Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations

We consider adaptive estimating the value of a linear functional from indirect white noise observations. For a flexible approach, the problem is embedded in an abstract Hilbert scale. We develop an adaptive estimator that is rate optimal within a logarithmic factor simultaneously over a wide collection of balls in the Hilbert scale. It is shown that the proposed estimator has the best possible adaptive properties for a wide range of linear functionals. The case of discretized indirect white noise observations is studied, and the adaptive estimator in this setting is developed. Received: 5 May 1999 / Revised version: 25 October 1999 / Published online: 5 September 2000     

Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces

Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. In this paper, we consider phaseless sampling and reconstruction of real-valued signals in a high-dimensional shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. The determination of a signal in a shift-invariant space, up to a sign, by its magnitude measurements on the whole Euclidean space has been shown in the literature to be equivalent to its nonseparability. In this paper, we introduce an undirected graph associated with the signal in a shift-invariant space and use connectivity of the graph to characterize nonseparability of the signal. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that nonseparable signals in the shift-invariant space can be reconstructed in a stable way from their phaseless samples taken on that set. In this paper, we also propose a reconstruction algorithm which provides an approximation to the original signal when its noisy phaseless samples are available only. Finally, numerical simulations are performed to demonstrate the robustness of the proposed algorithm to reconstruct box spline signals from their noisy phaseless samples.

     

Nonparametric regression with martingale increment errors

We consider the problem of adaptive estimation of the regression function in a framework where we replace ergodicity assumptions (such as independence or mixing) by another structural assumption on the model. Namely, we propose adaptive upper bounds for kernel estimators with data-driven bandwidth (Lepski’s selection rule) in a regression model where the noise is an increment of martingale. It includes, as very particular cases, the usual i.i.d. regression and auto-regressive models. The cornerstone tool for this study is a new result for self-normalized martingales, called “stability”, which is of independent interest. In a first part, we only use the martingale increment structure of the noise. We give an adaptive upper bound using a random rate, that involves the occupation time near the estimation point. Thanks to this approach, the theoretical study of the statistical procedure is disconnected from usual ergodicity properties like mixing. Then, in a second part, we make a link with the usual minimax theory of deterministic rates. Under a β-mixing assumption on the covariates process, we prove that the random rate considered in the first part is equivalent, with large probability, to a deterministic rate which is the usual minimax adaptive one.     

Derivation of linear estimation algorithms from measurements affected by multiplicative and additive noises

This paper addresses the problem of estimating signals from observation models with multiplicative and additive noises. Assuming that the state-space model is unknown, the multiplicative noise is non-white and the signal and additive noise are correlated, recursive algorithms are derived for the least-squares linear filter and fixed-point smoother. The proposed algorithms are obtained using an innovation approach and taking into account the information provided by the covariance functions of the process involved.     

Adaptive Wavelet Methods II—Beyond the Elliptic Case

This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet-based method developed in [17] for symmetric positive definite problems to indefinite or unsymmetric systems of operator equations. This is accomplished by first introducing techniques (such as the least squares formulation developed in [26]) that transform the original (continuous) problem into an equivalent infinite system of equations which is now well-posed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve the resulting infinite system of equations. This second step requires a significant modification of the ideas from [17]. The main departure from [17] is to develop an iterative scheme that directly applies to the infinite-dimensional problem rather than finite subproblems derived from the infinite problem. This rests on an adaptive application of the infinite-dimensional operator to finite vectors representing elements from finite-dimensional trial spaces. It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-best N -term approximation. An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of trial spaces, like the LBB condition, no longer arise.     

A Constructive Inversion Framework for Twisted Convolution

In this paper we develop constructive invertibility conditions for the twisted convolution. Our approach is based on splitting the twisted convolution with rational parameters into a finite number of weighted convolutions, which can be interpreted as another twisted convolution on a finite cyclic group. In analogy with the twisted convolution of finite discrete signals, we derive an anti-homomorphism between the sequence space and a suitable matrix algebra which preserves the algebraic structure. In this way, the problem reduces to the analysis of finite matrices whose entries are sequences supported on corresponding cosets. The invertibility condition then follows from Cramer’s rule and Wiener’s lemma for this special class of matrices. The problem results from a well known approach of studying the invertibility properties of the Gabor frame operator in the rational case. The presented approach gives further insights into Gabor frames. In particular, it can be applied for both the continuous (on \Bbb R^d{\Bbb R}^d ) and the finite discrete setting. In the latter case, we obtain algorithmic schemes for directly computing the inverse of Gabor frame-type matrices equivalent to those known in the literature.     

An accurate a posteriori error estimator for the Steklov eigenvalue problem and its applications

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine mesh and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.

     

Nonlinear Dynamical Systems and Adaptive Filters in Biomedicine

In this paper we present the application of a method of adaptive estimation using an algebra–geometric approach, to the study of dynamic processes in the brain. It is assumed that the brain dynamic processes can be described by nonlinear or bilinear lattice models. Our research focuses on the development of an estimation algorithm for a signal process in the lattice models with background additive white noise, and with different assumptions regarding the characteristics of the signal process. We analyze the estimation algorithm and implement it as a stochastic differential equation under the assumption that the Lie algebra, associated with the signal process, can be reduced to a finite dimensional nilpotent algebra. A generalization is given for the case of lattice models, which belong to a class of causal lattices with certain restrictions on input and output signals. The application of adaptive filters for state estimation of the CA3 region of the hippocampus (a common location of the epileptic focus) is discussed. Our areas of application involve two problems: (1) an adaptive estimation of state variables of the hippocampal network, and (2) space identification of the coupled ordinary equation lattice model for the CA3 region.     

A new adaptive approach of the Metropolis-Hastings algorithm applied to structural damage identification using time domain data

In the present work, the formulation and solution of the inverse problem of structural damage identification is presented based on the Bayesian inference, a powerful approach that has been widely used for the formulation of inverse problems in a statistical framework. The structural damage is continuously described by a cohesion field, which is spatially discretized by the finite element method, and the solution of the inverse problem of damage identification, from the Bayesian point of view, is the posterior probability densities of the nodal cohesion parameters. In this approach, prior information about the parameters of interest and the quantification of the uncertainties related to the magnitudes measured can be used to estimate the sought parameters. Markov Chain Monte Carlo (MCMC) method, implemented via the Metropolis-Hastings (MH) algorithm, is commonly used to sample such densities. However, the conventional MH algorithm may present some difficulties, for instance, in high dimensional problems or when the parameters of interest are highly correlated or the posterior probability density is very peaked. In order to overcome these difficulties, a new adaptive MH algorithm (P-AMH) is proposed in the present work. Numerical results related to an inverse problem of damage identification in a simply supported Euler-Bernoulli beam are presented. Synthetic experimental time domain data, obtained with different damage scenarios, and noise levels, were addressed with the aim at assessing the proposed damage identification approach. An adaptive MH algorithm (H-AMH) and the conventional MH algorithm, already consolidated in the literature, were also considered for comparison purposes. The numerical results show that both adaptive algorithms outperformed the conventional MH. Besides, the P-AMH provided Markov chains with faster convergence and better mixing than the ones provided by the H-AMH.     

Random sampling of bandlimited signals on graphs

We study the problem of sampling k-bandlimited signals on graphs. We propose two sampling strategies that consist in selecting a small subset of nodes at random. The first strategy is non-adaptive, i.e., independent of the graph structure, and its performance depends on a parameter called the graph coherence. On the contrary, the second strategy is adaptive but yields optimal results. Indeed, no more than $O(k\log ?(k))$ measurements are sufficient to ensure an accurate and stable recovery of all k-bandlimited signals. This second strategy is based on a careful choice of the sampling distribution, which can be estimated quickly. Then, we propose a computationally efficient decoder to reconstruct k-bandlimited signals from their samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we conduct several experiments to test these techniques.     

Dynamical sampling: Time–space trade-off

We consider the problem of spatiotemporal sampling in which an initial state of an evolution process is to be recovered from a set of samples at different time levels. We are particularly interested in lossless trade-off between spatial and temporal samples. We show that for a special class of signals it is possible to recover the initial state using a reduced number of measuring devices activated more frequently. We present several algorithms for this kind of recovery and describe their robustness to noise.     

Convergence of adaptive FEM for some elliptic obstacle problem

In this work, we treat the convergence of adaptive lowest-order FEM for some elliptic obstacle problem with affine obstacle. For error estimation, we use a residual error estimator from [D. Braess, C. Carstensen, and R. Hoppe, Convergence analysis of a conforming adaptive finite element method for an obstacle problem, Numer. Math. 107 (2007), pp. 455–471]. We extend recent ideas from [J. Cascon, C. Kreuzer, R. Nochetto, and K. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 (2008), pp. 2524–2550] for the unrestricted variational problem to overcome the lack of Galerkin orthogonality. The main result states that an appropriately weighted sum of energy error, edge residuals and data oscillations satisfies a contraction property within each step of the adaptive feedback loop. This result is superior to a prior result from Braess et al. (2007) in two ways: first, it is unnecessary to control the decay of the data oscillations explicitly; second, our analysis avoids the use of some discrete local efficiency estimate so that the local mesh-refinement is fairly arbitrary.     

Near optimality of stochastic control in systems with unknown parameter processes

Stochastic control for systems with an unknown parameter is considered in this paper. The underlying problem is to minimize a functional subject to a system described by a singularly perturbed differential equation with an unknown parameter process driven by fast fluctuating random disturbances. This problem arises in the context of stochastic adaptive control, adaptive signal processing, and failure-prone manufacturing systems. Due to the nature of the wide-bandwidth noise processes, identifying the parameter process for eacht is very hard since the driving noise changes very rapidly. An alternative approach is used, and an auxiliary control problem is introduced to overcome the difficulties. By means of weak convergence methods and comparison control techniques, nearly optimal controls are obtained.This research was supported in part by the National Science Foundation under Grant DMS-9022139 and DMS-9224372.     

A Constructive Inversion Framework for Twisted Convolution

In this paper we develop constructive invertibility conditions for the twisted convolution. Our approach is based on splitting the twisted convolution with rational parameters into a finite number of weighted convolutions, which can be interpreted as another twisted convolution on a finite cyclic group. In analogy with the twisted convolution of finite discrete signals, we derive an anti-homomorphism between the sequence space and a suitable matrix algebra which preserves the algebraic structure. In this way, the problem reduces to the analysis of finite matrices whose entries are sequences supported on corresponding cosets. The invertibility condition then follows from Cramer’s rule and Wiener’s lemma for this special class of matrices. The problem results from a well known approach of studying the invertibility properties of the Gabor frame operator in the rational case. The presented approach gives further insights into Gabor frames. In particular, it can be applied for both the continuous (on ) and the finite discrete setting. In the latter case, we obtain algorithmic schemes for directly computing the inverse of Gabor frame-type matrices equivalent to those known in the literature.     

An adaptive estimation of MAVE

Minimum average variance estimation (MAVE, Xia et al. (2002) [29]) is an effective dimension reduction method. It requires no strong probabilistic assumptions on the predictors, and can consistently estimate the central mean subspace. It is applicable to a wide range of models, including time series. However, the least squares criterion used in MAVE will lose its efficiency when the error is not normally distributed. In this article, we propose an adaptive MAVE which can be adaptive to different error distributions. We show that the proposed estimate has the same convergence rate as the original MAVE. An EM algorithm is proposed to implement the new adaptive MAVE. Using both simulation studies and a real data analysis, we demonstrate the superior finite sample performance of the proposed approach over the existing least squares based MAVE when the error distribution is non-normal and the comparable performance when the error is normal.     

Local sampling set conditions in weighted shift-invariant signal spaces

How to represent a continuous signal in terms of a discrete sequence is a fundamental problem in sampling theory. Most of the known results concern global sampling in shift-invariant signal spaces. But in fact, the local reconstruction from local samples is one of the most desirable properties for many applications in signal processing, e.g. for implementing real-time reconstruction numerically. However, the local reconstruction problem has not been given much attention. In this article, we find conditions on a finite sampling set X such that at least in principle a continuous signal on a finite interval is uniquely and stably determined by their sampling value on the finite sampling set X in shift-invariant signal spaces.     

Localized nonlinear functional equations and two sampling problems in signal processing

Let 1 ≤ p ≤ ∞. In this paper, we consider solving a nonlinear functional equation f (x) = y, where x, y belong to ? ^p and f has continuous bounded gradient in an inverse-closed subalgebra of ? (?²), the Banach algebra of all bounded linear operators on the Hilbert space ? ². We introduce strict monotonicity property for functions f on Banach spaces ? ^p so that the above nonlinear functional equation is solvable and the solution x depends continuously on the given data y in ? ^p . We show that the Van-Cittert iteration converges in ? ^p with exponential rate and hence it could be used to locate the true solution of the above nonlinear functional equation. We apply the above theory to handle two problems in signal processing: nonlinear sampling termed with instantaneous companding and subsequently average sampling; and local identification of innovation positions and qualification of amplitudes of signals with finite rate of innovation.     

Stagnation in the p-version of the finite element method

A special feature of the p-version of the finite element method for solving a differential boundary value problem stated in the form of minimizing a quadratic functional on a certain set is studied. This special feature results in approximate solutions remaining unchanged on finite numbers of increasing finite-dimensional subsets of increasing dimension, in which solutions are sought. Necessary and sufficient conditions for the existence of this feature are found, and the stagnation effect is interpreted for a specially constructed example. For the adaptive p-version of the finite element approach, a modified strategy is proposed that takes this feature into account and thus improves the reliability of the method.     

Adaptive refinement procedure for the finite difference method

An adaptive refinement procedure consisting of a localized error estimator and a physically based approach to mesh refinement is developed for the finite difference method. The error estimator is a variation of a successful finite element error estimator. The errors are estimated by computing an error energy norm in terms of discontinuous and continuous stress fields formed from the finite difference results for plane stress problems. The error measure identifies regions of high error which are subsequently refined to improve the result. The local refinement procedure utilizes a recently developed approach for developing finite difference templates to produce a graduated mesh. The adaptive refinement procedure is demonstrated with a problem that contains a well-defined singularity. The results are compared to finite element and uniformly refined finite difference results.     

Multiscale Flow Simulation by Adaptive Finite Volume Particle Methods

We report on recent developments concerning adaptive finite particle methods, which are used for the numerical simulation of multiscale phenomena in time-dependent evolution processes. The proposed concept relies on a finite volume approach, which we combine with WENO reconstruction from particle average values. In this method, polyharmonic splines are key tools for both the optimal recovery from scattered particle averages and the construction of customized adaption rules. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)