solution of inverse problems in image processing by wavelet expansion we describe a wavelet based approach to linear inverse problems in image processing in this approach both the images and the linear operator to be inverted are represented by wavelet expansions leading to a multiresolution sparse matrix representation of the inverse problem the constraints for a regularized solution are enforced through wavelet expansion coefficients a unique feature of the wavelet approach is a general and consistent scheme for representing an operator in different resolutions an important problem in multigrid multiresolution processing this and the sparseness of the representation induce a multigrid algorithm the proposed approach was tested on image restoration problems and produced good results

hybrid variable fidelity optimization by using a kriging based scaling function solving design problems that rely on very complex and computationally expensive calculations using standard optimization methods might not be feasible given design cycle time constraints variable fidelity methods address this issue by using lower fidelity models and a scaling function to approximate the higher fidelity models in a provably convergent framework in the past scaling functions have mainly been either first order multiplicative or additive corrections these are being extended to second order in this investigation variable metric approaches for calculating second order scaling information are developed a kriging based scaling function is introduced to better approximate the high fidelity response on a more global level an adaptive hybrid method is also developed in this investigation the adaptive hybrid method combines the additive and multiplicative approaches so that the designer does not have to determine which is more suitable prior to optimization the methodologies developed in this research are compared to existing methods using two demonstration problems the first problem is analytic whereas the second involves the design of a supercritical high lift airfoil the results demonstrate that the krigingbased scaling methods improve computational expense by lowering the number of high fidelity function calls required for convergence the results also indicate the hybrid method is both robust and effective

optimized estimation of moments for nonstationary signals the behavior of nonstationary moment estimators that are based on linear time invariant filtering is analyzed the performance of such estimators is evaluated in terms of their time averaged variance and time averaged squared bias optimal estimators that minimize a convex combination of bias and variance are derived the superiority of such optimally weighted filtering over the conventional exponentially windowed moment estimation technique is demonstrated by means of a simple example the same example also serves to illustrate the difficulties encountered when the construction of optimal estimators relies on uncertain prior information as well as to demonstrate the feasibility of overcoming such difficulties by using appropriately designed robust optimal estimators we also include a brief discussion of the relevance of the proposed moment estimation technique to identification of linear time variant systems and to estimation of time variant spectra

sampling based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures we define a risk averse nonanticipative feasible policy for multistage stochastic programs and propose a methodology to implement it the approach is based on dynamic programming equations written for a risk averse formulation of the problem this formulation relies on a new class of multiperiod risk functionals called extended polyhedral risk measures dual representations of such risk functionals are given and used to derive conditions of coherence in the one period case conditions for convexity and consistency with second order stochastic dominance are also provided the risk averse dynamic programming equations are specialized considering convex combinations of one period extended polyhedral risk measures such as spectral risk measures to implement the proposed policy the approximation of the risk averse recourse functions for stochastic linear programs is discussed in this context we detail a stochastic dual dynamic programming algorithm which converges to the optimal value of the risk averse p

a wavelet based kl like expansion for wide sense stationary random processes we describe a wavelet based series expansion for wide sense stationary processes the expansion coefficients are uncorrelated random variables a property similar to that of a karhunen loeve kl expansion unlike the kl expansion however the wavelet based expansion does not require the solution of the eigen equation and does not require that the process be time limited this expansion also has advantages over fourier series which is often used as an approximation to the kl expansion in that it completely eliminates correlation and that the computation for its coefficients are more stable over large time intervals the basis functions of this expansion can be obtained easily from wavelets of the lemaire meyer 1990 type and the power spectral density of the process finally the expansion can be extended to some nonstationary processes such as those with wide sense stationary increments

provably convergent multifidelity optimization algorithm not requiring high fidelity derivatives functions does not alter the con vergence proof of the optimization algorithm and is shown to be robust to poor low delity information the algorithm is compared with an unconstrained single delity quasi newton algorithm and two

estimation of 1 f noise several models have emerged for describing 1 f sup spl gamma noise processes based on these various techniques for estimating the properties of such processes have been developed this paper provides theoretical analysis of a new wavelet based approach which has the advantages of having low computational complexity and being able to handle the case where the 1 f sup spl gamma noise might be embedded in a further white noise process however the analysis conducted here shows that these advantages are balanced by the fact that the wavelet based scheme is only consistent for spectral exponents spl gamma in the range spl gamma spl isin 0 1 this is in contradiction to the results suggested in previous empirical studies when spl gamma spl isin 0 1 this paper also establishes that wavelet based maximum likelihood methods are asymptotically gaussian and efficient finally the asymptotic rate of mean square convergence of the parameter estimates is established and is shown to slow as spl gamma approaches one combined with a survey of non wavelet based methods these new results give a perspective on the various tradeoffs to be considered when modeling and estimating 1 f sup spl gamma noise processes

entropy based algorithms for best basis selection adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals it permits efficient compression of a variety of signals such as sound and images the predefined libraries of modulated waveforms include orthogonal wavelet packets and localized trigonometric functions and have reasonably well controlled time frequency localization properties the idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost the method relies heavily on the remarkable orthogonality properties of the new libraries all expansions in a given library conserve energy and are thus comparable several cost functionals are useful one of the most attractive is shannon entropy which has a geometric interpretation in this context

a karhunen loeve like expansion for 1 f processes via wavelets while so called 1 f or scaling processes emerge regularly in modeling a wide range of natural phenomena as yet no entirely satisfactory framework has been described for the analysis of such processes orthonormal wavelet bases are used to provide a new construction for nearly 1 f processes from a set of uncorrelated random variables

on the multichannel rendezvous problem fundamental limits optimal hopping sequences and bounded time to rendezvous one of the fundamental problems in a cognitive radio network known as the multichannel rendezvous problem is for two secondary users to find a common channel that is not blocked by primary users the basic idea for solving such a problem in most works in the literature is for the two users to select their own channel hopping sequences and then rendezvous when they both hop to a common unblocked channel at the same time in this paper we focus on the fundamental limits of the multichannel rendezvous problem and formulate such a problem as a constrained optimization problem where the selection of the random hopping sequences of the two secondary users must satisfy certain constraints we derive various lower bounds for the expected respectively maximum time to rendezvous under certain constraints for some of these lower bounds we are also able to construct optimal channel hopping sequences that achieve the lower bounds inspired by the constructions of quorum systems and relative difference sets our constructions of the channel hopping sequences are based on the mathematical theories of finite projective planes orthogonal latin squares and sawtooth sequences the use of such theories in the constructions of channel hopping sequences appear to be new and better than other existing schemes in terms of minimizing the expected respectively maximum time to rendezvous

spectral representation of fractional brownian motion in n dimensions and its properties fractional brownian motion fbm provides a useful model for processes with strong long term dependence such as 1 f sup spl beta spectral behavior however fbm s are nonstationary processes so that the interpretation of such a spectrum is still a matter of speculation to facilitate the study of this problem another model is provided for the construction of fbm from a white noise like process by means of a stochastic or ito integral in frequency of a stationary uncorrelated random process also a generalized power spectrum of the nonstationary fbm process is defined this new approach to fbm can be used to compute all of the correlations power spectra and other properties of fbm in this paper a number of these fbm properties are developed from this model such as the t sup h law of scaling the power law of fractional order the correlation of two arbitrary fbm s and the evaluation of the fractal dimension under various transformations this new treatment of fbm using a spectral representation is extended also for the first time to two or more topological dimensions in order to analyze the features of isotropic n dimensional fbm

an improved method for 2 d self similar image synthesis we propose a new method called incremental fourier synthesis to generate 2 d self similar images based on a 2d fractional brownian motion fbm model with this method the stationary increments of fbm are created by a fourier synthesis method and the increments are added up to generate the nonstationary 2d fbm process since the new method takes advantage of the fft its computational complexity is only o n sup 2 log sub 2 n and its memory requirement is only o n sup 2 for a self similar image of size n spl times n

martingale characterizations of risk averse stochastic optimization problems this paper addresses risk awareness of stochastic optimization problems nested risk measures appear naturally in this context as they allow beneficial reformulations for algorithmic treatments the reformulations presented extend usual dynamic equations by involving risk awareness in the problem formulation nested risk measures are built on risk measures which originate by conditioning on the history of a stochastic process we derive martingale properties of these risk measures and use them to prove continuity it is demonstrated that stochastic optimization problems which incorporate risk awareness via nesting risk measures are continuous with respect to the natural distance governing these optimization problems the nested distance

orthogonal time varying filter banks and wavelet packets considers the construction of orthogonal time varying filter banks by examining the time domain description of the two channel orthogonal filter bank the authors find it possible to construct a set of orthogonal boundary filters which allows to apply the filter bank to one sided or finite length signals without redundancy or distortion the method is constructive and complete there is a whole space of orthogonal boundary solutions and there is considerable freedom for optimization this may be used to generate subband tree structures where the tree varies over time and to change between different filter sets the authors also show that the iteration of discrete time time varying filter banks gives continuous time bases just as in the stationary case this gives rise to wavelet or wavelet packet bases for half line and interval regions

rectangular sets of probability measures in this paper we consider the notion of rectangularity of a set of probability measures from a somewhat different point of view we define rectangularity as a property of dynamic decomposition of a distributionally robust stochastic optimization problem and show how it relates to the modern theory of coherent risk measures consequently we discuss robust formulations of multistage stochastic optimization problems in frameworks of stochastic programming stochastic optimal control and markov decision processes

the wavelet transform of stochastic processes with stationary increments and its application to fractional brownian motion the wavelet transform of random processes with wide sense stationary increments is shown to be a wide sense stationary process whose correlation function and spectral distribution are determined the second order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained applications to the spectral analysis and to the synthesis of fractional brownian motion are given

conditional risk mappings we introduce an axiomatic definition of a conditional convex risk mapping and we derive its properties in particular we prove a representation theorem for conditional risk mappings in terms of conditional expectations we also develop dynamic programming relations for multistage optimization problems involving conditional risk mappings

analysis and synthesis of 1 f processes via shannon wavelets 1 f processes can be very useful in modeling processes with long term correlation we propose analysis and synthesis procedures to express these processes in terms of the shannon wavelet unlike previous techniques our analysis procedure generates uncorrelated decomposition coefficients for the 1 f process this is done by taking onto account and then removing the residual correlation between the wavelet components the analysis procedure is the major contribution of this work the proposed synthesis algorithm which is a byproduct of the proposed analysis algorithm is competitive with other techniques

correlation structure of the discrete wavelet coefficients of fractional brownian motion it is shown that the discrete wavelet coefficients of fractional brownian motion at different scales are correlated and that their auto and cross correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional brownian motion itself the rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet

a matrix valued wavelet kl like expansion for wide sense stationary random processes matrix valued wavelet series expansions for wide sense stationary processes are studied in this paper the expansion coefficients a are uncorrelated matrix random process which is a property similar to that of a matrix karhunen loe spl grave ve mkl expansion unlike the mkl expansion however the matrix wavelet expansion does not require the solution of the eigen equation this expansion also has advantages over the fourier series which is often used as an approximation to the mkl expansion in that it completely eliminates correlation the basis functions of this expansion can be obtained easily from wavelets of the matrix valued lemarie spl acute meyer type and the power spectral density of the process

tilings of the time frequency plane construction of arbitrary orthogonal bases and fast tiling algorithms the authors consider expansions which give arbitrary orthonormal tilings of the time frequency plane these differ from the short time fourier transform wavelet transform and wavelet packets tilings in that they change over time they show how this can be achieved using time varying orthogonal tree structures which preserve orthogonality even across transitions the method is based on the construction of boundary and transition filters these allow us to construct essentially arbitrary tilings time varying modulated lapped transforms are a special case where both boundary and overlapping solutions are possible with filters obtained by modulation they present a double tree algorithm which for a given signal decides on the best binary segmentation in both time and frequency that is it is a joint optimization of time and frequency splitting the algorithm is optimal for additive cost functions e g rate distortion and results in time varying best bases the main application of which is for compression of nonstationary signals experiments on test signals are presented

on solving multistage stochastic programs with coherent risk measures we consider a class of multistage stochastic linear programs in which at each stage a coherent risk measure of future costs is to be minimized a general computational approach based on dynamic programming is derived that can be shown to converge to an optimal policy by computing an inner approximation to future cost functions we can evaluate an upper bound on the cost of an optimal policy and an outer approximation delivers a lower bound the approach we describe is particularly useful in sampling based algorithms and a numerical example is provided to show the efficacy of the methodology when used in conjunction with stochastic dual dynamic programming

multiscale autoregressive processes ii lattice structures for whitening and modeling for pt i see ibid vol 40 no 8 p 1915 34 1992 lattice structures for the whitening and modeling of isotropic processes on trees are developed and a result relating the stability properties of these models to the reflection coefficient sequence introduced in pt i is presented this framework allows one to obtain a detailed analysis of the wold decomposition of processes on trees one interesting aspect of this is that there is a significantly larger class of singular processes on dyadic trees than on the integers

self similarity part i splines and operators the central theme of this pair of papers parts i and ii in this issue is self similarity which is used as a bridge for connecting splines and fractals the first part of the investigation is deterministic and the context is that of l splines these are defined in the following terms s t is a cardinal l spline iff l s t sigma kisinza k delta t k where l is a suitable pseudodifferential operator our starting point for the construction of self similar splines is the identification of the class of differential operators l that are both translation and scale invariant this results into a two parameter family of generalized fractional derivatives parttau gamma where gamma is the order of the derivative and tau is an additional phase factor we specify the corresponding l splines which yield an extended class of fractional splines the operator parttau gamma is used to define a scale invariant energy measure the squared l2 norm of the gammath derivative of the signal which provides a regularization functional for interpolating or fitting the noisy samples of a signal we prove that the corresponding variational or smoothing spline estimator is a cardinal fractional spline of order 2gamma which admits a stable representation in a b spline basis we characterize the equivalent frequency response of the estimator and show that it closely matches that of a classical butterworth filter of order 2gamma we also establish a formal link between the regularization parameter lambda and the cutoff frequency of the smoothing spline filter omega0aplambda 2gamma finally we present an efficient computational solution to the fractional smoothing spline problem it uses the fast fourier transform and takes advantage of the multiresolution properties of the underlying basis functions

risk forms representation disintegration and application to partially observable two stage systems we introduce the concept of a risk form which is a real functional of two arguments a measurable function on a polish space and a measure on that space we generalize the duality theory and the kusuoka representation to this setting for a risk form acting on a product of polish spaces we define marginal and conditional forms and we prove a disintegration formula which represents a risk form as a composition of its marginal and conditional forms we apply the proposed approach to two stage stochastic programming problems with partial information and decision dependent observation distribution

on kusuoka representation of law invariant risk measures in this paper we discuss representations of law invariant coherent risk measures in a form of integrals of the average value at risk measures we show that such an integral representation exists iff the dual set of the considered risk measure is generated by one of its elements and this representation is uniquely defined on the other hand representation of risk measures as a maximum of such integral forms is not unique the suggested approach gives a constructive way for writing such representations

wavelet based representations for a class of self similar signals with application to fractal modulation a potentially important family of self similar signals based upon a deterministic scale invariance characterization is introduced these signals which are referred to as dy homogeneous signals because they generalize the well known homogeneous functions have highly convenient representations in terms of orthonormal wavelet bases in particular wavelet representations can be exploited to construct orthonormal self similar bases for these signals the spectral and fractal characteristics of dy homogeneous signals make them appealing candidates for use in a number of applications as one potential example their use in a communications based context is considered specifically a strategy for embedding information into a dy homogeneous waveform on multiple time scales is developed this multirate modulation strategy called fractal modulation is potentially well suited for use with noisy channels of simultaneously unknown duration and bandwidth

on the convergence of decomposition methods for multistage stochastic convex programs we prove the almost sure convergence of a class of sampling based nested decomposition algorithms for multistage stochastic convex programs in which the stage costs are general convex functions of the decisions and uncertainty is modelled by a scenario tree as special cases our results imply the almost sure convergence of stochastic dual dynamic programming cutting plane and partial sampling cupps algorithm and dynamic outer approximation sampling algorithms when applied to problems with general convex cost functions

fractal estimation from noisy data via discrete fractional gaussian noise dfgn and the haar basis the authors show that the application of the discrete wavelet transform dwt using the haar basis to the increments of fractional brownian motion fbm also known as discrete fractional gaussian noise dfgn yields coefficients which are weakly correlated and have a variance that is exponentially related to scale similar results were derived by flandrin 1989 tewfik and kim for a continuous time fbm going through a continuous wavelet transform cwt the new theoretical results justify an improvement to a fractal estimation algorithm recently proposed by wornell and oppenheim the performance of the new algorithm is compared with that of wornell and oppenheim s see ieee trans signal processing vol 40 p 611 623 mar 1992 algorithm in numerical simulation

on the spectrum of fractional brownian motions fractional brownian motions fbms provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order however due to the nonstationary nature of these processes the precise meaning of such spectra remains generally unclear two complementary approaches are proposed which are intended to clarify this point the first one based on a time frequency analysis takes into account the nonstationary nature of fbm and puts emphasis on time averaged measurements the second one based on a time scale analysis is matched to self similarity properties of fbm and reveals an underlying stationary structure relative to each time scaling

new fast smoothers for multiscale systems we consider the smoothing problem for multiscale stochastic systems based on the wavelet transform these models involve processes indexed by the nodes of a dyadic tree each level of the dyadic tree represents one scale or resolution of the process therefore moving upward on the tree divides the resolution by 2 whereas moving downward multiplies it by 2 the processes are built according to a recursion in scale from coarse to fine to which random details are added to operate the change in scale one must perform an interpolation this is achieved using the qmf pair of operators attached to a wavelet transform these models have proved to be of great value to capture textures or fractal like processes as well as to perform multiresolution sensor fusion an example of which is given here up to now however only subclasses of multiscale systems were amenable to fast algorithms and through different formalisms those relying on haar s wavelet and those involving only one of the two wavelet interpolators we provide here a unifying framework that handles any system based on orthogonal wavelets a smoothing theory is presented to define the field of fast algorithms for markov random fields and give intuition on how to design them this theory reveals the difficulties arising with general multiscale systems we then prove that orthogonality properties of wavelets are the gate to fastness

a note on the discrete wavelet transform of second order processes some properties of second order random processes such as stationarity and scaling are characterized via corresponding multiscale properties of their discrete wavelet transform the single scale characterizations valid in the continuous setting do not extend to the discrete setting some extensions of these results are also indicated

efficient multiscale regularization with applications to the computation of optical flow a new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical flow fields in an image sequence the solution of the new problem formulation is computed with an efficient multiscale algorithm experiments on several image sequences demonstrate the substantial computational savings that can be achieved due to the fact that the algorithm is noniterative and in fact has a per pixel computational complexity that is independent of image size the new approach also has a number of other important advantages specifically multiresolution flow field estimates are available allowing great flexibility in dealing with the tradeoff between resolution and accuracy multiscale error covariance information is also available which is of considerable use in assessing the accuracy of the estimates in particular these error statistics can be used as the basis for a rational procedure for determining the spatially varying optimal reconstruction resolution furthermore if there are compelling reasons to insist upon a standard smoothness constraint the new algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation finally the usefulness of the approach should extend to a wide variety of ill posed inverse problems in which variational techniques seeking a smooth solution are generally used

matching wavelet packets to gaussian random processes we consider the problem of approximating a set of arbitrary discrete time gaussian random processes by a single representative wavelet based gaussian process we measure the similarity between the original processes and the wavelet based process with the bhattacharyya 1943 coefficient by manipulating the bhattacharyya coefficient we reduce the task of defining the representative process to finding an optimal unitary matrix of wavelet based eigenvectors an associated diagonal matrix of eigenvalues and a mean vector the matching algorithm we derive maximizes the nonadditive bhattacharyya coefficient in three steps a migration algorithm that determines the best basis by searching through a wavelet packet tree for the optimal unitary matrix of wavelet based eigenvectors and two separate fixed point algorithms that derive an appropriate set of eigenvalues and a mean vector we illustrate the method with two different classes of processes first order markov and bandlimited the technique is also applied to the problem of robust terrain classification in polarimetric sar images

best wavelet packet bases in a rate distortion sense a fast rate distortion r d optimal scheme for coding adaptive trees whose individual nodes spawn descendents forming a disjoint and complete basis cover for the space spanned by their parent nodes is presented the scheme guarantees operation on the convex hull of the operational r d curve and uses a fast dynamic programing pruning algorithm to markedly reduce computational complexity applications for this coding technique include r coefman et al s yale univ 1990 generalized multiresolution wavelet packet decomposition iterative subband coders and quadtree structures applications to image processing involving wavelet packets as well as discrete cosine transform dct quadtrees are presented

a multiscale guide to brownian motion we revise the levy s construction of brownian motion as a simple though rigorous approach to operate with various gaussian processes a brownian path is explicitly constructed as a linear combination of wavelet based geometrical features at multiple length scales with random weights such a wavelet representation gives a closed formula mapping of the unit interval onto the functional space of brownian paths this formula elucidates many classical results about brownian motion e g non differentiability of its path providing intuitive feeling for non mathematicians the illustrative character of the wavelet representation along with the simple structure of the underlying probability space is different from the usual presentation of most classical textbooks similar concepts are discussed for brownian bridge fractional brownian motion ornstein uhlenbeck process gaussian free field and fractional gaussian fields wavelet representations and dyadic decompositions form the basis of many highly efficient numerical methods to simulate gaussian processes and fields including brownian motion and other diffusive processes in confining domains

modeling and estimation of multiresolution stochastic processes an overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image in particular it is shown how the wavelet transform directly suggests such a modeling paradigm this perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time like variable the investigation of models on homogeneous trees is emphasized the framework examined here allows for consideration in a very natural way of the fusion of data from sensors with differing resolutions also thanks to the fact that wavelet transforms do an excellent job of compressing large classes of covariance kernels it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

unified framework for training point selection and error estimation for surrogate models a unified framework for surrogate model training point selection and error estimation is proposed building auxiliary local surrogate models over subdomains of the global surrogate model forms the basis of the proposed framework a discrepancy function defined as the absolute difference between response predictions from local and global surrogate models for randomly chosen test candidates drives the framework thereby not requiring any additional exact function evaluations the benefits of this new approach are demonstrated with analytical test functions and the construction of a two dimensional aerodynamic database the results show that the proposed training point selection approach improves the convergence monotonicity and produces more accurate surrogate models compared to random and quasi random training point selection strategies the introduced root mean square discrepancy and maximum absolute discrepancy exhibit close agreement with the actual root mean square error and maximum absolute error r

fractal function estimation via wavelet shrinkage in scientific studies objects are often very rough mathematically these rough objects are modelled by fractal functions and the fractal dimension is usually used to measure their roughness this paper investigates fractal function estimation by wavelet shrinkage it is shown that wavelet shrinkage can estimate fractal functions with their fractal dimensions virtually preserved

1 f noise in diffuse optical imaging and wavelet based response estimation in diffuse optical imaging doi data analysis the functional response is contaminated with physiological noise as in functional magnetic resonance imaging fmri in this work we extend a previously proposed method for fmri to estimate the parameters of a linear model of doi time series the regression is performed in the wavelet domain to infer drift coefficients at different scales and to estimate the strength of the hemodynamic response function hrf this multiresolution approach benefits from the whitening property of the discrete wavelet transform dwt which approximately decorrelates long memory noise processes we also show that a more accurate estimation is obtained by removing some regressors correlating with the protocol moreover we observe that this improvement is related to a quantitative measure of 1 f noise the performances of the method are first evaluated against a standard spline cosine drift approach with simulated hrf and real background physiology lastly the technique is applied to experimental event related data acquired by near infrared spectroscopy nirs

multiscale systems kalman filters and riccati equations an algorithm analogous to the rauch tung striebel algorithm spl minus consisting of a fine to coarse kalman filter like sweep followed by a coarse to fine smoothing step spl minus was developed previously by the authors ibid vol 39 no 3 p 464 78 1994 in this paper they present a detailed system theoretic analysis of this filter and of the new scale recursive riccati equation associated with it while this analysis is similar in spirit to that for standard kalman filters the structure of the dyadic tree leads to several significant differences in particular the structure of the kalman filter error dynamics leads to the formulation of an ml version of the filtering equation and to a corresponding smoothing algorithm based on triangularizing the hamiltonian for the smoothing problem in addition the notion of stability for dynamics requires some care as do the concepts of reachability and observability using these system theoretic constructs the stability and steady state behavior of the fine to coarse kalman filter and its riccati equation are analysed

improved two point function approximations for design optimization two point approximations are developed by utilizing both the function and gradient information of two data points the objective of this work is to build a high quality approximation to realize computational savings in solving complex optimization and reliability analysis problems two developments are proposed in the new twopoint approximations 1 calculation of a correction term by matching with the previous known function value and supplementing it to the first order approximation for including the effects of higher order terms and 2 development of a second order approximation without the actual calculation of second order derivatives by using an approximate hessian matrix several highly nonlinear functions and structural examples are used for demonstrating the new two point approximations that improved the accuracy of existing first order methods evelopmentof new and improved constraint approximations has been an active area of research in the mathematical optimization community over the last two decades 1 3 more accurate function approximations reduce the finite element analysis cost and increase the search domain in each iteration of an optimization run recent developments in constraint approximations addressed methods for eigenvalues dynamic response static forces structural reliability etc some of the typical methods are summarized in the following kirsch4 developed a scaling factor for a stiffness matrix and approximated the displacements stresses and forces with respect to sizing and topology variables vanderplaats and salajegheh5 used two levels of approximations for discrete sizing and shape design of structures first for solving the continuous problem and next the discrete one using a dual theory canfield6 developed an approach for eigenvalues by approximating the modal strain and kinetic energies independently thomas et al 7 presented an approximation for the frequency response of damped structures by using the concept of ref 6 a multivariate spline approximation was developed by wang and grandhi8 making use of the previously generated exact finite element analyses similarly a hermite approximation for n dimensional problems used several data points and had the property of reproducing the function and gradient values at the known data points 9 canfield10 used multipoint data in building an approximate hessian matrix and constructed a second order approximation for improving the accuracy toropov et al 11 approximated functions as a linear combination or products of variables and the coefficients of the polynomial were computed by applying a least squared approach on multiple data points fadel et al 12 considered intermediate variables in terms of exponentials and they were computed by matching the approximate function gradients with the previous point s exact values this approach did not compare the function values of the previous point wang and grandhi13 used a single exponential for all of the variables and calculated it by matching the approximate function with the previous point exact value and the gradient information of the previous point was not utilized the proposed paper develops an improved two point approximation utilizing both function and gradient information of two data points in the new two point approximation two developments are proposed 1 calculation of a correction term based on two function values and

multifidelity quasi newton method for design optimization multifidelity approaches are frequently used in design when high fidelity models are too expensive to use directly and lower fidelity models of reasonable accuracy exist in optimization corrected

risk averse dynamic programming for markov decision processes we introduce the concept of a markov risk measure and we use it to formulate risk averse control problems for two markov decision models a finite horizon model and a discounted infinite horizon model for both models we derive risk averse dynamic programming equations and a value iteration method for the infinite horizon problem we develop a risk averse policy iteration method and we prove its convergence we also propose a version of the newton method to solve a nonsmooth equation arising in the policy iteration method and we prove its global convergence finally we discuss relations to min max markov decision models

multi fidelity optimization via surrogate modelling this paper demonstrates the application of correlated gaussian process based approximations to optimization where multiple levels of analysis are available using an extension to the geostatistical method of co kriging an exchange algorithm is used to choose which points of the search space to sample within each level of analysis the derivation of the co kriging equations is presented in an intuitive manner along with a new variance estimator to account for varying degrees of computational noise in the multiple levels of analysis a multi fidelity wing optimization is used to demonstrate the methodology

wavelet based estimation of a semiparametric generalized linear model of fmri time series addresses the problem of detecting significant changes in fmri time series that are correlated to a stimulus time course this paper provides a new approach to estimate the parameters of a semiparametric generalized linear model of the fmri time series the fmri signal is described as the sum of two effects a smooth trend and the response to the stimulus the trend belongs to a subspace spanned by large scale wavelets the wavelet transform provides an approximation to the karhunen loeve transform for the long memory noise and we have developed a scale space regression that permits one to carry out the regression in the wavelet domain while omitting the scales that are contaminated by the trend in order to demonstrate that our approach outperforms the state of the art detrending technique we evaluated our method against a smoothing spline approach experiments with simulated data and experimental fmri data demonstrate that our approach can infer and remove drifts that cannot be adequately represented with splines

version independence and nested distributions in multistage stochastic optimization the amount of stagewise available information is crucial in multistage stochastic optimization but unlike data which directly enter the profit loss functions of a decision problem information is invariant w r t bijective transformations the usual concept to deal with information in multistage stochastic programming is by introducing filtrations i e increasing sequences of sigma algebras to which the decisions must be adapted for the definition of filtrations one has to fix a certain probability space while random variables are typically given by their distributions only and all realizations of this distribution on some probability space are equivalent we introduce here the new concept of nested distributions to describe the information structure as well as the scenario process of a stochastic optimization program in a way which is independent of specific versions of probability spaces and random variables the setting is totally in distribution two stochastic programs with identical objective function and constraints are equivalent if and only if the scenario processes have the same nested distribution as a byproduct we analyze the question of whether introducing extra randomness by defining randomized decisions would lead to improvement in the objective value in the language of information this would mean that enlarging the filtration based on available information by conditionally independent additional random variables would have a positive effect we show that in general the answer is yes while for compound convex objectives the answer is no finally we define a distance between nested distributions which generalizes the well known kantorovich distance of probability distributions and demonstrates that this distance may be used in quantifying the quality of approximation between a continuous stochastic program and a tree discretization or between two tree discretizations

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

wavelet representations of stochastic processes and multiresolution stochastic models deterministic signal analysis in a multiresolution framework through the use of wavelets has been extensively studied very successfully in recent years in the context of stochastic processes the use of wavelet bases has not yet been fully investigated we use compactly supported wavelets to obtain multiresolution representations of stochastic processes with paths in l sup 2 defined in the time domain we derive the correlation structure of the discrete wavelet coefficients of a stochastic process and give new results on how and when to obtain strong decay in correlation along time as well as across scales we study the relation between the wavelet representation of a stochastic process and multiresolution stochastic models on trees proposed by basseville et al see ieee trans inform theory vol 38 p 766 784 mar 1992 we propose multiresolution stochastic models of the discrete wavelet coefficients as approximations to the original time process these models are simple due to the strong decorrelation of the wavelet transform experiments show that these models significantly improve the approximation in comparison with the often used assumption that the wavelet coefficients are completely uncorrelated

multiscale autoregressive processes i schur levinson parametrizations in many applications it is of interest to analyze and recognize phenomena occurring at different scales the recently introduced wavelet transforms provide a time and scale decomposition of signals that offers the possibility of such analysis a corresponding statistical framework to support the development of optimal multiscale statistical signal processing algorithms is described the theory of multiscale signal representation leads naturally to models of signals on trees and this provides the framework for investigation in particular the class of isotropic processes on homogeneous trees is described and a theory of autoregressive models is developed in this context this leads to generalizations of schur and levinson recursions associated properties of the resulting reflection coefficients and the initial pieces in a system theory for multiscale modeling