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.

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.

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.

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.

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. >

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.

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.

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.

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. >

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.

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.

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. >

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.

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.

Abstract: For integral equations of the second kind with appropriate smoothness of both kernel and exact solution, it is known that the iterated–collocation method based on discontinuous piecewise polynomials of order r, and on collocation at the Gauss points, can exhibit a uniform (superconvergent) error of order $2r$. This paper shows that the order of the iterated–collocation approximation on an arbitrary mesh can be further improved by one step of Richardson extrapolation, assuming the calculation to be repeated with each subinterval halved.

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.

Abstract: The study of the finite element approximation to nonlinear second order elliptic boundary value problems with discontinuous coefficients is presented in the case of mixed Dirichlet-Neumann boundary conditions. The change in domain and numerical integration are taken into account. With the assumptions which guarantee that the corresponding operator is strongly monotone and Lipschitz-continuous the following convergence results are proved: 1. the rate of convergenceO(h ?) if the exact solutionu?H 1 (Ω) is piecewise of classH 1+? (0