Abstract: The authors consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation. Examples of signals with a finite rate of innovation include streams of Diracs (e.g., the Poisson process), nonuniform splines, and piecewise polynomials. Even though these signals are not bandlimited, we show that they can be sampled uniformly at (or above) the rate of innovation using an appropriate kernel and then be perfectly reconstructed. Thus, we prove sampling theorems for classes of signals and kernels that generalize the classic "bandlimited and sinc kernel" case. In particular, we show how to sample and reconstruct periodic and finite-length streams of Diracs, nonuniform splines, and piecewise polynomials using sinc and Gaussian kernels. For infinite-length signals with finite local rate of innovation, we show local sampling and reconstruction based on spline kernels. The key in all constructions is to identify the innovative part of a signal (e.g., time instants and weights of Diracs) using an annihilating or locator filter: a device well known in spectral analysis and error-correction coding. This leads to standard computational procedures for solving the sampling problem, which we show through experimental results. Applications of these new sampling results can be found in signal processing, communications systems, and biological systems.

Abstract: We propose an algorithm for regression tree construction called GUIDE. It is specifically designed to eliminate variable selection bias, a problem that can undermine the reliability of inferences from a tree structure. GUIDE controls bias by employing chi-square analysis of residuals and bootstrap calibration of signif- icance probabilities. This approach allows fast computation speed, natural ex- tension to data sets with categorical variables, and direct detection of local two- variable interactions. Previous algorithms are not unbiased and are insensitive to local interactions during split selection. The speed of GUIDE enables two further enhancements—complex modeling at the terminal nodes, such as polynomial or best simple linear models, and bagging. In an experiment with real data sets, the prediction mean square error of the piecewise constant GUIDE model is within ±20% of that of CART r � . Piecewise linear GUIDE models are more accurate; with bagging they can outperform the spline-based MARS r � method.

Abstract: Presents a stability analysis method for piecewise discrete-time linear systems based on a piecewise smooth Lyapunov function. It is shown that the stability of the 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 (LMIs) that is numerically feasible with commercially available software.

Abstract: We present algorithms for time-series gene expression analysis that permit the principled estimation of unobserved time-points, clustering, and dataset alignment. Each expression profile is modeled as a cubic spline (piecewise polynomial) that is estimated from the observed data and every time point influences the overall smooth expression curve. We constrain the spline coefficients of genes in the same class to have similar expression patterns, while also allowing for gene specific parameters. We show that unobserved time-points can be reconstructed using our method with 10-15% less error when compared to previous best methods. Our clustering algorithm operates directly on the continuous representations of gene expression profiles, and we demonstrate that this is particularly effective when applied to non-uniformly sampled data. Our continuous alignment algorithm also avoids difficulties encountered by discrete approaches. In particular, our method allows for control of the number of degrees of freedom of the warp through the specification of parameterized functions, which helps to avoid overfitting. We demonstrate that our algorithm produces stable low-error alignments on real expression data and further show a specific application to yeast knockout data that produces biologically meaningful results.

Abstract: In this article, we define timed regular expressions , a formalism for specifying discrete behaviors augmented with timing information, and prove that its expressive power is equivalent to the timed automata of Alur and Dill. This result is the timed analogue of Kleene Theorem and, similarly to that result, the hard part in the proof is the translation from automata to expressions. This result is extended from finite to infinite (in the sense of Büchi) behaviors. In addition to these fundamental results, we give a clean algebraic framework for two commonly accepted formalisms for timed behaviors, time-event sequences and piecewise-constant signals.

Abstract: One of the most important properties of neural nets (NNs) for control purposes is the universal approximation property. Unfortunately,, this property is generally proven for continuous functions. In most real industrial control systems there are nonsmooth functions (e.g., piecewise continuous) for which approximation results in the literature are sparse. Examples include friction, deadzone, backlash, and so on. It is found that attempts to approximate piecewise continuous functions using smooth activation functions require many NN nodes and many training iterations, and still do not yield very good results. Therefore, a novel neural-network structure is given for approximation of piecewise continuous functions of the sort that appear in friction, deadzone, backlash, and other motion control actuator nonlinearities. The novel NN consists of neurons having standard sigmoid activation functions, plus some additional neurons having a special class of nonsmooth activation functions termed "jump approximation basis function." Two types of nonsmooth jump approximation basis functions are determined- a polynomial-like basis and a sigmoid-like basis. This modified NN with additional neurons having "jump approximation" activation functions can approximate any piecewise continuous function with discontinuities at a finite number of known points. Applications of the new NN structure are made to rigid-link robotic systems with friction nonlinearities. Friction is a nonlinear effect that can limit the performance of industrial control systems; it occurs in all mechanical systems and therefore is unavoidable in control systems. It can cause tracking errors, limit cycles, and other undesirable effects. Often, inexact friction compensation is used with standard adaptive techniques that require models that are linear in the unknown parameters. It is shown here how a certain class of augmented NN, capable of approximating piecewise continuous functions, can be used for friction compensation.

Abstract: The saturation assumption asserts that the best approximation error in \(H^1_0\) with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements. We establish a link between this assumption and the oscillation of \(f=-\Delta u\), and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption. We also show that this condition is necessary, and asymptotically valid provided \(f\in L^2\).

Abstract: The design of a finite impulse response (FIR) filter often involves a spectral "mask" that the magnitude spectrum must satisfy. The mask specifies upper and lower bounds at each frequency and, hence, yields an infinite number of constraints. In current practice, spectral masks are often approximated by discretization, but in this paper, we derive a result that allows us to precisely enforce piecewise constant and piecewise trigonometric polynomial masks in a finite and convex manner via linear matrix inequalities. While this result is theoretically satisfying in that it allows us to avoid the heuristic approximations involved in discretization techniques, it is also of practical interest because it generates competitive design algorithms (based on interior point methods) for a diverse class of FIR filtering and narrowband beamforming problems. The examples we provide include the design of standard linear and nonlinear phase FIR filters, robust "chip" waveforms for wireless communications, and narrowband beamformers for linear antenna arrays. Our main result also provides a contribution to system theory, as it is an extension of the well-known positive-real and bounded-real lemmas.

Abstract: In this article, we are studying the non-linear effects in a single-phase H-bridge inverter. The PWM control is related to a current feedback control. We are proposing an analytical model, which is a piecewise linear map. The distinctive feature of this study lies in the investigation of the map's properties. This investigation allows for the analytical determination of the fixed points, their domains of stability, and of the bifurcation points. More precisely, we will show that some of these bifurcations are discontinuous. The analysis is performed while keeping in mind the current controller's tuning. In this particular setting, we will show that all the bifurcations are of a certain type: border collision bifurcations. Although we are treating the appearance of chaos in a converter, the work presented stays close to the preoccupations of the engineer, because the particularities of the digital control are shown as an advantage. Moreover, we have strived to comment on the different modes observed, perio...

Abstract: We propose a new model for non-stationary random processes to represent time series with a time-varying spectral structure. Our SLEX model can be considered as a discrete time-dependent Cramer spectral representation. It is based on the so-called Smooth Localized complex EXponential basis functions which are orthogonal and localized in both time and frequency domains. Our model delivers a finite sample size representation of a SLEX process having a SLEX spectrum which is piecewise constant over time segments. In addition, we embed it into a sequence of models with a limit spectrum, a smoothly in time varying “evolutionary” spectrum. Hence, we develop the SLEX model parallel to the Dahlhaus (1997, Ann. Statist., 25, 1–37) model of local stationarity, and we show that the two models are asymptotically mean square equivalent. Moreover, to define both the growing complexity of our model sequence and the regularity of the SLEX spectrum we use a wavelet expansion of the spectrum over time. Finally, we develop theory on how to estimate the spectral quantities, and we briefly discuss how to form inference based on resampling (bootstrapping) made possible by the special structure of the SLEX model which allows for simple synthesis of non-stationary processes.

Abstract: Real-world financial data is often nonlinear, comprises high-frequency multipolynomial components, and is discontinuous (piecewise continuous). Not surprisingly, it is hard to model such data. Classical neural networks are unable to automatically determine the optimum model and appropriate order for financial data approximation. We address this problem by developing neuron-adaptive higher order neural-network (NAHONN) models. After introducing one-dimensional (1-D), two-dimensional (2-D), and n-dimensional NAHONN models, we present an appropriate learning algorithm. Network convergence and the universal approximation capability of NAHONNs are also established. NAHONN Group models (NAHONGs) are also introduced. Both NAHONNs and NAHONGs are shown to be "open box" and as such are more acceptable to financial experts than classical (closed box) neural networks. These models are further shown to be capable of automatically finding not only the optimum model, but also the appropriate order for specific financial data.

Abstract: When estimating hydraulic transmissivity the question of parametrization is of great importance. The transmissivity is assumed to be a piecewise constant space-dependent function and the unknowns are both the transmissivity values and the zonation, the partition of the domain whose parts correspond to the zones where the transmissivity is constant. Refinement and coarsening indicators, which are easy to compute from the gradient of the least squares misfit function, are introduced to construct iteratively the zonation and to prevent overparametrization.

Abstract: It is shown that a piecewise linear function on a convex domain in R d can be represented as a boolean polynomial in terms of its linear components.

Abstract: In this paper we propose an observer design procedure for a class of bi-modal piece-wise affine systems. The designed observers have the characteristic feature that they do not require information on the currently active dynamics of the piecewise linear system. A design procedure which guarantees global asymptotic stability of the estimation error is presented. It is shown that the applicability of the presented procedure is limited to continuous piece-wise affine systems. Therefore, we present an observer design procedure, applicable also to discontinuous systems, which guarantees that the estimation error is bounded, with respect to the state bounds, asymptotically. Sliding motions in the observed system, and the observer are discussed. The presented theory is illustrated with an example.

Abstract: A nonlinear H/sub /spl infin// guidance law based on a fuzzy model is proposed for tactical missiles pursuing maneuvering targets in three-dimensional (3-D) space. In the proposed guidance scheme, the relative motion equations between the missile and target are first interpolated piecewise by Takagi-Sugeno linear fuzzy models. Then, a nonlinear fuzzy H/sub /spl infin// guidance law is designed to eliminate the effects of approximation error and external disturbances to achieve the desired goal. The linear matrix inequality (LMI) technique is then employed to treat this H/sub /spl infin// optimal guidance design in consideration of control constraints. Finally, the problem is further transformed into a standard eigenvalue problem so that it can be efficiently solved via a convex optimization algorithm, which is available from a numerical computation software.

Abstract: In this paper, we propose and study different mixed variational methods in order to approximate with finite elements the unilateral problems arising in contact mechanics. The discretized unilateral conditions at the candidate contact interface are expressed by using either continuous piecewise linear or piecewise constant Lagrange multipliers in the saddle-point formulation. A priori error estimates are established and several numerical studies corresponding to the different choices of the discretized unilateral conditions are achieved.

Abstract: Boundary conditions are an essential part of the approximations used in the numerical solution of the transport equation. The collision probability and the characteristic methods are considered, and exact and approximated tracking methods to be used in the implementation of geometrical motions and albedo conditions are analyzed. The analysis of the exact boundary-condition treatment is carried out for finite domains and infinite lattices, where periodic trajectories must be used. Albedo-like boundary conditions may be used to approximate exact geometrical motions via spatially piecewise constant and either piecewise constant or discrete angular approximations for the boundary fluxes. We also have examined angular product quadrature formulas and shown that the recently proposed Bickley-Naylor quadratures do not respect particle conservation in the presence of anisotropy of scattering. Numerical examples show that the approximated albedo-type boundary method converges toward the results obtained with the exact boundary treatment. However, because of problems related to the multigroup implementation, numerical extra burden in group iterations prevents the efficient use of approximated boundary conditions for multigroup calculations. Nevertheless, this method remains a candidate of choice for use in multidomain calculations via interface boundary fluxes.

Abstract: We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the data is globally smooth, while the presence of jump discontinuities is responsible for spurious O (1) Gibbs oscillations in the neighborhood of edges and an overall deterioration of the unacceptable first-order convergence in rate. The purpose is to regain the superior accuracy in the piecewise smooth case, and this is achieved by mollification. Here we utilize a modified version of the two-parameter family of spectral mollifiers introduced by Gottlieb and Tadmor [GoTa85]. The ubiquitous one-parameter, finite-order mollifiers are based on dilation . In contrast, our mollifiers achieve their high resolution by an intricate process of high-order cancellation . To this end, we first implement a localization step using an edge detection procedure [GeTa00a, b]. The accurate recovery of piecewise smooth data is then carried out in the direction of smoothness away from the edges, and adaptivity is responsible for the high resolution. The resulting adaptive mollifier greatly accelerates the convergence rate, recovering piecewise analytic data within exponential accuracy while removing the spurious oscillations that remained in [GoTa85]. Thus, these adaptive mollifiers offer a robust, general-purpose ``black box'' procedure for accurate post-processing of piecewise smooth data.

Abstract: The restoration of the spatial structure of heterogeneous media, such as composites, porous materials, microemulsions, ceramics, or polymer blends from two-point correlation functions, is a problem of relevance to several areas of science. In this contribution we revisit the question of the uniqueness of the restoration problem. We present numerical evidence that periodic, piecewise uniform structures with smooth boundaries are completely specified by their two-point correlation functions, up to a translation and, in some cases, inversion. We discuss the physical relevance of the results.

Abstract: One of the most popular pairs of finite elements for solving mixed formulations of the Stokes and Navier-Stokes problem is the Qk - Pk-1disc element. Two possible versions of the discontinuous pressure space can be considered: one can either use an unmapped version of the Pk-1disc space consisting of piecewise polynomial functions of degree at most k - 1 on each cell or define a mapped version where the pressure space is defined as the image of a polynomial space on a reference cell. Since the reference transformation is in general not affine but multilinear, the two variants are not equal on arbitrary meshes. It is well-known, that the inf-sup condition is satisfied for the first variant. In the present paper we show that the latter approach satisfies the inf-sup condition as well for k ≥ 2 in any space dimension.

Abstract: The objective of this paper is to contribute to the methodology available for dealing with the detection and the estimation of the location of discontinuities in one-dimensional piecewise smooth regression functions observed in white Gaussian noise over an interval. Our approach is nonparametric in nature because the unknown function is not assumed to have any specific form. Our method relies upon a wavelet analysis of the observed signal and belongs to the class of "indirect" methods, where one detects and locates the change points prior to fitting the curve, and then uses ones favorite function estimation technique on each segment to recover the curve. We show that, provided discontinuities can be detected and located with sufficient accuracy, detection followed by wavelet smoothing enjoys optimal rates of convergence.

Abstract: In this paper, we propose a simple and efficient shock-fitting solution algorithm for the LWR model assuming a linear speed–density relationship or parabolic fundamental diagram. The solution is exact if the boundary conditions for density variable on the spatial axis are piecewise linear and those on the time axis are piecewise constant. Discontinuities are explicitly handled. The method utilizes the concept that for a linear speed–density relationship, a linear density variation along the spatial axis remains linear if not interrupted by shocks. Explicit expressions for the nonlinear shock path trajectory between two linear density functions are also derived. Two numerical examples are used to illustrate the effectiveness of the proposed method.

Abstract: We study Spectral Measures of Risk from the perspective of portfolio optimization. We derive exact results which extend to general Spectral Measures M_phi the Pflug--Rockafellar--Uryasev methodology for the minimization of alpha--Expected Shortfall. The minimization problem of a spectral measure is shown to be equivalent to the minimization of a suitable function which contains additional parameters, but displays analytical properties (piecewise linearity and convexity in all arguments, absence of sorting subroutines) which allow for efficient minimization procedures. In doing so we also reveal a new picture where the classical risk--reward problem a la Markowitz (minimizing risks with constrained returns or maximizing returns with constrained risks) is shown to coincide to the unconstrained optimization of a single suitable spectral measure. In other words, minimizing a spectral measure turns out to be already an optimization process itself, where risk minimization and returns maximization cannot be disentangled from each other.

Abstract: The paper addresses the aspects of control of real time systems with varying sampling rate. An example is given in which a stable continuous system is sampled at two different sampling rates. Two controllers are designed to minimize the same continuous quadratic loss function with the same weights. It is shown that although the design leads to stable controlled closed loop systems, for both discretizations, the resulting system can be unstable due to variations in sampling rate. To avoid that problem, we suggest an optimal controller design in which a bound on the cost, for all possible sampling rate variations, is computed. This results in a piecewise constant state feedback control law and guarantees stability regardless of the variations in sampling rate. The controller synthesis is cast into an LMI, which conveniently solves the synthesis problem. To illustrate the procedure, the introduction example is revised using the proposed LMI synthesis method and the stable control law is given, which is robustly stable against variations in sampling rate.