain decomposition algorithm for the solution of time harmonic electromagnetic fields arising in three dimensional finite size photonic band gap and electromagnetic band gap structures the method is based on the finite element approximation and a nonoverlapping domain decomposition method a set of cement unknowns on the inter domain interfaces has been explicitly introduced to enforce the appropriate field continuities the introduction of these extra unknowns allows for nonconforming nonmatching triangulations across domain eliminating the need for periodic mesh to ensure and improve the convergence of the outer iteration loop robin transmission condition is used to communicate information across interfaces the resulting system of equations is solved with a fast algorithm that loosely resembles the well known finite element tearing and interconnecting algorithm in this algorithm the method solves in the preprocessing step for the robin primal subdomain problems multiple times by exciting one dual unknown at a time this step generates an iteration matrix that is then used to update the dual unknowns in the outer loop iteration the present method becomes extremely efficient for problems with geometric repetitions such as photonic and electromagnetic band gap structures

kernel methods for ill posed range based localization problems a kernel regression technique for the ill posed range based localization problem is proposed we introduce a generic kernel design method which relies on a probabilistic modeling of the respective physical environment in terms of a class of stochastic differential equations the combination of dynamic modeling with physical and stochastic interpretability leads to a unique solution in a reproducing kernel hilbert space and thus eliminates the need for a time discrete representation of the localization solution by design of the problem formulation an extended kalman filter naturally evolves as an iterative optimization method as a practical example we tackle an ill posed localization problem that stems from a preinstalled sensor network of unknown geometry providing nonsynchronous range information to a moving beacon to obtain a unique solution for the beacon positions and the network geometry we apply the proposed kernel regression technique and kernel design method the proposed approach eventually leads to a least squares optimization problem that can be interpreted as a maximum a posteriori estimation problem while the methodology is not restricted to localization problems only its validity is shown by a demonstrator which is comprised of a mobile robot and a network of multiple nonmoving beacons that provide time of arrival measurements of ultra wide band signals

stochastic evolution equations with fractional brownian motion in this paper linear stochastic evolution equations driven by infinite dimensional fractional brownian motion are studied a necessary and sufficient condition for the existence and uniqueness of the solution is established and the spatial regularity of the solution is analyzed separate proofs are required for the cases of hurst parameter above and below 1 2 the particular case of the laplacian on the circle is discussed in detail

on weakly bounded empirical processes let f be a class of functions on a probability space and let x 1 x k be independent random variables distributed according to we establish an upper bound that holds with high probability on rm sup f in f i f x i geq t for every t 0 and that depends on a natural geometric parameter associated with f we use this result to analyze the supremum of empirical processes of the form z f left k 1 sum i 1 k f p x i mathbb e f p right for p 1 using the geometry of f we also present some geometric applications of this approach based on properties of the random operator gamma k 1 2 sum i 1 k x i e i where x i i 1 k are sampled according to an isotropic log concave measure on mathbb r n

the flatness theorem for nonsymmetric convex bodies via the local theory of banach spaces let l be a lattice in rn and k a convex body disjoint from l the classical flatness theorem asserts that then w k l the l width of k does not exceed some bound depending only on the dimension n this fact was later found relevant to questions in integer programming kannan and lovasz 1988 showed that under the above assumptions w k l cn2 where c is a universal constant banaszczyk 1996 proved that w k l cn 1 log n if k has a centre of symmetry in the present paper we show that w k l cn3 2 for an arbitrary k it is conjectured that the exponent 3 2 may be replaced by 1 perhaps at the cost of a logarithmic factor we prove that for some naturally arising classes of bodies

numerical evaluation of reproducing kernel hilbert space inner products a new approach to numerically evaluate an inner product or a norm in an arbitrary reproducing kernel hilbert space rkhs is considered the proposed methodology enables us to approximate the rkhs inner product with the desired accuracy avoiding analytical expressions furthermore its implementation is illustrated by means of some classic examples and compared with the standard iterative method provided by weiner for this purpose finally applications in both the problem of representing approximately second order stochastic processes by means of series expansions and in the problem of signal detection are studied