Abstract: A class of fourth-order partial differential equations (PDEs) are proposed to optimize the trade-off between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a cost functional which is an increasing function of the absolute value of the Laplacian of the image intensity function. Since the Laplacian of an image at a pixel is zero if the image is planar in its neighborhood, these PDEs attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image. Piecewise planar images look more natural than step images which anisotropic diffusion (second order PDEs) uses to approximate an observed image. So the proposed PDEs are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion, while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion. Although both approaches seem to be comparable in removing speckles in the observed images, speckles are more visible in images processed by the proposed PDEs, because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges. Speckles can be easily removed by simple algorithms such as the one presented in this paper.

Abstract: The total variation (TV) denoising method is a PDE-based technique that preserves edges well but has the sometimes undesirable staircase effect, namely, the transformation of smooth regions ( ramps) into piecewise constant regions ( stairs). In this paper we present an improved model, constructed by adding a nonlinear fourth order diffusive term to the Euler--Lagrange equations of the variational TV model. Our technique substantially reduces the staircase effect, while preserving sharp jump discontinuities (edges). We show numerical evidence of the power of resolution of this novel model with respect to the TV model in some 1D and 2D numerical examples.

Abstract: We study mathematical programs with complementarity constraints. Several stationarity concepts, based on a piecewise smooth formulation, are presented and compared. The concepts are related to stationarity conditions for certain smooth programs as well as to stationarity concepts for a nonsmooth exact penalty function. Further, we present Fiacco-McCormick type second order optimality conditions and an extension of the stability results of Robinson and Kojima to mathematical programs with complementarity constraints.

Abstract: The use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control. Lower bounds on the optimal control cost are obtained by semidefinite programming based on the Bellman inequality. This also gives an approximation to the optimal control law. An upper bound to the optimal cost is obtained by another convex optimization problem using the given control law. A compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy.

Abstract: We study the convergence behavior of a sequence of stationary points of a parametric NLP which regularizes a mathematical program with equilibrium constraints (MPEC) in the form of complementarity conditions. Accumulation points are feasible points of the MPEC; they are C-stationary if the MPEC linear independence constraint qualification holds; they are M-stationary if, in addition, an approaching subsequence satisfies second order necessary conditions, and they are B-stationary if, in addition, an upper level strict complementarity condition holds. These results complement recent results of Fukushima and Pang [Convergence of a smoothing continuation method for mathematical programs with equilibrium constraints, in Ill-posed Variational Problems and Regularization Techniques, Springer-Verlag, New York, 1999]. We further show that every local minimizer of the MPEC which satisfies the linear independence, upper level strict complementarity, and a second order optimality condition can be embedded into a locally unique piecewise smooth curve of local minimizers of the parametric NLP.

Abstract: In this paper we derive global W1,∞ and piecewise C1,α estimates for solutions to divergence form elliptic equations with piecewise Holder continuous coefficients. The novelty of these estimates is that, even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients, they are independent of the distance between these surfaces.

Abstract: A compensation scheme is presented for general nonlinear actuator deadzones of unknown width. The compensator uses two neural networks (NNs), 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. Rigorous proofs of closed-loop stability for the deadzone compensator are provided and yield tuning algorithms for the weights of the two NNs. The technique provides a general procedure for using NNs to determine the preinverse of an unknown right-invertible function.

Abstract: Many boundary value problems (BVPs) or initial BVPs have nonsmooth solutions, with jumps along lower-dimensional interfaces. The explicit-jump immersed interface method (EJIIM) was developed following Li's fast iterative immersed interface method (FIIIM), recognizing that the foundation for the efficient solution of many such problems is a good solver for elliptic BVPs. EJIIM generalizes the class of problems for which FIIIM is applicable. It handles interfaces between constant and variable coefficients and extends the immersed interface method (IIM) to BVPs on irregular domains with Neumann and Dirichlet boundary conditions. Proofs of second order convergence for a one-dimensional (1D) problem with piecewise constant coefficients and for two-dimensional (2D) problems with singular sources are given. Other problems are reduced to the singular sources case, with additional equations determining the source strengths. The advantages of EJIIM are high quality of solutions even on coarse grids and easy adaptation to many problems with complicated geometries, while still maintaining the efficiency of the FIIIM.

Abstract: In this paper we present various algorithms both for stability analysis and state-feedback design for discrete-time piecewise affine systems. Our approach hinges on the use of piecewise quadratic Lyapunov functions that can be computed as the solution of a set of linear matrix inequalities. We show that the continuity of the Lyapunov function is not required in the discrete-time case. Moreover, the basic algorithms are made less conservative by exploiting the switching structure of piecewise affine systems and by using relaxation procedures.

Abstract: The discontinuous Galerkin finite element method (DGFEM) for the time discretization of parabolic problems is analyzed in the context of the hp-version of the Galerkin method. Error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp-DGFEM gives spectral convergence in problems with smooth time dependence. In conjunction with geometric time partitions it is proved that the hp-DGFEM results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms. For the h-version DGFEM algebraically graded time partitions are determined that give the optimal algebraic convergence rates. A fully discrete hp scheme is discussed exemplarily for the heat equation. The use of certain mesh-design principles for the spatial discretizations yields exponential rates of convergence in time and space. Numerical examples confirm the theoretical results.

Abstract: Convergence is established for a scalar finite difference scheme, based on the Godunov or Engquist--Osher (EO) flux, for scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient. Other works in this direction have established convergence for methods employing the solution of 2 × 2 Riemann problems. The algorithm discussed here uses only scalar Riemann solvers. Satisfaction of a set of Kruzkov-type entropy inequalities is established for the limit solution, from which geometric entropy conditions follow. Assuming a piecewise constant coefficient, it is shown that these conditions imply L1-contractiveness for piecewise C1 solutions, thus extending a well-known theorem.

Abstract: Maximum likelihood parameter estimation of the continuous time linear stochastic state space model is considered on the basis of largeN discrete time data using a structural equation modeling (SEM) program. Random subject effects are allowed to be part of the model. The exact discrete model (EDM) is employed which links the discrete time model parameters to the underlying continuous time model parameters by means of nonlinear restrictions. The EDM is generalized to cover not only time-invariant parameters but also the cases of stepwise time-varying (piecewise time-invariant) parameters and parameters varying continuously over time according to a general polynomial scheme. The identification of the continuous time parameters is discussed and an educational example is presented.

Abstract: It is a common practice for industries to price the same products at different levels. For example, airlines charge various fares for a common pool of seats. Seasonal products are sold at full or discount prices during different phases of the season. This article presents a model that reflects this yield management problem. The model assumes that (1) products are offered at multiple predetermined prices over time; (2) demand is price sensitive and obeys the Poisson process; and (3) price is allowed to change monotonically, i.e., either the markup or markdown policy is implemented. To maximize the expected revenue, management needs to determine the optimal times to switch between prices based on the remaining season and inventory. Major results in this research include (1) an exact solution for the continuous-time model; (2) piecewise concavity of the value function with respect to time and inventory; and (3) monotonicity of the optimal policy. The implementation of optimal policies is fairly facile because of the existence of threshold points embedded in the value function. The value function and time thresholds can be solved with a reasonable computation effort. Numerical examples are provided.

Abstract: We propose a procedure for synthesizing piecewise linear optimal controllers for hybrid systems and investigate conditions for closed-loop stability. Hybrid systems are modeled in discrete-time within the mixed logical dynamical framework, or, equivalently, as piecewise affine systems. A stabilizing controller is obtained by designing a model predictive controller, which is based on the minimization of a weighted 1//spl infin/-norm of the tracking error and the input trajectories over a finite horizon. The control law is obtained by solving a mixed-integer linear program (MILP) which depends on the current state. Although efficient branch and bound algorithms exist to solve MILPs, these are known to be NP-hard problems, which may prevent their online solution if the sampling-time is too small for the available computation power. Rather than solving the MILP online, we propose a different approach where all the computation is moved off line, by solving a multiparametric MILP. As the resulting control law is piecewise affine, online computation is drastically reduced to a simple linear function evaluation. An example of piecewise linear optimal control of a heat exchange system shows the potential of the method.

Abstract: Let Ω be a region in the plane which contains the origin, is star-shaped with respect to the origin and has a piecewise C 1 boundary. For each integer Q≥ 1, we consider the integer lattice points from which are visible from the origin and prove that the 1 st consecutive spacing distribution of the angles formed with the origin exists. This is a probability measure supported on an interval [m Ω,∞), with m Ω >0. Its repartition function is explicitly expressed as the convolution between the square of the distance from origin function and a certain kernel.

Abstract: Given a process whose output is a dynamic function of time, the traditional discrete event specification (DEVS) approximates the input, output, and state trajectories through piecewise constant segments, where the segments correspond to discrete time intervals that are not necessarily equal in length. For processes that defy accurate modeling through piecewise constant segments, this paper presents GDEVS, a generalized discrete event specification, wherein the trajectories are organized through piecewise polynomial segments. The utilization of arbitrary polynomial functions for segments promises higher accuracies in modeling continuous processes as discrete event abstractions. In general, discrete event systems including DEVS and GDEVS execute faster on host computers because executions occur corresponding to significant changes in the system unlike in continuous simulations where execution is on a continuous basis. GDEVS' superiority over DEVS lies in its ability to discretize a system characteristic. A key contribution of GDEVS is that it permits the development of a uniform simulation environment for hybrid, i.e. both continuous and discrete, systems. GDEVS is illustrated for a first order system and a hybrid system, with piecewise linear segments. Two representative systems have been modeled under GDEVS and executed on a simulator developed for GDEVS.

Abstract: We develop the general a priori error analysis of residual-free bubble finite element approximations to linear elliptic convection-dominated diffusion problems subject to homogeneous Dirichlet boundary condition. Optimal-order error bounds are derived in various norms, using piecewise polynomial finite elements of degree greater than or equal to 1.

Abstract: Using new systems capable of making synchronized phasor measurements, the real-time stability assessment of a transient event in power systems has become an important area of investigation. Using these phasor measurements as input conditions for computing a relatively good, simplified dynamic model can yield accurate and real-time transient stability prediction in a central location equipped with high-speed computers. In an effort to reduce the computing time for integrating the differential/algebraic equation (DAE) model of postfault power system dynamics for prediction use, this paper presents a faster implicitly decoupled PQ integration technique. Two piecewise dynamic equivalents are also proposed, i.e., piecewise constant current load equivalent and piecewise constant transfer admittance equivalent. These equivalents can eliminate the algebraic equations by approximating the load flow solution piecewisely such that only internal generator buses are preserved, while approximately retaining the characteristics of the nonlinear loads. The proposed techniques have been tested on two sample power systems with promising simulation results.

Abstract: Regression spline smoothing involves modelling a regression function as a piecewise polynomial with a high number of pieces relative to the sample size. Because the number of possible models is so large, efficient strategies for choosing among them are required. In this paper we review approaches to this problem and compare them through a simulation study. For simplicity and conciseness we restrict attention to the univariate smoothing setting with Gaussian noise and the truncated polynomial regression spline basis.

Abstract: In process analytical applications it is not always possible to keep the measurement conditions constant. However, fluctuations in external variables such as temperature can have a strong influence on measurement results. For example, nonlinear temperature effects on near-infrared (NIR) spectra may lead to a strongly biased prediction result from multivariate calibration models such as PLS. A new method, called Continuous Piecewise Direct Standardization (CPDS) has been developed for the correction of such external influences. It represents a generalization of the discrete PDS calibration transfer method and is able to adjust for continuous nonlinear influences such as the temperature effects on spectra. It was applied to short-wave NIR spectra of ethanol/water/2-propanol mixtures measured at different temperatures in the range 30−70 °C. The method was able to remove, almost completely, the temperature effects on the spectra, and prediction of the mole fractions of the chemical components was close to the...

Abstract: We propose an extension, called Lp+, of the temporal logic LTL, which enables talking about finitely many register values: the models are infinite words over tuples of integers (resp. real numbers). The formulas of Lp+ are flat: on the left of an until, only atomic formulas or LTL formulas are allowed. We prove, in the spirit of the correspondence between automata and temporal logics, that the models of a Lp+ formula are recognized by a piecewise flat counter machine; for each state q, at most one loop of the machine on q may modify the register values. Emptiness of (piecewise) flat counter machines is decidable (this follows from a result in [9]). It follows that satisfiability and model-checking the negation of a formula are decidable for Lp+. On the other hand, we show that inclusion is undecidable for such languages. This shows that validity and model-checking positive formulas are undecidable.

Abstract: Given a function f on [0,1] and a wavelet-type expansion of f , we introduce a new algorithm providing an approximation $\tilde f of f with a prescribed number D of nonzero coefficients in its expansion. This algorithm depends only on the number of coefficients to be kept and not on any smoothness assumption on f . Nevertheless it provides the optimal rate D -α of approximation with respect to the L q -norm when f belongs to some Besov space B α p,∈fty whenever α>(1/p-1/q) + . These results extend to more general expansions including splines and piecewise polynomials and to multivariate functions. Moreover, this construction allows us to compute easily the metric entropy of Besov balls.

Abstract: A method and apparatus for mesh-free engineering analysis of geometric models is described. The method and apparatus, which are preferably software-based and implemented on personal computers or other programmable processing devices, represent geometric models by implicit mathematical functions. The implicit functions allow interpolation of all desired boundary conditions over the geometry without meshing, and the boundary conditions may then may be combined with a piecewise continuous model of the solution structure (i.e., the analysis problem). By solving for elements of the solution structure (its basis or coordinate functions) which satisfy the given boundary conditions either exactly or approximately, the solution structure will define the behavior and boundary conditions (exactly or approximately) throughout the geometric model.

Abstract: Presents a stable nonparametric adaptive control approach using a piecewise local linear approximator. The continuous piecewise linear approximator is developed and its universal approximation capability is proved. The controller architecture is based on adaptive feedback linearization plus sliding mode control. A time varying activation region is introduced for efficient self-organization of the approximator during operation. We modify the adaptive control approach for piecewise linear approximation and self-organizing structures. In addition, we provide analyses of asymptotic stability of the tracking error and parameter convergence for the proposed adaptive control scheme with the online self-organizing structure. The method with a deadzone is also discussed to prevent a high-frequency input which might excite the unmodeled dynamics in practical applications. The application of the piecewise linear adaptive control method is demonstrated by a computational simulation.

Abstract: An algorithm for general nonlinearly constrained optimization is presented, which solves an unconstrained piecewise quadratic subproblem and a quadratic programming subproblem at each iterate. The algorithm is robust since it can circumvent the difficulties associated with the possible inconsistency of QP subproblem of the original SQP method. Moreover, the algorithm can converge to a point which satisfies a certain first-order necessary optimality condition even when the original problem is itself infeasible, which is a feature of Burke and Han's methods [Math. Programming, 43 (1989), pp. 277--303]. Unlike Burke and Han's methods, our algorithm does not introduce additional bound constraints. The algorithm solves the same subproblems as the Han--Powell SQP algorithm at feasible points of the original problem. Under certain assumptions, it is shown that the algorithm coincides with the Han--Powell method when the iterates are sufficiently close to the solution. Some global convergence results are proved and locally superlinear convergence results are also obtained. Preliminary numerical results are reported.

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 the right hand side of the differential equations consists of both continuous and discrete valued variables. The equations then run in combination with a finite state machine, and their interatcion is specified via an interface. This framework allows a compact decription 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 swithced system fulfills specifications given either in terms of sets of good or bad discrete states, or as formulas in temporal logic specifying sesired 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 two different automated procedures devised in this thesis. Both methods can be seen as special instances of using invariant sets and Lyapunov theory to guarantee either that bad states are avoided and/or that goal states are reached in finite time.The former discretization method 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.The second method utilized invariant sets of a special type called power cones, of which quadratic sets such as cones and paraboloids are special instances. This approach results in thighter approximations while still allowing for efficient implementation using Linear Matrix Inequalities and convex optimization.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 one of our methods to two examples, a chemical reactor model and a model of the landing gear of a fighter aircraft.

Abstract: A general theory of piecewise multiharmonic splines is constructed for a class of fractals (post-critically finite) that includes the familiar Sierpinski gasket, based on Kigami's theory of Laplacians on these fractals. The spline spaces are the analogues of the spaces of piecewise Cj polynomials of degree 2j + 1 on an interval, with nodes at dyadic rational points. We give explicit algorithms for effectively computing multiharmonic functions (solutions of Δj+1u = 0) and for constructing bases for the spline spaces (for general fractals we need to assume that j is odd), and also for computing inner products of these functions. This enables us to give a finite element method for the approximate solution of fractal differential equations. We give the analogue of Simpson's method for numerical integration on the Sierpinski gasket. We use splines to approximate functions vanishing on the boundary by functions vanishing in a neighbourhood of the boundary.

Abstract: We describe nonlinear deterministic versus stochastic methodology, their applications to EEG research and the neurophysiological background underlying both approaches. Nonlinear methods are based on the concept of attractors in phase space. This concept on the one hand incorporates the idea of an autonomous (stationary) system, on the other hand implicates the investigation of a long time evolution. It is an unresolved problem in nonlinear EEG research that nonlinear methods per se give no feedback about the stationarity aspect. Hence, we introduce a combined strategy utilizing both stochastic and nonlinear deterministic methods. We propose, in a first step to segment the EEG time series into piecewise quasi-stationary epochs by means of nonparametric change point analysis. Subsequently, nonlinear measures can be estimated with higher confidence for the segmented epochs fulfilling the stationarity condition. Language: en

Abstract: It is shown that the problem of computing astronomical refraction for any value of the zenith angle may be reduced to a simple, nonsingular, numerical quadrature when the proper choice is made for the independent variable of integration. The angle between the radius vector and the light ray is such a choice. The implementation of the quadrature method is discussed in its general form and illustrated by means of an application to a piecewise polytropic atmosphere. The flexibility, simplicity, and computational efficiency of the method are evident.