a shape calculus based method for a transmission problem with a random interface the present work is devoted to an approximation of the statistical moments of the solution of a class of elliptic transmission problems in r 3 with uncertainly located transmission interfaces in this model the diffusion coefficient has a jump discontinuity across the random transmission interface which models linear diffusion in two different media separated by an uncertain surface we apply the shape calculus approach to approximate the solution perturbation by the so called shape derivative correspondingly statistical moments of the solution are approximated by the moments of the shape derivative we characterize the shape derivative as a solution of a related homogeneous transmission problem with nonzero jump conditions which is solved by the boundary integral equation method a rigorous theoretical framework is developed and the theoretical findings are supported by and illustrated in two particular examples

multilevel frames for sparse tensor product spaces for au f with an elliptic differential operator a mathcal h rightarrow mathcal h and stochastic data f the m point correlation function mathcal m m u of the random solution u satisfies a deterministic equation with the m fold tensor product operator a m of a sparse tensor products of hierarchic fe spaces in mathcal h are known to allow for approximations to mathcal m m u which converge at essentially the rate as in the case m 1 i e for the deterministic problem they can be realized by wavelet type fe bases von petersdorff and schwab in appl math 51 2 145 180 2006 schwab and todor in computing 71 43 63 2003 if wavelet bases are not available we show here how to achieve the fast computation of sparse approximations of mathcal m m u for galerkin discretizations of a by multilevel frames such as bpx or other multilevel preconditioners of any standard fem approximation for a numerical examples illustrate feasibility and scope of the method

sparse second moment analysis for elliptic problems in stochastic domains we consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations assuming normal perturbations with small amplitude and known mean field and two point correlation function we derive using a second order shape calculus deterministic equations for the mean field and the two point correlation function of the random solution for a model dirichlet problem which are 3rd order accurate in the boundary perturbation size using a variational boundary integral equation formulation on the unperturbed nominal boundary and a wavelet discretization we present and analyze an algorithm to approximate the random solution s mean and its two point correlation function at essentially optimal order in essentially mathcal o n work and memory where n denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain

shape optimization for quadratic functionals and states with random right hand sides we consider shape optimization problems under uncertainties on the input parameters the presented theory applies to the minimization of the expectation of a quadratic objective for a state function that depends linearly on a random input parameter it covers important objectives such as tracking type functionals for elliptic second order partial differential equations and the compliance in linear elasticity we show that the robust objective and its gradient are completely determined by low order moments of the random input we derive a cheap deterministic algorithm to minimize this objective and present model cases in structural optimization

analytic regularity and collocation approximation for elliptic pdes with random domain deformations in this work we consider the problem of approximating the statistics of a given quantity of interest qoi that depends on the solution of a linear elliptic pde defined over a random domain parameterized by n random variables the elliptic problem is remapped onto a corresponding pde with a fixed deterministic domain we show that the solution can be analytically extended to a well defined region in c n with respect to the random variables a sparse grid stochastic collocation method is then used to compute the mean and variance of the qoi finally convergence rates for the mean and variance of the qoi are derived and compared to those obtained in numerical experiments

sparse finite elements for elliptic problems with stochastic loading we formulate elliptic boundary value problems with stochastic loading in a bounded domain d d we show well posedness of the problem in stochastic sobolev spaces and we derive a deterministic elliptic pde in d d for the spatial correlation of the random solution we show well posedness and regularity results for this pde in a scale of weighted sobolev spaces with mixed highest order derivatives discretization with sparse tensor products of any hierarchic finite element fe spaces in d yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data multilevel preconditioning in d d allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equation

functional bregman divergence and bayesian estimation of distributions a class of distortions termed functional bregman divergences is defined which includes squared error and relative entropy a functional bregman divergence acts on functions or distributions and generalizes the standard bregman divergence for vectors and a previous pointwise bregman divergence that was defined for functions a recent result showed that the mean minimizes the expected bregman divergence the new functional definition enables the extension of this result to the continuous case to show that the mean minimizes the expected functional bregman divergence over a set of functions or distributions it is shown how this theorem applies to the bayesian estimation of distributions estimation of the uniform distribution from independent and identically drawn samples is presented as a case study

karhunen loeve approximation of random fields by generalized fast multipole methods kl approximation of a possibly instationary random field a x l2 dp l d subject to prescribed meanfield ea x a x dp and covariance va x x a x ea x a x ea x dp in a polyhedral domain d rd is analyzed we show how for stationary covariances va x x ga x x with ga z analytic outside of z 0 an m term approximate kl expansion am x of a x can be computed in log linear complexity the approach applies in arbitrary domains d and for nonseparable covariances ca it involves galerkin approximation of the kl eigenvalue problem by discontinuous finite elements of degree p 0 on a quasiuniform possibly unstructured mesh of width h in d plus a generalized fast multipole accelerated krylov eigensolver the approximate kl expansion am x of a x has accuracy o exp bm1 d if ga is analytic at z 0 and accuracy o m k d if ga is ck at zero it is obtained in o mn logn b operations where n o h d

the frisch scheme in dynamic system identification abstract the use of the frisch scheme in the identification of linear dynamic systems is investigated in order to describe the whole family of models that can explain given input output noisy sequences unlike the algebraic case it is shown that in general only a single model is compatible with the data these results are first proposed for single input single output systems and then generalized to the multivariable case

covariance regularity and mathcal h h matrix approximation for rough random fields in an open bounded domain mathrm d subset mathbb r n d rn with smooth boundary partial mathrm d d or on a smooth closed and compact riemannian n manifold mathcal m subset mathbb r n 1 m rn 1 we consider the linear operator equation a u f au f where a is a boundedly invertible strongly elliptic pseudodifferential operator of order r in mathbb r r r with analytic coefficients covering all linear second order elliptic pdes as well as their boundary reductions here f in l 2 omega h t f l2 ht is an h t ht valued random field with finite second moments with h t ht denoting the isotropic sobolev space of not necessarily integer order t modelled on the domain mathrm d d or manifold mathcal m m respectively we prove that the random solution s covariance kernel k u a 1 otimes a 1 k f ku a 1 a 1 kf on mathrm d times mathrm d d d resp mathcal m times mathcal m m m is an asymptotically smooth function provided that the covariance function k f kf of the random data is a schwartz distributional kernel of an elliptic pseudodifferential operator as a consequence numerical mathcal h h matrix calculus allows the deterministic approximation of singular covariances k u ku of the random solution u a 1 f in l 2 omega h t r u a 1f l2 ht r in mathrm d times mathrm d d d text resp mathcal m times mathcal m resp m m with work versus accuracy essentially equal to that for the mean field approximation with splines of fixed order mathrm d d text resp mathcal m resp m overcoming the curse of dimensionality in this case

kernels for longitudinal data with variable sequence length and sampling intervals we develop several kernel methods for classification of longitudinal data and apply them to detect cognitive decline in the elderly we first develop mixed effects models a type of hierarchical empirical bayes generative models for the time series after demonstrating their utility in likelihood ratio classifiers and the improvement over standard regression models for such classifiers we develop novel fisher kernels based on mixture of mixed effects models and use them in support vector machine classifiers the hierarchical generative model allows us to handle variations in sequence length and sampling interval gracefully we also give nonparametric kernels not based on generative models but rather on the reproducing kernel hilbert space we apply the methods to detecting cognitive decline from longitudinal clinical data on motor and neuropsychological tests the likelihood ratio classifiers based on the neuropsychological tests perform better than than classifiers based on the motor behavior discriminant classifiers performed better than likelihood ratio classifiers for the motor behavior tests

compression techniques for boundary integral equations asymptotically optimal complexity estimates matrix compression techniques in the context of wavelet galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense the fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying galerkin method at a computational expense that is proven to stay proportional to the number of unknowns key issues are the second compression which reduces the near field complexity significantly and an additional a posteriori compression the latter is based on a general result concerning an optimal work balance that applies in particular to the quadrature used to compute the compressed stiffness matrix with sufficient accuracy in linear time

on the fast matrix multiplication in the boundary element method by panel clustering the boundary element method bem leads to a system of linear equations with a full matrix while fem yields sparse matrices this fact seems to require much computational work for the definition of the matrix for the solution of the system and in particular for the matrix vector multiplication which always occurs as an elementary in this paper a method for the approximate matrix vector multiplication is described which requires much less arithmetical work in addition the storage requirements are strongly reduced

speech enhancement combining optimal smoothing and errors in variables identification of noisy ar processes in the framework of speech enhancement several parametric approaches based on an a priori model for a speech signal have been proposed when using an autoregressive ar model three issues must be addressed 1 how to deal with ar parameter estimation indeed due to additive noise the standard least squares criterion leads to biased estimates of ar parameters 2 can an estimation of the variance of the additive noise for each speech frame be obtained a voice activity detector is often used for its estimation 3 which estimation rules and techniques filtering smoothing etc can be considered to retrieve the speech signal our contribution in this paper is threefold first we propose to view the identification of the noisy ar process as an errors in variables problem this blind method has the advantage of providing accurate estimations of both the ar parameters and the variance of the additive noise second we propose an alternative algorithm to standard kalman smoothing based on a constrained minimum variance estimation procedure with a lower computational cost third the combination of these two steps is investigated it provides better results than some existing speech enhancement approaches in terms of signal to noise ratio snr segmental snr and informal subjective tests

certain models from uncertain data the algebraic case abstract the problem of deriving possible linear relations from data affected by additive noise has received remarkable attention in recent years particularly regarding the assumptions prejudices behind the procedures leading to unique models unlike traditional approaches leading to unique solutions least squares maximum likelihood etc the frisch scheme belonging to the family of errors in variables ev schemes leads to a whole family of models compatible with a set of noisy data and is considered as mildly affected by prejudices this paper shows how also under the assumptions of the frisch scheme it is possible to obtain a unique model from uncertain data and also to derive the actual amount of noise when the noiseless data are linked by a single linear relation the more general ev case of non independent additive noises is then considered and it is shown how also under these assumptions it is possible to obtain the unique set of parameters linking the noiseless data and how the family of compatible noise covariance matrices is defined in this case by the infinite elements of a linear variety

mathcal h matrix based second moment analysis for rough random fields and finite element discretizations we consider the efficient solution of strongly elliptic partial differential equations with random load based on the finite element method the solution s two point correlation can efficiently be approximated by means of an mathcal h matrix in particular if the correlation length is rather short or the correlation kernel is nonsmooth since the inverses of the finite element matrices which correspond to the differential operator under consideration can likewise efficiently be approximated in the mathcal h matrix format we can solve the correspondent mathcal h matrix equation in essentially linear time by using the mathcal h matrix arithmetic numerical experiments for three dimensional finite element discretizations for several correlation lengths and different smoothness are provided they validate the presented method and demonstrate that the computation times do not increase for nonsmooth or shortly correlated data

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

an interpolation based fast multipole method for higher order boundary elements on parametric surfaces in this article a black box higher order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis due to the exact surface representation the rate of convergence of higher order methods is not limited by approximation errors of the surface an element wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying galerkin method by performing the interpolation for the fast multipole method directly on the reference domain the cost complexity in the polynomial degree is reduced by one order this gain is independent of the application of either mathcal h or mathcal h 2 matrices in fact several simplifications in the construction of mathcal h 2 matrices are pointed out which are a by product of the surface representation extensive numerical examples are provided in order to quantify and qualify the proposed method in this article a black box higher order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis due to the exact surface representation the rate of convergence of higher order methods is not limited by approximation errors of the surface an element wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying galerkin method by performing the interpolation for the fast multipole method directly on the reference domain the cost complexity in the polynomial degree is reducedby one order this gain is independent of the application of either h or h 2 matrices in fact several simplificationsin the construction of h 2 matrices are pointed out which are a by product of the surface representation extensive numerical examples are provided in order to quantify and qualify the proposed method

paper identification of scalar errors in variables models with dynamics the paper considers the task of identifying a causal linear dynamic system excited by stationary gaussian zero mean noise of unknown spectrum and given measurements of the system input and output contaminated by independent additive stationary noise signals of unknown spectra while the solution is normally not unique finite dimensional parameterizations of the solution set are given in some cases involving just one parameter even though the various spectra may not be rational

numerical methods for differential equations in random domains physical phenomena in domains with rough boundaries play an important role in a variety of applications often the topology of such boundaries cannot be accurately described in all of its relevant detail due to either insufficient data or measurement errors or both this topological uncertainty can be efficiently handled by treating rough boundaries as random fields so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains to deal with this class of problems we propose a novel computational framework which is based on using stochastic mappings to transform the original deterministic stochastic problem in a random domain into a stochastic problem in a deterministic domain the latter problem has been studied more extensively and existing analytical numerical techniques can be readily applied in this paper we employ both a stochastic galerkin method and monte carlo simulations to solve the transformed stochastic problem we demonstrate our approach by applying it to an elliptic problem in single and double connected random domains and comment on the accuracy and convergence of the numerical methods

a sparse h matrix arithmetic general complexity estimates in a preceding paper hackbusch computing 62 1999 89 108 a class of matrices h matrices has been introduced which are data sparse and allow an approximate matrix arithmetic of almost linear complexity several types of h matrices have been analysed in hackbusch computing 62 1999 89 108 and hackbusch and khoromskij preprint mpi no 22 leipzig 1999 computing 64 2000 21 47 which are able to approximate integral nonlocal operators in fem and bem applications in the case of quasi uniform unstructured meshes in the present paper the general construction of h matrices on rectangular and triangular meshes is proposed and analysed first the reliability of h matrices in bem is discussed then we prove the optimal complexity of storage and matrix vector multiplication in the case of rather arbitrary admissibility parameters 1 and for nite elements up to the order 1 dened on quasi uniform rectangular triangular meshes in r d d 1 2 3 the almost linear complexity of the matrix addition multiplication and inversion of h matrices is also veried c 2000 elsevier science b v all rights reserved msc 65f05 65f30 65f50

analysis of the domain mapping method for elliptic diffusion problems on random domains in this article we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains in particular based on the decay of the karhunen loeve expansion of the domain perturbation field we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension for the implementation of a related approximation scheme like quasi monte carlo quadrature stochastic collocation etc we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field this simplifies the implementation and yields a non intrusive approach having this machinery at hand we can easily transfer it to stochastic interface problems the theoretical findings are complemented by numerical examples for both stochastic interface problems and boundary value problems on random domains

hierarchical matrix approximation for the uncertainty quantification of potentials on random domains abstract computing statistical quantities of interest of the solution of partial differential equations on random domains is an important and challenging task in engineering we consider the computation of these quantities by the perturbation approach especially we discuss how third order accurate expansions of the mean and the correlation can numerically be computed these expansions become even fourth order accurate for certain types of boundary variations the correction terms are given by the solution of correlation equations in the tensor product domain which can efficiently be computed by means of h matrices they have recently been shown to be an efficient tool to solve correlation equations with rough data correlations that is with low sobolev smoothness or small correlation length in almost linear time numerical experiments in three dimensions for higher order ansatz spaces show the feasibility of the proposed algorithm the application to a non smooth domain is also included