Showing papers on "Piecewise published in 1990"

Reaction Path Following in Mass-Weighted Internal Coordinates

Abstract: Our previous algorithm for following reaction paths downhill (J. Chem. Phys. 1989, 90, 2154), has been extended to use mass-weighted internal coordinates. Points on the reaction path are round by constrained optimizations involving the internal degrees or freedom or the molecule. The points are optimized so that the segment or the reaction path between any two adjacent points is described by an arc or a circle in mass-weighted internal coordinates, and so that the gradients (in mass-weighted internals) at the end points or the arc are tangent to the path. The algorithm has the correct tangent vector and curvature vectors in the limit or small step size but requires only the transition vector and the energy gradients; the resulting path is continuous, differentiable, and piecewise quadratic

Approximation scheme with applications to computational fluid-dynamics-- i surface approximations and partial derivative estimates

Abstract: A~traet--We present a powerful, enhanced multiquadrics (MQ) scheme developed for spatial approximations. MQ is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions. It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy. Monotonicity and convexity are observed properties as a result of such high accuracy. Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation, but also for partial derivative estimates. MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning. In the second paper of this series, MQ is applied to parabolic, hyperbolic and elliptic partial differential equations. The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme. We show that MQ is also exceptionally accurate and efficient. The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results. 1. BACKGROUND The study of arbitrarily shaped curves, surfaces and bodies having arbitrary data orderings has immediate application to computational fluid-dynamics. The governing equations not only include source terms but gradients, divergences and Laplacians. In addition, many physical processes occur over a wide range of length scales. To obtain quantitatively accurate approximations of the physics, quantitatively accurate estimates of the spatial variations of such variables are required. In two and three dimensions, the range of such quantitatively accurate problems possible on current multiprocessing super computers using standard finite difference or finite element codes is limited. The question is whether there exist alternative techniques or combinations of techniques which can broaden the scope of problems to be solved by permitting steep gradients to be modelled using fewer data points. Toward that goal, our study consists of two parts. The first part will investigate a new numerical technique of curve, surface and body approximations of exceptional accuracy over an arbitrary data arrangement. The second part of this study will use such techniques to improve parabolic, hyperbolic or elliptic partial differential equations. We will demonstrate that the study of function approximations has a definite advantage to computational methods for partial differential equations. One very important use of computers is the simulation of multidimensional spatial processes. In this paper, we assumed that some finite physical quantity, F, is piecewise continuous in some finite domain. In many applications, F is known only at a finite number of locations, {xk: k = 1, 2 ..... N} where xk = x~ for a univariate problem, and Xk = (x~,yk .... )X for the multivariate problem. From a finite amount of information regarding F, we seek the best approximation which can not only supply accurate estimates of F at arbitrary locations on the domain, but will also provide accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain. The domain of F will consist of points, {xk }, of arbitrary ordering and sub-clustering. A rectangular grid is a very special case of a data ordering. Let us assume that an interpolation function, f, approximates F in the sense that

Finite element interpolation of nonsmooth functions satisfying boundary conditions

Abstract: In this paper, we propose a modified Lagrange type interpolation operator to approximate functions in Sobolev spaces by continuous piecewise polynomials. In order to define interpolators for "rough" functions and to preserve piecewise polynomial boundary conditions, the approximated functions are averaged appropriately either on dor (d 1)-simplices to generate nodal values for the interpolation operator. This combination of averaging and interpolation is shown to be a projection, and optimal error estimates are proved for the projection error.

Nonlinear theory of the switched reluctance motor for rapid computer-aided design

Abstract: The nonlinear magnetisation characteristics of the switched reluctance motor are modelled analytically by piecewise first- or second-order functions of flux linkage against rotor position, with current as an undetermined parameter. This model is more efficient than previous models based on flux linkage against current (with rotor position as a parameter). It also avoids the concept of inductance, which is, perhaps, unnecessary and inappropriate in a machine with such pronounced magnetic nonlinearities. The model is suitable, and has been widely use, for CAD and performance analysis, particularly at the stage of ‘sizing’, or initial estimation, where accuracy can be traded for speed of computation. The model includes all the significant electromagnetic and dynamic characteristics of the SR motor. Its accuracy can be enhanced by means of correction factors derived from only two or three points on magnetisation curves that have been accurately measured or calculated by finite elements, permitting economy in the use of data that is expensive to generate. Because the magnetisation curves do not need to be precalculated, stored or curvefitted, the algorithms are extremely fast. The model is computationally linked with a very accurate companion based on cubic-spline models of externally generated magnetisation curves. A piecewise analytical formula for instantaneous torque is also included, permitting the rapid (though approximate) calculation of mathematically smooth torque waveforms. Because the magnetisation curves do not need to be precalculated, stored or curve-fitted, the algorithms are extremely fast. The paper also presents a new method for calculating the unaligned magnetisation curve, based on dual-energy principles. Results are compared with test data for a range of motors.

Ray tracing trimmed rational surface patches

Abstract: This paper presents a new algorithm for computing the points at which a ray intersects a rational Bezier surface patch, and also an algorithm for determining if an intersection point lies within a region trimmed by piecewise Bezier curves. Both algorithms are based on a recent innovation known as Bezier clipping, described herein. The intersection algorithm is faster than previous methods for which published performance data allow reliable comparison. It robustly finds all intersections without requiring special preprocessing.

Piecewise solenoidal vector fields and the Stokes problem

Abstract: Nonconforming finite element approximations to solutions of the Stokes equations are constructed. Optimal rates of convergence are proved for the velocity and pressure approximations. For the pressure approximation, $C^0 $ piecewise polynomial functions are used. The class of vector fields used to approximate the velocity field have piecewise polynomial components, discontinuous across interelement boundaries. On each “triangle” these vector fields satisfy the incompressibility condition pointwise. It is shown that these piecewise solenoidal vector fields possess optimal approximation properties to smooth solenoidal vector fields on domains with curved boundaries.

Optimal finite-thrust spacecraft trajectories using collocation and nonlinear programming

Abstract: A new method is described for the determination of optimal spacecraft trajectories in an inverse-square field using finite, fixed thrust. The method employs a recently developed optimization technique which uses a piecewise polynomial representation for the state and controls, and collocation, thus converting the optimal control problem into a nonlinear programming problem, which is solved numerically. This technique has been modified to provide efficient handling of those portions of the trajectory which can be determined analytically, i.e., the coast arcs. Among the problems that have been solved using this method are optimal rendezvous and transfer (including multirevolution cases) and optimal multiburn orbit insertion from hyperbolic approach.

Approximating and learning unknown mappings using multilayer feedforward networks with bounded weights

Abstract: It is shown that feedforward networks having bounded weights are not undesirable restricted, but are in fact universal approximators, provided that the hidden-layer activation function belongs to one of several suitable broad classes of functions: polygonal functions, certain piecewise polynomial functions, or a class of functions analytic on some open interval. These results are obtained by trading bounds on network weights for possible increments to network complexity, as indexed by the number of hidden nodes. The hidden-layer activation functions used include functions not admitted by previous universal approximation results, so the present results also extend the already broad class of activation functions for which universal approximation results are available. A theorem which establishes the approximate ability of these arbitrary mappings to learn when examples are generated by a stationary ergodic process is given

Zeta functions and transfer operators for piecewise monotone transformations

Abstract: Given a piecewise monotone transformationT of the interval and a piecewise continuous complex weight functiong of bounded variation, we prove that the Ruelle zeta function ζ(z) of (T, g) extends meromorphically to {∣z∣<θ-1} (where θ=lim ∥g°Tn-1...g°Tg∥ ∞ 1/n ) and thatz is a pole of ζ if and only ifz −1 is an eigenvalue of the corresponding transfer operator L. We do not assume that L leaves a reference measure invariant.

Ellipticity and deformations with discontinuous gradients in finite elastostatics

Abstract: Loss of ellipticity of the equilibrium equations of finite elastostatics is closely related to the possible emergence of elastostatic shocks, i.e., deformations with discontinuous gradients. In certain situations where constitutive response functions are essentially one-dimentional, such as anti-plane shear or bar theories, strong ellipticity is closely related to convexity of the elastic potential and invertibility of certain constitutive response functions. The present work addresses the analogous issues within the context of three dimensional elastostatics of compressible but not necessarily isotropic hyperelastic materials. A certain direction-dependent resolution of the deformation gradient is introduced and its existence and uniqueness for a given direction are established. The elastic potential is expressed as a function of kinematic variables arising from this resolution. Strong ellipticity is shown to be equivalent to the positive definiteness of the Hessian matrix of this function, thus sufficing for its strict convexity. The underlying variables are interpretable physically as simple shears and extensions. Their work-conjugates define a traction response mapping. It is shown that discontinuous deformation gradients are sustainable if and only if this mapping fails to be invertible. This result is explicit, in the sense that it characterizes the set of all possible piecewise homogeneous deformations given the elastic potential function.

On Shape-Preserving Interpolation and Semi-Lagrangian Transport.

Abstract: A large number of interpolation schemes are evaluated in terms of their relative accuracy. The large number of schemes arises by considering combinations of interpolating forms (piecewise cubic polynomials, piecewise rational quadratic and cubic polynomials, and piecewise quadratic Bernstein polynomials), derivative estimates (Akima, Hyman, arithmetic, geometric and harmonic means, and Fritsch–Butland), and modification of these estimates required to ensure monotonicity and/or convexity upon the interpolant. Shape-preserving methods maintain in the interpolant the monotonicity and/or convexity implied in the discrete data.The schemes are first compared by evaluating their ability to interpolate evenly spaced data drawn from three test shapes (Gaussian, cosine bell, and triangle) at two resolutions. Details of the cosine bell tests are presented in this paper. Details of the other tests are presented in a companion technical report. Of the monotonic interpolants, the following are the most accurate: (1) The Hermite cubic interpolant with the derivative estimate of Hyman modified to produce monotonicity as suggested by de Boor and Swartz. (2) The second version of the rational cubic spline suggested by Delbourgo and Gregory, with the derivative estimate of Hyman modified to produce monotonicity. (3) The piecewise quadratic Bernstein polynomials suggested by McAllistor and Roulier with the derivative estimate of Hyman again modified. Imposing strict monotonicity at discrete extrema introduces significant errors. More accurate interpolations result if this requirement is relaxed at extrema. The Hermite cubic interpolant is improved by relaxing the strict monotonicity constraint to one suggested by Hyman at extrema. In a like manner, the accuracy of the rational and piecewise quadratic Bernstein polynomial interpolants can be improved by requiring only that convexity/concavity be satisfied rather than monotonicity.Some of the more accurate interpolants are incorporated into the semi-Lagrangian transport method and tested by examining the accuracy of the solution to one-dimensional advection of test shapes in a uniform velocity field. The semi-Lagrangian method using monotonic interpolators provides monotonic solutions. The semi-Lagrangian method using interpolators that maintain convex/concave constraints give solutions that are essentially nonoscillatory. The monotonic forms damp the solution with time, more so for narrow than broad structures. The best monotonic forms are the Hermite cubic interpolant with the Akima or Hyman derivative estimates modified to produce monotonicity with $C^0 $ continuity. The corresponding $C^1 $ continuous forms have unacceptable phase errors with the Hermite interpolant. The rational cubic with the Hyman derivative estimate modified to produce monotonicity is comparable to the $C^0 $ Hermite form described above. The $C^1 $ rational form does not have the phase error seen in the $C^1 $ Hermite interpolant. The essentially nonoscillatory forms damp much less than the monotonic forms. The solutions that used rational cubic interpolants with a Hyman derivative estimate modified to satisfy a convexity/concavity constraint were the most satisfactory of the shape-preserving schemes.

A new optimal multirate control of linear periodic and time-invariant systems

Abstract: The optimal multirate design of linear, continuous-time, periodic and time-invariant systems is considered. It is based on solving the continuous linear quadratic regulation (LQR) problem with the control being constrained to a certain piecewise constant feedback. Necessary and sufficient conditions for the asymptotic stability of the resulting closed-loop system are given. An explicit multirate feedback law that requires the solution of an algebraic discrete Riccati equation is presented. Such control is simple and can be easily implemented by digital computers. When applied to linear time-invariant systems, multirate optimal feedback optimal control provides a satisfactory response even if the state is sampled relatively slowly. Compared to the classical single-rate sampled-data feedback in which the state is always sampled at the same rate, the multirate system can provide a better response with a considerable reduction in the optimal cost. In general, the multirate scheme offers more flexibility in choosing the sampling rates. >

Error analysis of some finite element methods for the Stokes problem

Abstract: We prove the optimal order of convergence for some two-dimensional finite element methods for the Stokes equations. First we consider methods of the Taylor-Hood type: the triangular P3 P2 element and the Qk Qk-1 I k > 2, family of quadrilateral elements. Then we introduce two new low-order methods with piecewise constant approximations for the pressure. The analysis is performed using our macroelement technique, which is reviewed in a slightly altered form.

On shape-preserving interpolation and semi-Lagrangian transport

Abstract: A large number of interpolation schemes are evaluated in terms of their relative accuracy. The large number of schemes arises by considering combinations of interpolating forms (piecewise cubic polynomials, piecewise rational quadratic and cubic polynomials, and piecewise quadratic Bernstein polynomials), derivative estimates (Akima, Hyman, arithmetic, geometric and harmonic means, and Fritsch–Butland), and modification of these estimates required to ensure monotonicity and/or convexity upon the interpolant. Shape-preserving methods maintain in the interpolant the monotonicity and/or convexity implied in the discrete data.The schemes are first compared by evaluating their ability to interpolate evenly spaced data drawn from three test shapes (Gaussian, cosine bell, and triangle) at two resolutions. Details of the cosine bell tests are presented in this paper. Details of the other tests are presented in a companion technical report. Of the monotonic interpolants, the following are the most accurate: (1) The Hermite cubic interpolant with the derivative estimate of Hyman modified to produce monotonicity as suggested by de Boor and Swartz. (2) The second version of the rational cubic spline suggested by Delbourgo and Gregory, with the derivative estimate of Hyman modified to produce monotonicity. (3) The piecewise quadratic Bernstein polynomials suggested by McAllistor and Roulier with the derivative estimate of Hyman again modified. Imposing strict monotonicity at discrete extrema introduces significant errors. More accurate interpolations result if this requirement is relaxed at extrema. The Hermite cubic interpolant is improved by relaxing the strict monotonicity constraint to one suggested by Hyman at extrema. In a like manner, the accuracy of the rational and piecewise quadratic Bernstein polynomial interpolants can be improved by requiring only that convexity/concavity be satisfied rather than monotonicity.Some of the more accurate interpolants are incorporated into the semi-Lagrangian transport method and tested by examining the accuracy of the solution to one-dimensional advection of test shapes in a uniform velocity field. The semi-Lagrangian method using monotonic interpolators provides monotonic solutions. The semi-Lagrangian method using interpolators that maintain convex/concave constraints give solutions that are essentially nonoscillatory. The monotonic forms damp the solution with time, more so for narrow than broad structures. The best monotonic forms are the Hermite cubic interpolant with the Akima or Hyman derivative estimates modified to produce monotonicity with $C^0 $ continuity. The corresponding $C^1 $ continuous forms have unacceptable phase errors with the Hermite interpolant. The rational cubic with the Hyman derivative estimate modified to produce monotonicity is comparable to the $C^0 $ Hermite form described above. The $C^1 $ rational form does not have the phase error seen in the $C^1 $ Hermite interpolant. The essentially nonoscillatory forms damp much less than the monotonic forms. The solutions that used rational cubic interpolants with a Hyman derivative estimate modified to satisfy a convexity/concavity constraint were the most satisfactory of the shape-preserving schemes.

Optimal control of piecewise deterministic Markov processes.

Abstract: This thesis describes a complete theory of optimal control of piecewise deterministic Markov processes under weak assumptions. The theory consists of a description of the processes, a nonsmooth stochastic maximum principle as a necessary optimality condition, a generalized Bellman-Hamilton-Jacobi necessary and sufficient optimality condition involving the Clarke generalized gradient, existence results and regularity properties of the value function. The impulse control problem is transformed to an equivalent optimal dynamic control problem. Cost functions are subject only to growth conditions.

On the G 1 continuity of piecewise Be´zier surfaces: a review with new results

Abstract: The tensor product Bezier patch is currently one of the most widely used models in CAGD for free-form surface modelling. In a piecewise representation the patches are distributed on a mesh. If any piecewise surface has to be modelled using non-degenerate Bezier patches, it is necessary to use a mesh of unrestricted topology, i.e. with any number of patches meeting at a node. In order to obtain a smooth surface the geometric continuities between adjacent patches must be controlled. A lot of research has been devoted to this problem and various solutions have been proposed. This paper reviews the various studies dealing with the G 1 smooth connection between adjacent Bezier patches and those dealing with the techniques of free-form surface modelling using Bezier patches. First, the constraints guaranteeing G 1 continuity between two adjacent Bezier patches are analysed. This analysis reveals several important geometric properties hidden in these constraints, usually expressed analytically. From these results the G 1 smooth connection between N ( N > 2) patches meeting at a common corner is studied. The resulting G 1 constraints are deduced, and it is shown how to satisfy them in the definition of the control points of the Bezier patches. Degeneration problems around a four-patch corner adjacent to a non-four-patch corner are then analysed, and the supplementary conditions to be satisfied are developed in order to guarantee the G 1 continuity around a degenerate four-patch corner. After that, the various methods proposed to model complex surfaces using Bezier patches are reviewed. Based on this analysis, new alternative approaches for modelling free-form G 1 continuous surfaces are presented.

Generalizing smoothness constraints from discrete samples

Abstract: We study how certain smoothness constraints, for example, piecewise continuity, can be generalized from a discrete set of analog-valued data, by modifying the error backpropagation, learning algorithm. Numerical simulations demonstrate that by imposing two heuristic objectives — (1) reducing the number of hidden units, and (2) minimizing the magnitudes of the weights in the network — during the learning process, one obtains a network with a response function that smoothly interpolates between the training data.

On the periodicity of symbolic observations of piecewise smooth discrete-time systems

Abstract: A study is made of the behavior of discrete-time systems composed of a set of smooth transition maps coupled by a quantized feedback function. The feedback function partitions the state space into disjoint regions and assigns a smooth transition function to each region. The main result is that under a constraint on the norm of the derivative of the transition maps, a bounded state trajectory with limit points in the interior of the switching regions leads to a region index sequence that is eventually periodic. Under these assumptions, it is shown that eventually the feedback function is determined by a finite state automaton. A similar result is proved in the case of finite state dynamic feedback. >

Analog hardware for detecting discontinuities in early vision

Abstract: The detection of discontinuities in motion, intensity, color, and depth is a well-studied but difficult problem in computer vision [6]. We discuss the first hardware circuit that explicitly implements either analog or binary line processes in a deterministic fashion. Specifically, we show that the processes of smoothing (using a first-order or membrane type of stabilizer) and of segmentation can be implemented by a single, two-terminal nonlinear voltage-controlled resistor, the “resistive fuse”; and we derive its current-voltage relationship from a number of deterministic approximations to the underlying stochastic Markov random fields algorthms. The concept that the quadratic variation functionals of early vision can be solved via linear resistive networks minimizing power dissipation [37] can be extended to non-convex variational functionals with analog or binary line processes being solved by nonlinear resistive networks minimizing the electrical co-content. We have successfully designed, tested, and demonstrated an analog CMOS VLSI circuit that contains a 1D resistive network of fuses implementing piecewise smooth surface interpolation. We furthermore demonstrate the segmenting abilities of these analog and deterministic “line processes” by numerically simulating the nonlinear resistive network computing optical flow in the presence of motion discontinuities. Finally, we discuss various circuit implementations of the optical flow computation using these circuits.

Optimal control computation for differential-algebraic process systems with general constraints

Abstract: This paper is concerned with the combined problem of optimal parameter selection and control for differential-algebraic system (DASs) involving various constraints. The control parameterization technique of using piecewise constant functions is used to approximate the original problem into a sequence of finite-dimensional parameter selection problems. Based on characterizing the Hamiltonian function and adjoint system associated with a DAS and transforming different types of constraints into a cononical function which has the same form as the cost functional, a unified gradient computation algorithm for the controls and parameters is derived. This algorithm makes the resulting approximate problems be effectively solved by gradient-based optimization methods. As a specific example, the singular control problem of finding the optimal feeding policy for a fed-batch fermentation process governed by the product and substrate inhibited specific growth and product formation kinetics is solved. The computed resul...

Observability analysis of piece-wise constant systems with application to inertial navigation

Abstract: A method for analyzing the observability of time-varying linear systems which can be modeled as piecewise constant systems is presented. An observability matrix for such systems is developed for continuous and discrete time representations. A stripped observability matrix is introduced which simplifies the analysis in cases where the use of this matrix is legitimate. This approach circumvents the difficulty associated with the investigation of the observability Gramian of time-varying linear systems. It is shown that instead of investigating a Gramian, only a constant observability matrix needs to be investigated. Moreover, it is shown that if certain conditions on the null space of the dynamics matrix of the system are met, the observability matrix can be greatly simplified. A step-by-step observability analysis procedure is presented for this case. The method is applied to the analysis of in-flight alignment of inertial navigation systems whose estimability is known to be enhanced by maneuvers. >

Computational schemes for large-scale problems in extended linear-quadratic programming

Abstract: Numerical approaches are developed for solving large-scale problems of extended linear-quadratic programming that exhibit Lagrangian separability in both primal and dual variables simultaneously. Such problems are kin to large-scale linear complementarity models as derived from applications of variational inequalities, and they arise from general models in multistage stochastic programming and discrete-time optimal control. Because their objective functions are merely piecewise linear-quadratic, due to the presence of penalty terms, they do not fit a conventional quadratic programming framework. They have potentially advantageous features, however, which so far have not been exploited in solution procedures. These features are laid out and analyzed for their computational potential. In particular, a new class of algorithms, called finite-envelope methods, is described that does take advantage of the structure. Such methods reduce the solution of a high-dimensional extended linear-quadratic program to that of a sequence of low-dimensional ordinary quadratic programs.

Deterministic minimal time vessel routing

Abstract: We develop general methodologies for the minimal time routing problem of a vessel moving in stationary or time dependent environments, respectively. Local optimality considerations, combined with global boundary conditions, result in piecewise continuous optimal policies. In the stationary case, the velocity of the traveling vessel within each subregion depends only on the direction of motion. Variational calculus is used to derive the geometry of piecewise linear extremals. For the time dependent problem, the speed of the vessel within each subregion is assumed to be a known function of time and the direction of motion. Optimal control theory is used to reveal the nature of piecewise continuous optimal policies.

Segmentation as the search for the best description of the image in terms of primitives

Abstract: A paradigm is presented for the segmentation of images into piecewise continuous patches. Data aggregation is performed via model recovery in terms of variable-order bivariate polynomials using iterative regression. All the recovered models are candidates for the final description of the data. Selection of the models is achieved through a maximization of the quadratic Boolean problem. The procedure can be adapted to prefer certain kinds of descriptions (one which describes more data points, or has smaller error, or has a lower order model). A fast optimization procedure for model selection is discussed. The approach combines model extraction and model selection in a dynamic way. Partial recovery of the models is followed by the optimization (selection) procedure where only the best models are allowed to develop further. The results are comparable with the results obtained when using the selection module only after all the models are fully recovered, while the computational complexity is significantly reduced. The procedure was tested on real range and intensity images. >

Smooth mesh interpolation with cubic patches

Abstract: An algorithm for the local interpolation of a mesh of cubic curves with 3- and 4-sided facets by a piecewise cubic C 1 surface is stated and illustrated by an implementation. Precise necessary and sufficient conditions for oriented tangent-plane continuity between adjacent patches are derived, and the explicit constructions are characterized by the degree of the three scalar weight functions that relate the versal to the two transversal derivatives. The algorithm fully exploits the possibility of reparametrization by choosing all three weight functions nonconstant and not just degree-raising polynomials. The construction is local and consists mainly of averaging. The only systems to be solved are linear and of size 2 × 2. The algorithm guarantees interpolating surfaces without cusps and has a simple, implemented extension to n -sided facets.

A Spectral General Circulation Model Using a Piecewise-Constant Finite-Element Representation on a Hybrid Vertical Coordinate System

Abstract: A numerical scheme for the vertical discretization of primitive equations in a generalized pressure-type coordinate is developed through application of the Galerkin formalism with piecewise-constant finite elements: this methodology affords an elegant—and direct—mean of formulating conservative discretization schemes without the arbitrariness that usually characterizes the development of finite differences. The form of the resulting semidiscrete equations is equivalent to some second-order accurate finite-difference approximation to the continuous equations. Flexibility of this scheme in the choice of different layers for projecting the thermodynamic and momentum variables effectively allows for staggering of these variables in the vertical. Numerical integrations performed with this scheme at various vertical resolutions have revealed the sensitivity of the simulated circulation to resolution in the lower stratosphere. We found that application of the “lid” upper boundary condition at a finite h...

Smoothed perturbation analysis for a class of discrete-event systems

Abstract: A gradient-estimation procedure for a general class of stochastic discrete-event systems is developed. In contrast to most previous work, the authors focus on performance measures whose realizations are inherently discontinuous (in fact, piecewise constant) functions of the parameter of differentiation. Two broad classes of finite-horizon discontinuous performance measures arising naturally in applications are considered. Because of their discontinuity, these important classes of performance measures are not susceptible to infinitesimal perturbation analysis (IPA). Instead, the authors apply smoothed perturbation analysis, formalizing it and generalizing it in the process. Smoothed perturbation analysis uses conditional expectations to smooth jumps. The resulting gradient estimator involves two factors: the conditional rate at which jumps occur, and the expected effect of a jump. Among the types of performance measures to which the methods can be applied are transient state probabilities, finite-horizon throughputs, distributions on arrival, and expected terminal cost. >

Hyperbolicity and invariant measures for general C2 interval maps satisfying the Misiurewicz condition

Abstract: In this paper we will show that piecewise C2 mappings / on [0,1] or S1 satisfying the so-called Misiurewicz conditions are globally expanding (in the sense defined below) and have absolute continuous invariant probability measures of positive entropy. We do not need assumptions on the Schwarzian derivative of these maps. Instead we need the conditions that / is piecewise C2, that all critical points of / are "non-flat," and that / has no periodic attractors. Our proof gives an algorithm to verify this last condition. Our result implies the result of Misiurewicz in [Mi] (where only maps with negative Schwarzian derivatives are considered). Moreover, as a byproduct, the present paper implies (and simplifies the proof of) the results of Mane in [Ma], who considers general C2 maps (without conditions on the Schwarzian derivative), and restricts attention to points whose forward orbit stay away from the critical points. One of the main complications will be that in this paper we want to prove the existence of invariant measures and therefore have to consider points whose iterations come arbitrarily close to critical points. Misiurewicz deals with this problem using an assumption on the Schwarzian derivative of the map. This assumption implies very good control of the non-linearity of /", even for high n. In order to deal with this non-linearity, without an assumption on the Schwarzian derivative, we use the tools of [M.S.]. It will turn out that the estimates we obtain are so precise that the existence of invariant measures can be proved in a very simple way (in some sense much simpler than in [Mi]). The existence of these invariant measures under such general conditions was already conjectured a decade ago.