Showing papers on "Piecewise published in 2004"

New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities

Abstract: This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C 2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle-shaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2 -j , each element has an envelope that is aligned along a ridge of length 2 -j/2 and width 2 -j . We prove that curvelets provide an essentially optimal representation of typical objects f that are C 2 except for discontinuities along piecewise C 2 curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the n-term partial reconstruction f C n obtained by selecting the n largest terms in the curvelet series obeys ∥f - f C n ∥ 2 L2 ≤ C . n -2 . (log n) 3 , n → ∞. This rate of convergence holds uniformly over a class of functions that are C 2 except for discontinuities along piecewise C 2 curves and is essentially optimal. In comparison, the squared error of n-term wavelet approximations only converges as n -1 as n → ∞, which is considerably worse than the optimal behavior.

Collocation Methods for Volterra Integral and Related Functional Differential Equations

Abstract: Collocation based on piecewise polynomial approximation represents a powerful class of methods for the numerical solution of initial-value problems for functional differential and integral equations arising in a wide spectrum of applications, including biological and physical phenomena. The present book introduces the reader to the general principles underlying these methods and then describes in detail their convergence properties when applied to ordinary differential equations, functional equations with (Volterra type) memory terms, delay equations, and differential-algebraic and integral-algebraic equations. Each chapter starts with a self-contained introduction to the relevant theory of the class of equations under consideration. Numerous exercises and examples are supplied, along with extensive historical and bibliographical notes utilising the vast annotated reference list of over 1300 items. In sum, Hermann Brunner has written a treatise that can serve as an introduction for students, a guide for users, and a comprehensive resource for experts.

ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems

Abstract: We propose an efficient, hybrid Fourier-wavelet regularized deconvolution (ForWaRD) algorithm that performs noise regularization via scalar shrinkage in both the Fourier and wavelet domains. The Fourier shrinkage exploits the Fourier transform's economical representation of the colored noise inherent in deconvolution, whereas the wavelet shrinkage exploits the wavelet domain's economical representation of piecewise smooth signals and images. We derive the optimal balance between the amount of Fourier and wavelet regularization by optimizing an approximate mean-squared error (MSE) metric and find that signals with more economical wavelet representations require less Fourier shrinkage. ForWaRD is applicable to all ill-conditioned deconvolution problems, unlike the purely wavelet-based wavelet-vaguelette deconvolution (WVD); moreover, its estimate features minimal ringing, unlike the purely Fourier-based Wiener deconvolution. Even in problems for which the WVD was designed, we prove that ForWaRD's MSE decays with the optimal WVD rate as the number of samples increases. Further, we demonstrate that over a wide range of practical sample-lengths, ForWaRD improves on WVD's performance.

A numerical scheme for BSDEs

Abstract: In this paper we propose a numerical scheme for a class of backward stochastic differential equations (BSDEs) with possible path-dependent terminal values. We prove that our scheme converges in the strong $L^2$ sense and derive its rate of convergence. As an intermediate step we prove an $L^2$-type regularity of the solution to such BSDEs. Such a notion of regularity, which can be thought of as the modulus of continuity of the paths in an $L^2$ sense, is new. Some other features of our scheme include the following: (i) both components of the solution are approximated by step processes (i.e., piecewise constant processes); (ii) the regularity requirements on the coefficients are practically "minimum"; (iii) the dimension of the integrals involved in the approximation is independent of the partition size.

Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions

Abstract: This paper presents a stability analysis method for discrete-time Takagi-Sugeno fuzzy dynamic systems based on a piecewise smooth Lyapunov function. It is shown that the stability of the fuzzy dynamic system can be established if a piecewise Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of linear matrix inequalities that is numerically feasible with commercially available software. It is also demonstrated via numerical examples that the stability result based on the piecewise quadratic Lyapunov functions is less conservative than that based on the common quadratic Lyapunov functions.

An intuitive framework for real-time freeform modeling

Abstract: We present a freeform modeling framework for unstructured triangle meshes which is based on constraint shape optimization. The goal is to simplify the user interaction even for quite complex freeform or multiresolution modifications. The user first sets various boundary constraints to define a custom tailored (abstract) basis function which is adjusted to a given design task. The actual modification is then controlled by moving one single 9-dof manipulator object. The technique can handle arbitrary support regions and piecewise boundary conditions with smoothness ranging continuously from C0 to C2. To more naturally adapt the modification to the shape of the support region, the deformed surface can be tuned to bend with anisotropic stiffness. We are able to achieve real-time response in an interactive design session even for complex meshes by precomputing a set of scalar-valued basis functions that correspond to the degrees of freedom of the manipulator by which the user controls the modification.

A Bayesian approach to identification of hybrid systems

Abstract: In this paper, we present a novel procedure for the identification of hybrid systems in the class of piecewise ARX systems. The presented method facilitates the use of available a priori knowledge on the system to be identified, but can also be used as a black-box method. We treat the unknown parameters as random variables, described by their probability density functions. The identification problem is posed as the problem of computing the a posteriori probability density function of the model parameters, and subsequently relaxed until a practically implementable method is obtained. A particle filtering method is used for a numerical implementation of the proposed procedure. A modified version of the multicategory robust linear programming classification procedure, which uses the information derived in the previous steps of the identification algorithm, is used for estimating the partition of the piecewise ARX map. The proposed procedure is applied for the identification of a component placement process in pick-and-place machines.

Superconvergence Properties of Optimal Control Problems

Abstract: An optimal control problem for a two-dimensional (2-d) elliptic equation is investigated with pointwise control constraints. This paper is concerned with discretization of the control by piecewise constant functions. The state and the adjoint state are discretized by linear finite elements. Approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order h2.

Image restoration subject to a total variation constraint

Abstract: Total variation has proven to be a valuable concept in connection with the recovery of images featuring piecewise smooth components. So far, however, it has been used exclusively as an objective to be minimized under constraints. In this paper, we propose an alternative formulation in which total variation is used as a constraint in a general convex programming framework. This approach places no limitation on the incorporation of additional constraints in the restoration process and the resulting optimization problem can be solved efficiently via block-iterative methods. Image denoising and deconvolution applications are demonstrated.

A two-layer approach to wave modelling

Abstract: A set of model equations for water–wave propagation is derived by piecewise integration of the primitive equations of motion through two arbitrary layers. Within each layer, an independent velocity profile is derived. With two separate velocity profiles, matched at the interface of the two layers, the resulting set of equations has three free parameters, allowing for an optimization with known analytical properties of water waves. The optimized model equations show good linear wave characteristics up to kh ≈ 6 , while the second–order nonlinear behaviour is captured to kh ≈ 6 as well. A numerical algorithm for solving the model equations is developed and tested against one– and two–horizontal–dimension cases. Agreement with laboratory data is excellent.

The p‐Version of the Finite Element Method

Abstract: In the p-version of the finite element method, the triangulation is fixed and the degree p, of the piecewise polynomial approximation, is progressively increased until some desired level of precision is reached. In this paper, we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation. These properties lead to an a priori estimate of the asymptotic rate of convergence of the p-version. The estimate shows that the p-version gives results which are not worse than those obtained by the conventional finite element method (called the h-version, in which h represents the maximum diameter of the elements), when quasi-uniform triangulations are employed and the basis for comparison is the number of degrees of freedom. Furthermore, in the case of a singularity problem, we show (under conditions which are usually satisfied in practice) that the rate of convergence of the p-version is twice that of the h-version with quasi-uniform mesh. Inverse approximation theorems which determine the smoothness of a function based on the rate at which it is approximated by piecewise polynomials over a fixed triangulation are proved for both singular and nonsingular problems. We present numerical examples which illustrate the effectiveness of the p-version for a simple one-dimensional problem and for two problems in two-dimensional elasticity.We also discuss roundott error and computational costs associated with the p-version. Finally, we describe some important features, such as hierarchic basis functions, which have been utilized in COMET-X, an experimental computer implemen- tation of the p-version.

Combinatorial yamabe flow on surfaces

Abstract: In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics In the case of surfaces, we define the combinatorial Yamabe flow on the space of all piecewise flat metrics associated to a triangulated surface We show that the flow either develops removable singularities or converges exponentially fast to a constant combinatorial curvature metric If the singularity develops, we show that the singularity is always removable by a surgery procedure on the triangulation We conjecture that after finitely many such surgery changes on the triangulation, the flow converges to the constant combinatorial curvature metric as time approaches infinity

Probabilistic space-time video modeling via piecewise GMM

Abstract: In this paper, we describe a statistical video representation and modeling scheme. Video representation schemes are needed to segment a video stream into meaningful video-objects, useful for later indexing and retrieval applications. In the proposed methodology, unsupervised clustering via Gaussian mixture modeling extracts coherent space-time regions in feature space, and corresponding coherent segments (video-regions) in the video content. A key feature of the system is the analysis of video input as a single entity as opposed to a sequence of separate frames. Space and time are treated uniformly. The probabilistic space-time video representation scheme is extended to a piecewise GMM framework in which a succession of GMMs are extracted for the video sequence, instead of a single global model for the entire sequence. The piecewise GMM framework allows for the analysis of extended video sequences and the description of nonlinear, nonconvex motion patterns. The extracted space-time regions allow for the detection and recognition of video events. Results of segmenting video content into static versus dynamic video regions and video content editing are presented.

On Spatial Adaptive Estimation of Nonparametric Regression

Abstract: The paper is devoted to developing spatial adaptive estimates for restoring functions from noisy observations. We show that the traditional least square (piecewise polynomial) estimate equipped with adaptively adjusted window possesses simultaneously many attractive adaptive properties, namely, 1) it is near– optimal within lnn–factor for estimating a function (or its derivative) at a single point; 2) it is spatial adaptive in the sense that its quality is close to that one which could be achieved if smoothness of the underlying function was known in advance; 3) it is optimal in order (in the case of “strong” accuracy measure) or near–optimal within lnn–factor (in the case of “weak” accuracy measure) for estimating whole function (or its derivative) over wide range of the classes and global loss functions. We demonstrate that the “spatial adaptive abilities” of our estimate are, in a sense, the best possible. Besides this, our adaptive estimate is computationally efficient and demonstrates reasonable practical behavior.

Decay of correlations for piecewise smooth maps with indifferent fixed points

Abstract: We consider a piecewise smooth expanding map f on the unit interval that has the form $f(x)=x+x^{1+\gamma}+o(x^{1+\gamma})$ near 0, where $0 . We prove by showing both lower and upper bounds that the rate of decay of correlations with respect to the absolutely continuous invariant probability measure $\mu$ is polynomial with the same degree $1/\gamma-1$ for Lipschitz functions. We also show that the density function h of $\mu$ has the order $x^{-\gamma}$ as $x\to 0$ . Perron–Frobenius operators are the main tool used for proofs.

Evans Functions for Integral Neural Field Equations with Heaviside Firing Rate Function

Abstract: In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the firing rate function is a Heaviside. This allows a discussion of wave stability and bifurcation as a function of system parameters, including the speed and strength of synaptic coupling and the speed of axonal signals. The theory is illustrated with the construction and stability analysis of front solutions to a scalar neural field model and a limiting case is shown to recover recent results of L. Zhang [On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential and Integral Equations, 16, (2003), pp.513-536.]. Traveling fronts and pulses are considered in more general models possessing either a linear or piecewise constant recovery variable. We establish the stability of coexisting traveling fronts beyond a front bifurcation and consider parameter regimes that support two stable traveling fronts of different speed. Such fronts may be connected and depending on their relative speed the resulting region of activity can widen or contract. The conditions for the contracting case to lead to a pulse solution are established. The stability of pulses is obtained for a variety of examples, in each case confirming a previously conjectured stability result. Finally we show how this theory may be used to describe the dynamic instability of a standing pulse that arises in a model with slow recovery. Numerical simulations show that such an instability can lead to the shedding of a pair of traveling pulses.

Quadratic stabilization of a switched affine system about a nonequilibrium point

Abstract: This work deals with the problem of quadratic stabilization of switched affine systems, where the state of the switched system has to be driven to a point ("switched equilibrium") which is not in the set of subsystems equilibria. Quadratic stability of the switched equilibrium is assessed using a continuous Lyapunov function, having piecewise continuous derivative. A necessary and sufficient condition is given for the case of two subsystems and a sufficient condition is given in the general case. Two switching rules are presented: a state feedback, in which sliding modes may occur, and an hybrid feedback, in which sliding modes can be avoided. Two examples illustrate our results.

Online event-driven subsequence matching over financial data streams

Abstract: Subsequence similarity matching in time series databases is an important research area for many applications. This paper presents a new approximate approach for automatic online subsequence similarity matching over massive data streams. With a simultaneous on-line segmentation and pruning algorithm over the incoming stream, the resulting piecewise linear representation of the data stream features high sensitivity and accuracy. The similarity definition is based on a permutation followed by a metric distance function, which provides the similarity search with flexibility, sensitivity and scalability. Also, the metric-based indexing methods can be applied for speed-up. To reduce the system burden, the event-driven similarity search is performed only when there is a potential event. The query sequence is the most recent subsequence of piecewise data representation of the incoming stream which is automatically generated by the system. The retrieved results can be analyzed in different ways according to the requirements of specific applications. This paper discusses an application for future data movement prediction based on statistical information. Experiments on real stock data are performed. The correctness of trend predictions is used to evaluate the performance of subsequence similarity matching.

Mixed Discontinuous Galerkin Approximation of the Maxwell Operator

Abstract: We introduce and analyze a discontinuous Galerkin discretization of the Maxwell operator in mixed form. Here, all the unknowns of the underlying system of partial differential equations are approximated by discontinuous finite element spaces of the same order. For piecewise constant coefficients, the method is shown to be stable and optimally convergent with respect to the mesh size. Numerical experiments highlighting the performance of the proposed method for problems with both smooth and singular analytical solutions are presented.

Convergence of the lax-friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux

Abstract: The authors give the first convergence proof for the Lax-Friedrichs finite difference scheme for non-convex genuinely nonlinear scalar conservation laws of the form where the coefficient k(x,t) is allowed to be discontinuous along curves in the (x,t) plane. In contrast to most of the existing literature on problems with discontinuous coefficients, here the convergence proof is not based on the singular mapping approach, but rather on the div-curl lemma (but not the Young measure) and a Lax type entropy estimate that is robust with respect to the regularity of k(x,t). Following [14], the authors propose a definition of entropy solution that extends the classical Kružkov definition to the situation where k(x,t) is piecewise Lipschitz continuous in the (x,t) plane, and prove the stability (uniqueness) of such entropy solutions, provided that the flux function satisfies a so-called crossing condition, and that strong traces of the solution exist along the curves where k(x,t) is discontinuous. It is shown that a convergent subsequence of approximations produced by the Lax-Friedrichs scheme converges to such an entropy solution, implying that the entire computed sequence converges.

Horseshoes in piecewise continuous maps

Abstract: This paper presents a framework on existence of horseshoes in piecewise continuous maps, which is more applicable to practical problems.

Model predictive control of continuous-time nonlinear systems with piecewise constant control

Abstract: A new model predictive control (MPC) algorithm for nonlinear systems is presented. The plant under control, the state and control constraints, and the performance index to be minimized are described in continuous time, while the manipulated variables are allowed to change at fixed and uniformly distributed sampling times. In so doing, the optimization is performed with respect to sequences, as in discrete-time nonlinear MPC, but the continuous-time evolution of the system is considered as in continuous-time nonlinear MPC.