Showing papers on "Piecewise published in 1987"
TL;DR: An hierarchy of uniformly high-order accurate schemes is presented which generalizes Godunov's scheme and its second- order accurate MUSCL extension to an arbitrary order of accuracy.
2,891 citations
TL;DR: In this article, an initial path estimate is perturbed using a geometric interpretation of the ray equations, and the travel time along the path is minimized in a piecewise fashion, iteratively performed until the travel times converges within a specified limit.
Abstract: A new approximate algorithm for two-point ray tracing is proposed and tested in a variety of laterally heterogeneous velocity models. An initial path estimate is perturbed using a geometric interpretation of the ray equations, and the travel time along the path is minimized in a piecewise fashion. This perturbation is iteratively performed until the travel time converges within a specified limit. Test results show that this algorithm successfully finds the correct travel time within typical observational error much faster than existing three-dimensional ray tracing programs. The method finds an accurate ray path in a fully three-dimensional form even where lateral variations in velocity are severe. Because our algorithm utilizes direct minimization of the travel time instead of solving the ray equations, a simple linear interpolation scheme can be employed to compute velocity as a function of position, providing an added computational advantage.
814 citations
TL;DR: This paper analyzes the performance of the piecewise polynomial solution representation of B-splines in COLSYS and implements a basis replacement, demonstrating the improvement in performance.
Abstract: The numerical approximation of mixed order systems of multipoint value ordinary differential equations by collocation requires appropriate representation of the piecewise polynomial solutions. B-splines were originally implemented in the general purpose code COLSYS, but better alternatives exist. One promising alternative as proposed by Osborne and discussed by Ascher, Pruess and Russell. In this paper we analyze the performance of the latter solution representation for cases not previously covered, where the mesh is not necessarily dense. This analysis and other considerations have led us to implement a basis replacement in COLSYS and we discuss some implementation details. Numerical results are given which demonstrate the improvement in performance of the code.
243 citations
TL;DR: For a model convection-dominated singularly perturbed convectiondiffusion prob- lem, it is shown in this article that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion.
Abstract: For a model convection-dominated singularly perturbed convection-diffusion prob- lem, it is shown that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion. The ensuing method with piecewise linear elements converges with a pointwise accuracy of almost h 5/4 under local smoothness assumptions. 1. Introduction. The streamline diffusion method is a finite element method for convection-dominated convection-diffusion problems which combines formal high accuracy with decent stability properties. The method was introduced in the case of stationary problems by Hughes and Brooks (7), cf. Raithby and Torrance (14) and Wahlbin (15) for earlier thoughts in this direction. The mathematical analysis of the method was started in Johnson and Navert (8) and continued with extensions to, e.g., time-dependent problems in Navert (12), Johnson, Navert and Pitkaranta (9) and Johnson and Saranen (10). In these papers local error estimates in L2 of order O(h k 1/2), in regions of smoothness, with piecewise polynomial finite elements of degree k, were derived, together with estimates stating, as a typical example, that in the zero diffusion limit a sharp discontinuity in the exact solution across a streamline will be captured in a numerical crosswind layer of width 0(h1/2), essentially. The purpose of the present paper is first to improve the result just mentioned on numerical crosswind smear to 0(h3/4). The improvement from 0(h1/2) to 0(h3/4) is obtained by adding a small amount, 0(h3/2), of artificial crosswind diffusion to the method. In the piecewise linear case (k = 1) this does not destroy the known O( h3/2) accuracy in L2 in smooth regions. Using our first result, we then obtain our second main result, localized pointwise error estimates of order 0(h5/4) in regions of smoothness. (The previously known best pointwise error estimate in the piecewise linear situation is 0(h1/2).) Another consequence is a global L,-estimate of order 0(h'/2) in the presence of typical crosswind and downwind singularities. We shall consider the model problem of finding u = u(x, y) such that
164 citations
TL;DR: In this article, the authors consider spaces of piecewise polynomials of degreed defined over a triangulation of a polygonal domain and possessing continuous derivatives globally.
Abstract: We consider spaces of piecewise polynomials of degreed defined over a triangulation of a polygonal domain and possessingr continuous derivatives globally. Morgan and Scott constructed a basis in the case wherer=1 andd≥5. The purpose of this paper is to extend the dimension part of their result tor≥0 andd≥4r+l. We use Bezier nets as a crucial tool in deriving the dimension of such spaces.
135 citations
01 Jul 1987
TL;DR: It is shown that the bound for the residual tracking error depends on the speed of the plant parameter variations in such a way that as these parameters become constant the bound reduces to zero.
Abstract: In this paper we consider the Model Reference Control and Model Reference Adaptive Control problem of a class of linear time-varying plants with bounded, piecewise smooth parameters. We first introduce a new controller structure that solves the Model Reference Control problem, with zero tracking error, in the case of smooth slowly time-varying plant parameters. When the plant parameters are unknown we use the robust adaptive law, proposed in [12], to adjust the controller parameters and establish boundedness of all signals in the adaptive loop and good tracking properties, provided that the smooth time-variations of the plant parameters are sufficiently slow and the minimum time between discontinuities is large enough. Furthermore, we show that when the plant parameters become constant, the tracking error converges asymptotically to zero.
132 citations
TL;DR: A technique based on the Clough-Tocher subtriangulation is described which determines piecewise cubic mapping functions for image registration which is a piecewise combination of local mapping functions.
104 citations
TL;DR: In this article, boundary approximation techniques for homogeneous self-adjoint elliptic equations are described and convergence of such approximations is proved and error estimates are derived in a natural norm.
Abstract: Boundary approximation techniques are described for solving homogeneous self-adjoint elliptic equations. Piecewise expansions into particular solutions are used which approximate both the boundary and interface conditions in a least squares sense. Convergence of such approximations is proved and error estimates are derived in a natural norm. Numerical experiments are reported for the singular Motz problem which yield extremely accurate solutions with only a modest computational effort.
101 citations
TL;DR: A technique is developed that captures this local structure while simultaneously locating the discontinuities, and it is demonstrated that these tasks are in fact inseparable.
Abstract: The detailed structure of intensities in the local neighborhood of an edge can often indicate the nature of the physical event givinig rise to that edge. We argue that the limit, as we approach arbitrarily close to either side of an edge, of such image parameters as type of texture, texture gradient, color, appropriate directional derivatives of intensity, etc., is a key aspect of this structure. However, the general problem of capturing this local structure is surprisingly complex. Thus, we restrict ourselves in this paper to a relatively simple domain?one-dimensional cuts through idealized images modeled by piecewise smooth (C1) functions corrupted by Gaussian noise. Within this domain, we define local structure to be the limit of the uncorrupted intensity and of its derivatives as we approach arbitrarily close to either side of a discontinuity. We develop a technique that captures this local structure while simultaneously locating the discontinuities, and demonstrate that these tasks are in fact inseparable. The technique is an extension, using estimation theory, of the classical definition of discontinuity. It handles, in a consistent fashion, both jump discontinuities in the function and jump discontinuities in its first derivative (so-called step-edges are a special case of the former and roof-edges of the latter). It also integrates, again in a consistent fashion, information derived from a number of different neighborhood sizes.
99 citations
TL;DR: In this paper, the authors consider a family of two-stage stochastic control problems with nonclassical information patterns, and show that whenever the performance index does not contain a product term between the decision variables, the optimal solution is linear in the observation variables.
Abstract: In this note we consider a parameterized family of two-stage stochastic control problems with nonclassical information patterns, which includes the well-known 1968 counterexample of Witsenhausen. We show that whenever the performance index does not contain a product term between the decision variables, the optimal solution is linear in the observation variables. The parameter space can be partitioned into two regions in one of which the optimal solution is linear, whereas in the other it is inherently nonlinear. Extensive computations using two-point piecewise constant policies and linear plus piecewise constant policies provide numerical evidence that nonlinear policies may indeed outperform linear policies when the product term is present.
96 citations
TL;DR: In this article, it was shown that a bounded semialgebraic function can be reparametrized in such a way that all the derivatives up to a fixed orderk, with respect to new coordinates, are small, and the number of pieces is effectively bounded.
Abstract: We prove that a bounded semialgebraic function can be (piecewise) reparametrized in such a way that all the derivatives up to a fixed orderk, with respect to new coordinates, are small, and the number of pieces is effectively bounded.
TL;DR: By subdividing eachn-simplex in an appropriate way, it is shown how to solve the interpolation problem using piecewise cubic polynomials.
Abstract: We consider the problem of C1 interpolation to data given at the vertices and mid-edge points of a tessellation in Rn. The given data are positional and gradient information at the vertices, together with the gradient at the mid-edge points. By subdividing eachn-simplex in an appropriate way, we show how to solve the interpolation problem using piecewise cubic polynomials. The subdivision process is the key to the method and is inductive in nature. It is systematically built up from the two-dimensional case where a variant of the well-known Clough-Tocher element is used.
TL;DR: In this paper, the dimension of the space of bivariate piecewise quartic polynomials defined on a triangulation of a connected polygonal domain is established.
Abstract: We establish the dimension of the space of $C^1 $ bivariate piecewise quartic polynomials defined on a triangulation of a connected polygonal domain. Our approach is to construct a minimal determining set and an associated explicit basis for the space. For general triangulations, the minimal determining set must be defined globally.
TL;DR: A new almost optimal error estimate is proved and a related new algorithm for automatic time step control is presented that is very simple but yet is efficient and gives control of the global error.
Abstract: In this note we extend results by one of the authors on time discretization error estimates and related automatic time step control for stiff ordinary differential equations to the case of a nonlinear parabolic problem. The method for time discretization is the so-called Discontinuous Galerkin method based on using piecewise polynomials of degree $q \geqq 0$. We consider in this note the case $q = 0$ corresponding to a variant of the backward Euler method. We prove a new almost optimal error estimate and present a related new algorithm for automatic time step control. This algorithm is very simple but yet is efficient and gives control of the global error.
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01 Jan 1987
TL;DR: Theoretical methods for piecewise polynomial surfaces and recursive algorithms forPolynomial evaluation are described.
Abstract: Preface A brief overview 1. Matrix subdivision 2: Stationary subdivision 3: Piecewise polynomial curves 4: Geometric methods for piecewise polynomial surfaces 5: Recursive algorithms for polynomial evaluation.
TL;DR: In this paper, a two-step Numerov algorithm for the Schrodinger equation was proposed. But the efficiency of the algorithm is not as good as that of the piecewise perturbation method.
TL;DR: It is shown that human subjects are incapable of producing with the arm, in free space, planned or extemporaneously drawn trajectories in which the plane of wrist motion changes smoothly or continuously.
01 Sep 1987
TL;DR: In this paper, the VSAERO low-order panel method is described for the calculation of subsonic aerodynamic characteristics of general configurations, based on piecewise constant doublet and source singularities.
Abstract: The VSAERO low order panel method formulation is described for the calculation of subsonic aerodynamic characteristics of general configurations. The method is based on piecewise constant doublet and source singularities. Two forms of the internal Dirichlet boundary condition are discussed and the source distribution is determined by the external Neumann boundary condition. A number of basic test cases are examined. Calculations are compared with higher order solutions for a number of cases. It is demonstrated that for comparable density of control points where the boundary conditions are satisfied, the low order method gives comparable accuracy to the higher order solutions. It is also shown that problems associated with some earlier low order panel methods, e.g., leakage in internal flows and junctions and also poor trailing edge solutions, do not appear for the present method. Further, the application of the Kutta conditions is extremely simple; no extra equation or trailing edge velocity point is required. The method has very low computing costs and this has made it practical for application to nonlinear problems requiring iterative solutions for wake shape and surface boundary layer effects.
TL;DR: A piecewise linear finite-element technique is proposed to solve the integral equations that relate charges to potentials and lead to the desired capacitance matrix and it is shown by example that the proposed method is both less complex than methods based on piecewise constant surface charge distributions and equally accurate.
Abstract: The problem of reducing the complexity of parasitic capacitance evaluation of interconnection lines in a multilevel stratified dielectric medium (a good approximation for VLSI) is considered. We start out with a review of the Green's function method for the Si-SiO 2 composite and its derivation via the Fourier integral approach. Next, a piecewise linear finite-element technique is proposed to solve the integral equations that relate charges to potentials and lead to the desired capacitance matrix. We show by example that the proposed method is both less complex than methods based on piecewise constant surface charge distributions and equally accurate. This supports the accuracy and usefulness of the technique for IC design.
TL;DR: In this paper, a simple superconvergent scheme which recovers the gradient when solving a second-order elliptic problem in the plane by the usual linear elements was proposed. But the recovered gradient globally approximates the true gradient even by one order of accuracy higher in the L 2 -norm than the piecewise constant gradient of the Ritz-Galerkin solution.
01 Jan 1987
TL;DR: Preliminary results on the extension of high-order accurate essentially non-oscillatory (ENO) schemes to the solution of hyperbolic systems of conservation laws in 2D are presented.
Abstract: In this paper we present preliminary results on the extension of high-order accurate essentially non-oscillatory (ENO) schemes to the solution of hyperbolic systems of conservation laws in 2D. The design involves an essentially non-oscillatory piecewise polynomial reconstruction of the solution from its cell averages, solution in the small of the resulting piecewise polynomial initial value problem, and averaging of this solution over each cell. Unlike standard finite difference methods this procedure uses an adaptive stencil of grid points and consequently the resulting schemes are highly nonlinear.
TL;DR: A new piecewise linear upper bound is presented on this function, called the network recourse function, which is compared to the standard Madansky bound, and is shown computationally to be a little weaker, but much faster to find.
Abstract: We consider the optimal value of a pure minimum cost network flow problem as a function of supply, demand and arc capacities. We present a new piecewise linear upper bound on this function, which is called the network recourse function. The bound is compared to the standard Madansky bound, and is shown computationally to be a little weaker, but much faster to find. The amount of work is linear in the number of stochastic variables, not exponential as is the case for the Madansky bound. Therefore, the reduction in work increases as the number of stochastic variables increases. Computational results are presented.
TL;DR: A natural approach to constructing these bases explicitly is presented, and it is shown that it succeeds for r ≤ 3 but fails for r > 3.
TL;DR: In this article, different representations of piecewise affine linear functions as a difference of convex convex piecewise linear functions are given, where the difference is defined as a convex function.
Abstract: We give different representations of piecewise affine linear functions as a difference of convex piecewise affine linear functions.
TL;DR: A method for blending implicitly defined surfaces is proposed where the function defining the blending surface is constructed from the functions defining the surfaces to be blended with the aid of a piecewise quadric function.
TL;DR: An asymptotic error analysis of the conventional discrete dynamic programming method is presented, and the error is shown to be of the order of the state discretization interval (AS), a result with significant implications in the optimization of multistate systems where the "curse of dimensionality" restricts the number of states to a relatively small number.
Abstract: An asymptotic error analysis of the conventional discrete dynamic programming (DDP) method is presented, and upper bounds of the error in the control policy (i.e., the difference of the estimated and true optimal control) at each operation period are computed. This error is shown to be of the order of the state discretization interval (AS), a result with significant implications in the optimization of multistate systems where the "curse of dimensionality" restricts the number of states to a relatively small number. The error in the optimal cost varies with AS 2. The analysis provides useful insights into the effects of state discretization on calculated control and cost functions, the comparability of results from different discretizations, and criteria about the required number of nodes. In an effort to reduce the discretization error in the case of smooth cost functions, a new discrete dynamic programming method, termed gradient dynamic programming (GDP), is proposed. GDP uses a piecewise Hermite interpolation of the cost-to-go function, at each stage, which preserves the values of the cost-to-go function and of its first derivatives at the discretization nodes. The error in the control policy is shown to be of the order of (AS) 3 and the error in the cost to vary with AS '. Thus as AS decreases, GDP converges to the true optimum much more rapidly than DDP. Another major advantage of the new methodology is that it facilitates the use of Newton-type iterative methods in the solution of the nonlinear optimization problems at each stage. The linear convergence of DDP and the superlinear convergence of GDP are illustrated in an example.
TL;DR: In this article, a comparison between flux-limited finite difference methods and characteristic Galerkin methods for approximating hyperbolic conservation laws is made, where linear advection, inviscid Burgers and Euler equations are explored.
TL;DR: In this paper, the authors identify admissible slopes at data points of various C 1 interpolants which ensure a desirable shape, in turn for the following function classes commonly used for shape-preserving interpolations: monotone polynomials, C 1 -monotone piecewise polynomial, convex polynomorphisms, parametric cubic curves and rational functions.
Abstract: In many applications, interpolation of experimental data exhibiting some geometric property such as nonnegativity, monotonicity or convexity is unacceptable unless the interpolant reflects these characteristics. This paper identifies admissible slopes at data points of variousC 1 interpolants which ensure a desirable shape. We discuss this question, in turn for the following function classes commonly used for shape preserving interpolations: monotone polynomials,C 1 monotone piecewise polynomials, convex polynomials, parametric cubic curves and rational functions.
TL;DR: In this paper, the authors state two new second order sufficient conditions for local strict optimality in semi-infinite programming, which do not require a strict complementarity condition for the local reduction of the feasible set.
Abstract: We state two new second order sufficient conditions for local strict optimality in semi-infinite programming, which do not require a strict complementarity condition for the local reduction of the feasible set. This set will be described (locally) by means of a finite number of local optimal value functions, Due to the lack of strict complementarity, the latter functions are only of differentiability class C 1We show how the unique solution of a parametric nonlinear programming problem with inequalities is glued together from C 1piece of unique solutions of problems with only equally constraints. As a consequence, the local optimal value functions will be piecewise of class C 2, which gives information of second order.