Abstract: This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach.

Abstract: I describe a new time-domain algorithm for detecting localized structures (bursts), revealing pulse shapes, and generally characterizing intensity variations. The input is raw counting data, in any of three forms: time-tagged photon events (TTE), binned counts, or time-to-spill (TTS) data. The output is the most probable segmentation of the observation into time intervals during which the photon arrival rate is perceptibly constant, i.e., has no statistically significant variations. The idea is not that the source is deemed to have this discontinuous, piecewise constant form, rather that such an approximate and generic model is often useful. Since the analysis is based on Bayesian statistics, I call the resulting structures Bayesian blocks. Unlike most, this method does not stipulate time bins—instead the data determine a piecewise constant representation. Therefore the analysis procedure itself does not impose a lower limit to the timescale on which variability can be detected. Locations, amplitudes, and rise and decay times of pulses within a time series can be estimated independent of any pulse-shape model—but only if they do not overlap too much, as deconvolution is not incorporated. The Bayesian blocks method is demonstrated by analyzing pulse structure in BATSE γ-ray data.

Abstract: This paper, which we dedicate to Lucien Le Cam for his seventieth birthday, has been written in the spirit of his pioneering works on the relationships between the metric structure of the parameter space and the rate of convergence of optimal estimators. It has been written in his honour as a contribution to his theory. It contains further developments of the theory of minimum contrast estimators elaborated in a previous paper. We focus on minimum contrast estimators on sieves. By a `sieve' we mean some approximating space of the set of parameters. The sieves which are commonly used in practice are D-dimensional linear spaces generated by some basis: piecewise polynomials, wavelets, Fourier, etc. It was recently pointed out that nonlinear sieves should also be considered since they provide better spatial adaptation (think of histograms built from any partition of D subintervals of [0,1] as a typical example). We introduce some metric assumptions which are closely related to the notion of finite-dimensional metric space in the sense of Le Cam. These assumptions are satisfied by the examples of practical interest and allow us to compute sharp rates of convergence for minimum contrast estimators.

Abstract: A compensation scheme is presented for general nonlinear actuator deadzones of unknown width. The compensator uses two neural networks (NN): one to estimate the unknown deadzone, and another to provide adaptive compensation in the feedforward path. The compensator NN has a special augmented form containing extra neurons whose activation functions provide a 'jump function basis set' for approximating piecewise continuous functions. Closed-loop stability analysis for the deadzone compensator is provided, and yield tuning algorithms for the weights of the two NN. The technique provides a general procedure for using NN to determine the pre-inverse of an unknown right-invertible function.

Abstract: Using Lyapunov functionals the global behaviour of the solutions of a reaction-diffusion system modelling chemotaxis is studied for bounded piecewise smooth domains in the plane. Geometric criteria can be given so that this dynamical system tends to a (not necessarily trivial) stationary state.

Abstract: A fast, second-order accurate iterative method is proposed for the elliptic equation \[ \grad\cdot(\beta(x,y) \grad u) =f(x,y) \] in a rectangular region $\Omega$ in two-space dimensions. We assume that there is an irregular interface across which the coefficient $\beta$, the solution u and its derivatives, and/or the source term f may have jumps. We are especially interested in the cases where the coefficients $\beta$ are piecewise constant and the jump in $\beta$ is large. The interface may or may not align with an underlying Cartesian grid. The idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by LeVeque and Li [ SIAM J. Numer. Anal., 4 (1994), pp. 1019--1044]. In order to take advantage of fast Poisson solvers on a rectangular region, an intermediate unknown function, the jump in the normal derivative across the interface, is introduced. Our discretization is equivalent to using a second-order difference scheme for a corresponding Poisson equation in the region, and a second-order discretization for a Neumann-like interface condition. Thus second-order accuracy is guaranteed. A GMRES iteration is employed to solve the Schur complement system derived from the discretization. A new weighted least squares method is also proposed to approximate interface quantities from a grid function. Numerical experiments are provided and analyzed. The number of iterations in solving the Schur complement system appears to be independent of both the jump in the coefficient and the mesh size.

Abstract: Digital video's increased popularity has been driven to a large extent by a flurry of international standards (MPEG-1, MPEG-2, H.263, etc). In most standards, the rate control scheme, which plays an important role in improving and stabilizing the decoding and playback quality, is not defined, and thus different strategies can be implemented in each encoder design. Several rate-distortion (R-D)-based techniques have been proposed aimed at the best possible quality for a given channel rate and buffer size. These approaches are complex because they require the R-D characteristics of the input data to be measured before making quantization assignment decisions. We show how the complexity of computing the R-D data can be reduced without significantly reducing the performance of the optimization procedure. We propose two methods which provide successive reductions in complexity by: (1) using models to interpolate the rate and distortion characteristics, and (2) using past frames instead of current ones to determine the models. Our first method is applicable to situations (e.g., broadcast video) where a long encoding delay is possible, while our second approach is more useful for computation-constrained interactive video applications. The first method can also be used to benchmark other approaches. Both methods can achieve over 1 dB peak signal-to-noise rate (PSNR) gain over simple methods like the MPEG Test Model 5 (TM5) rate control, with even greater gains during scene change transitions. In addition, both methods make few a priori assumptions and provide robustness in their performance over a range of video sources and encoding rates. In terms of complexity, our first algorithm roughly doubles the encoding time as compared to simpler techniques (such as TM5). However, the complexity is greatly reduced as compared to methods which exactly measure the R-D data. Our second algorithm has a complexity marginally higher than TM5 and a PSNR performance slightly lower than that of the first approach.

Abstract: We consider one-dimensional linear transport equations with bounded but possibly discontinuous coefficient a. The Cauchy problem is studied from two different points of view. In the first case we assume that a is piecewise continuous. We give an existence result and a precise description of the solutions on the lines of discontinuity. In the second case, we assume that a satisfies a one-sided Lipschitz condition. We give existence, uniqueness and general stability results for backward Lipschitz solutions and forward measure solutions, by using a duality method. We prove that the flux associated to these measure solutions is a product by some canonical representative â of a. Key-words. Linear transport equations, discontinuous coefficients, weak stability, duality, product of a measure by a discontinuous function, nonnegative solutions. 1991 Mathematics Subject Classification. Primary 35F10, 35B35, 34A12. To appear in Nonlinear Analysis, TMA ∗Departement de Mathematiques et Applications, UMR CNRS 8553, Ecole Normale Superieure et CNRS, 45 rue d’Ulm, 75230 Paris Cedex 05, France, francois.bouchut@ens.fr †Mathematiques, Applications et Physique Mathematique d’Orleans, UMR CNRS 6628, Universite d’Orleans, 45067 Orleans Cedex 2, France, james@math.cnrs.fr

Abstract: We are concerned with the retrieval of the unknown cross section of a homogeneous cylindrical obstacle embedded in a homogeneous medium and illuminated by time-harmonic electromagnetic line sources. The dielectric parameters of the obstacle and embedding materials are known and piecewise constant. That is, the shape (here, the contour) of the obstacle is sufficient for its full characterization. The inverse scattering problem is then to determine the contour from the knowledge of the scattered field measured for several locations of the sources and/or frequencies. An iterative process is implemented: given an initial contour, this contour is progressively evolved such as to minimize the residual in the data fit. This algorithm presents two main important points. The first concerns the choice of the transformation enforced on the contour. We will show that this involves the design of a velocity field whose expression only requires the resolution of an adjoint problem at each step. The second concerns the use of a level-set function in order to represent the obstacle. This level-set function will be of great use to handle in a natural way splitting or merging of obstacles along the iterative process. The evolution of this level-set is controlled by a Hamilton-Jacobi-type equation which will be solved by using an appropriate finite-difference scheme. Numerical results of inversion obtained from both noiseless and noisy synthetic data illustrate the behaviour of the algorithm for a variety of obstacles.

Abstract: We compute upper and lower bounds on the VC dimension and pseudodimension of feedforward neural networks composed of piecewise polynomial activation functions. We show that if the number of layers is fixed, then the VC dimension and pseudo-dimension grow as W log W, where W is the number of parameters in the network. This result stands in opposition to the case where the number of layers is unbounded, in which case the VC dimension and pseudo-dimension grow as W2 . We combine our results with recently established approximation error rates and determine error bounds for the problem of regression estimation by piecewise polynomial networks with unbounded weights.

Abstract: Standard wavelet shrinkage procedures for nonparametric regression are restricted to equispaced samples. There, data are transformed into empirical wavelet coefficients and threshold rules are applied to the coefficients. The estimators are obtained via the inverse transform of the denoised wavelet coefficients. In many applications, however, the samples are nonequispaced. It can be shown that these procedures would produce suboptimal estimators if they were applied directly to nonequispaced samples. We propose a wavelet shrinkage procedure for nonequispaced samples. We show that the estimate is adaptive and near optimal. For global estimation, the estimate is within a logarithmic factor of the minimax risk over a wide range of piecewise Holder classes, indeed with a number of discontinuities that grows polynomially fast with the sample size. For estimating a target function at a point, the estimate is optimally adaptive to unknown degree of smoothness within a constant. In addition, the estimate enjoys a smoothness property: if the target function is the zero function, then with probability tending to 1 the estimate is also the zero function.

Abstract: Segmentation of a nonstationary process consists in assuming piecewise stationarity and in detecting the instants of change. We consider the case where all the data is available at the same time and perform a global segmentation instead of a sequential procedure. We build a change process and define arbitrarily its prior distribution. This allows us to propose the MAP estimate as well as some minimum contrast estimate as a solution. One of the interests of the method is its ability to give the best solution, according to the resolution level required by the user, that is, to the prior distribution chosen. The method can address a wide class of parametric and nonparametric models. Simulations and applications to real data are proposed.

Abstract: This paper introduces a new class of simple nonlinear PID controllers and provides a formal treatment of their stability analysis. These controllers are comprised of a sector-bounded nonlinear gain in cascade with a linear fixed-gain P, PD, PI, or PID controller. Three simple nonlinear gains are proposed: the sigmoidal function, the hyperbolic function, and the piecewise]linear function. The systems to be controlled are assumed to be modeled or approximated by second-order transfer functions, which can represent many robotic applications. The stability of the closed-loop systems incorporating nonlinear P, PD, PI, and PID controllers are investigated using the Popov stability criterion. It is shown that for P and PD controllers, the nonlinear gain is unbounded for closed-loop stability. For PI and PID controllers, simple expressions are derived that relate the controller gains and system parameters to the maximum allowable nonlinear gain for stability. A numerical example is given for illustration. The stability of partially-nonlinear PID controllers is also discussed. Finally, the nonlinear PI controller is implemented as a force controller on a robotic arm and experimental results are presented. These results demonstrate the superior performance of the nonlinear PI controller relative to a fixed-gain PI controller. Q 1998

Abstract: The numerical solution of a parabolic equation with memory is considered. The equation is first discretized in time by means of the discontinuous Galerkin method with piecewise constant or piecewise linear approximating functions. The analysis presented allows variable time steps which, as will be shown, call then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data. The combination with finite element discretizat in space is also studied.

Abstract: A Dirichlet boundary value problem for a linear parabolic dierential equation is studied on a rectangular domain in the x t plane. The coecient of the second order space derivative is a small singular perturbation parameter, which gives rise to parabolic boundary layers on the two lateral sides of the rectangle. It is proved that a numerical method, comprising a standard nite dierence operator (centred in space, implicit in time) on a tted piecewise uniform mesh of NxNt elements condensing in the boundary layers, is uniform with respect to the small parameter, in the sense that its numerical solutions converge in the maximum norm to the exact solution uniformly well for all values of the parameter in the semi-open interval (0,1]. More specically, it is shown that the errors are bounded in the maximum norm by C((N 1 x lnNx) 2 +N 1 t ), where C is a constant independent not only of Nx and Nt but also of the small parameter. Numerical results are presented, which validate numerically this theoretical result and show that a numerical method consisting of the same nite dierence operator on a uniform mesh of NxNt elements is not uniform with respect to the small parameter.

Abstract: A new approach is introduced for the analysis and calculation of straight prismatic beams of piecewise constant cross-section under arbitrary loads. This theory can be called “exact” because it determines exact static and kinematic generalized quantities. Moreover, contrary to classical theories, it is not limited to high-aspect ratio (i.e. relatively slender) beams.

Abstract: In a chemical kinetics calculation, a solution-mapping procedure is applied to parametrize the solution of the initial-value ordinary differential equation system as a set of algebraic polynomial equations. To increase the accuracy, the parametrization is done piecewise, dividing the multidimensional chemical composition space into hypercubes and constructing polynomials for each hypercube. A differential equation solver is used to provide the solution at selected points throughout a hypercube, and from these solutions the polynomial coefficients are determined. Factorial design methods are used to reduce the required number of computed points. The polynomial coefficients for each hypercube are stored in a data structure for subsequent reuse, since over the duration of a flame simulation it is likely that a particular set of concentrations and temperature will occur repeatedly at different times and positions. The method is applied to H2–air combustion using an 8-species reaction set. After N2 is added as an inert species and enthalpy is considered, this results in a 10-dimensional chemical composition space. To add the capability of using a variable time-step, time-step is added as an additional dimension, making an 11-dimensional space. Reactive fluid dynamical simulations of a 1-D laminar premixed flame and a 2-D turbulent non-premixed jet are performed. The results are compared to identical control runs which use an ordinary differential equation solver to calculate the chemical kinetic rate equations. The resulting accuracy is very good, and a factor of 10 increase in computational efficiency is attained.

Abstract: By using extensive Molecular Dynamics simulations, we have determined the violation of the fluctuation-dissipation theorem in a Lennard-Jones liquid quenched to low temperatures. For this we have calculated $X(C)$, the ratio between a one particle time-correlation function $C$ and the associated response function. Our results are best fitted by assuming that $X(C)$ is a discontinuous, piecewise constant function. This is similar to what is found in spin systems with one step replica symmetry breaking. This strengthen the conjecture of a similarity between the phase space structure of structural glasses and such spin systems.

Abstract: An algorithm for obtaining a piecewise triangular approximation of a trimmed NURBS surface is presented. The algorithm is geometry based, i.e. the surface is subdivided into triangular facets based on its geometric characteristics and not on its parametrization. No assumption is made about the surface's parametrical representation; it does not have to be continuously differentiable, only Co continuity is assumed. The surface subdivision is performed in model space, however, the triangulation is carried out in parameter space using the parametric vertices of subdivision rectangles. Along with computing the triangulation, the method produces a compact database for browsing in the triangular irregular network, e.g. finding all neighbors of a given triangle.

Abstract: A method introduced by Arjas & Gasbarra (1994) and later modified by Arjas & Heikkinen (1997) for the non-parametric Bayesian estimation of an intensity on the real line is generalized to cover spatial processes. The method is based on a model approximation where the approximating intensities have the structure of a piecewise constant function. Random step functions on the plane are generated using Voronoi tessellations of random point patterns. Smoothing between nearby intensity values is applied by means of a Markov random field prior in the spirit of Bayesian image analysis. The performance of the method is illustrated in examples with both real and simulated data.

Abstract: We present a general method of solving differential-algebraic equations by expanding the solution as a Taylor series. It seems especially suitable for (piecewise) smooth problems of high index. We describe the method in general, discuss steps to be taken if the method, as initially applied, fails because it leads to a system of equations with identically singular Jacobian, and illustrate by solving two problems of index 5.

Abstract: We prove the global existence of solutions of the Navier–Stokes equations of compressible flow in one space dimension with minimal hypotheses on the initial data, the equation of state, and the external force. Specifically, we require of the initial data only that the density be bounded above and below away from zero, and that the density and velocity be in L 2, modulo constant states at $x=\infty$ and $x=-\infty$ , which may be different. There are no smallness hypotheses on either the data or on the external force. In particular, we include the important case that the initial data is piecewise constant with arbitrarily large jump discontinuities. Our results show that, even in this generality, neither vacuum states nor concentration states can form in finite time.

Abstract: The hysteresis behavior of a linear stroke magnetorheological damper is characterized for sinusoidal displacement excitation at 2.0 Hz (nominal). First, we characterize the linearized MR damper behavior using equivalent viscous damping and complex stifiness. Four difierent nonlinear modeling perspectives are then discussed for purposes of system identiflcation procedures, including: (1) nonlinear Bingham plastic model, (2) nonlinear biviscous model, (3) nonlinear hysteretic biviscous model, and (4) nonlinear viscoelastic-plastic model. The flrst three nonlinear models are piecewise continuous in velocity. The fourth model is piecewise smooth in velocity. By adding progressively more model parameters with which to better represent pre-yield damper behavior, the force vs. velocity hysteresis model is substantially improved. Of the three nonlinear piecewise continuous models, the nonlinear hysteretic biviscous model provides the best representation of force vs. velocity hysteresis. The nonlinear viscoelastic plastic model is superior for purposes of simulation to the hysteretic biviscous model because it is piecewise smooth in velocity, with a smooth transition from pre-yield to post-yield behaviors. The nonlinear models represent the force vs. displacement hysteresis behavior nearly equally well, although the nonlinear viscoelastic-plastic is quantiflably superior. Thus, any of the nonlinear damper models could be used equally successfully if only a prediction of energy dissipation or damping were of interest.

Abstract: This thesis deals with modeling and analysis of a special class of hybrid systems, i.e., systems displaying behavior of both continuous and discrete nature.We focus on the class of systems which are naturally modeled as having continuous trajectories, but where the trajectories evolve according to a system of first order differential equations with piecewise continuous right hand side. We further restrict our attention to models where the right hand side is piecewise affine in the continuous state variables. Systems which are naturally modeled in this fashion are called piecewise affine switched systems.The modeling is packaged into a modeling framework where we take a logical approach to writing down the right hand side of the differential equations. The equations then run in combination with a finite state machine, and their interaction is specified via an interface. This framework allows a compact description of a potentially large number of affine models, as well as a modular approach to modeling of complex systems.The analysis consists of verifying if a switched system fulfills specifications given either in terms of sets of good or bad discrete states, or as formulas in temporal logic specifying desired or undesired behavior. This kind of analysis is often applied to purely discrete systems, we therefore take the approach to discretize the switched system using conservative approximations. This is done using an automated procedure devised in this thesis. The discretization results in two different approximations, named acceptor and exceptor, which together can be utilized to obtain bounds on behavior which can be certified. We also introduce a discrete device, called reflector, which allows us to reduce the non-determinism resulting from the approximations. Furthermore, implementations of the procedure suggested, using either linear programming or quantifier elimination as computational tools, are discussed.Together, the discrete approximations, obtained automatically from a model of a switched system, can be used to verify properties of interest. This is illustrated as we apply our methods to two relatively large examples, a chemical reactor model and a model of the landing gear of a fighter aircraft.

Abstract: In this paper we present a method to optimize the smoothness order of subdivision algorithms generating surfaces of arbitrary topology. In particular we construct a subdivision algorithm with some negative weights producing G 2-surfaces. These surfaces are piecewise bicubic and are flat at their extraordinary points. The underlying ideas can also be used to improve the smoothness order of subdivision algorithms for surfaces of higher degree or triangular nets.

Abstract: In this paper we consider the problem of minimizing a (possibly nonconvex) quadratic function with a quadratic constraint. We point out some new properties of the problem. In particular, in the first part of the paper, we show that (i) given a KKT point that is not a global minimizer, it is easy to find a "better" feasible point; (ii) strict complementarity holds at the local-nonglobal minimizer. In the second part of this paper, we show that the original constrained problem is equivalent to the unconstrained minimization of a piecewise quartic merit function. Using the unconstrained formulation we give, in the nonconvex case, a new second order necessary condition for global minimizers. In the third part of this paper, algorithmic applications of the preceding results are briefly outlined and some preliminary numerical experiments are reported.