Piecewise Linear Interpolation Research Articles

Piecewise polynomial interpolations and approximations of one-dimensional functions through mixed integer linear programming

Piecewise polynomial functions are extensively used to approximate general nonlinear functions or sets of data. In this work, we propose a mixed integer linear programming (MILP) framework for generating optimal piecewise polynomial approximations of varying degrees to nonlinear functions of a single variable. The nonlinear functions may be represented by discrete exact samplings or by data corrupted by noise. We studied two distinct approaches to the problem: (1) the generation of interpolating piecewise polynomial functions, in which the approximating function values in the extremes of each polynomial segment coincide with the original function values; and (2) a de facto approximation strategy in which the polynomial segments are free, except for the enforcement of continuity of the overall approximation. Our results from the implemented models show that the procedure is capable of efficiently approximating nonlinear functions and it has the added capability of allowing for the straightforward implementation of further constraints on solutions, such as the convexity of polynomial segments. Finally, the models for the generation of piecewise linear approximations and interpolations were applied for the linearization of mixed integer nonlinear programming (MINLP) models extracted from the MINLP library. These models were linearized by the reformulation of their nonlinearities as piecewise linear functions with varying numbers of segments, resulting in MILP models that were solved to optimality and the solutions from the linearized models were compared with global optimal solutions from the original problems.

A robust grid equidistribution method for a one-dimensional singularly perturbed semilinear reaction-diffusion problem

The numerical solution of a singularly perturbed semilinear reaction-diffusion two-point boundary-value problem is addressed. The method considered is adaptive movement of a fixed number (N + 1) of mesh points by equidistribution of a monitor function that uses discrete second-order derivatives. We extend the analysis by Kopteva & Stynes (2001, SIAM J. Numer. Anal., 39, 1446-1467) to a new equation and a more intricate monitor function. It is proved that there exists a solution to the fully discrete equidistribution problem, i.e. a mesh exists that equidistributes the discrete monitor function computed from the discrete solution on this mesh. Furthermore, in the case when the boundary-value problem is linear, it is shown that after O(|1ne|/ In N) iterations of the algorithm, the piecewise linear interpolant of the computed solution achieves second-order accuracy in the maximum norm, uniformly in the diffusion coefficient e 2 . Numerical experiments are presented that support our theoretical results.

Stiffly accurate Runge–Kutta methods for nonlinear evolution problems governed by a monotone operator

Stiy accurate implicit Runge{Kutta methods are studied for the time discretisation of nonlinear rst-order evolution equations. The equation is supposed to be governed by a time-dependent hemicontinuous operator that is (up to a shift) monotone and coercive, and fullls a certain growth condi- tion. It is proven that the piecewise constant as well as the piecewise linear interpolant of the time-discrete solution converge towards the exact weak solu- tion, provided the Runge{Kutta method is consistent and satises a stability criterion that implies algebraic stability; examples are the Radau IIA and Lo- batto IIIC methods. The convergence analysis is also extended to problems involving a strongly continuous perturbation of the monotone main part.

Comparison of different chaotic maps in particle swarm optimization algorithm for long-term cascaded hydroelectric system scheduling

The goal of this paper is to present a novel chaotic particle swarm optimization (CPSO) algorithm and compares the efficiency of three one-dimensional chaotic maps within symmetrical region for long-term cascaded hydroelectric system scheduling. The introduced chaotic maps improve the global optimal capability of CPSO algorithm. Moreover, a piecewise linear interpolation function is employed to transform all constraints into restrict upriver water level for implementing the maximum of objective function. Numerical results and comparisons demonstrate the effect and speed of different algorithms on a practical hydro-system.

Approximation capabilities of multilayer fuzzy neural networks on the set of fuzzy-valued functions

This paper first introduces a piecewise linear interpolation method for fuzzy-valued functions. Based on this, we present a concrete approximation procedure to show the capability of four-layer regular fuzzy neural networks to perform approximation on the set of all d p continuous fuzzy-valued functions. This approach can also be used to approximate d ∞ continuous fuzzy-valued functions. An example is given to illustrate the approximation procedure.

A Parameterized Macromodeling Strategy With Uniform Stability Test

This paper presents a strategy for the construction of parameterized linear macromodels from tabulated port responses. These macromodels are able to reproduce the input-output behavior of the structure of interest both in terms of frequency and one or more design variables such as geometry and material parameters. A highly efficient combination of rational identification and piecewise linear interpolation leads to a macromodel form which can be cast as a polytopic descriptor form. This in turns enables the construction of a numerically robust testing procedure, based on linear matrix inequalities, for the assessment of uniform model stability within any prescribed region of the parameters space. Several numerical examples are used to illustrate the theory on practical application cases.

STOCHASTIC DYNAMICS OF VISCOELASTIC SKEINS: CONDENSATION WAVES AND CONTINUUM LIMITS

Skeins (one-dimensional queues) of migrating birds show typical fluctuations in swarm length and frequent events of "condensation waves" starting at the leading bird and traveling backward within the moving skein, similar to queuing traffic waves in car files but more smooth.These dynamical phenomena can be fairly reproduced by stochastic ordinary differential equations for a "multi-particle" system including the individual tendency of birds to attain a preferred speed as well as mutual interaction "forces" between neighbors, induced by distance-dependent attraction or repulsion as well as adjustment of velocities. Such a one-dimensional system constitutes a so-called "stochastic viscoelastic skein."For the simple case of nearest neighbor interactions we define the density between individualsu = u(t, x) as a step function inversely proportional to the neighbor distance, and the velocity function v = v(t, x) as a standard piecewise linear interpolation between individual velocities. Then, in the limit of infinitely many birds in a skein of finite length, with mean neighbor distance δ converging to zero and after a suitable scaling, we obtain continuum mass and force balance equations that constitute generalized nonlinear compressible Navier–Stokes equations. The resulting density-dependent stress functions and viscosity coefficients are directly derived from the parameter functions in the original model.We investigate two different sources of additive noise in the force balance equations: (1) independent stochastic accelerations of each bird and (2) exogenous stochastic noise arising from pressure perturbations in the interspace between them. Proper scaling of these noise terms leads, under suitable modeling assumptions, to their maintenance in the continuum limit δ → 0, where they appear as (1) uncorrelated spatiotemporal Gaussian noise or, respectively, (2) certain spatially correlated stochastic integrals. In both cases some a priori estimates are given which support convergence to the resulting stochastic Navier–Stokes system.Natural conditions at the moving swarm boundaries (along characteristics of the hyperbolic system) appear as singularly perturbed zero-tension Neumann conditions for the velocity function v. Numerical solutions of this free boundary value problem are compared to multi-particle simulations of the original discrete system. By analyzing its linearization around the constant swarm state, we can characterize several properties of swarm dynamics. In particular, we compute approximating values for the averaged speed and length of typical condensation waves.

Nonlinear stability of general linear methods for neutral delay differential equations

This paper is concerned with the numerical solution of neutral delay differential equations (NDDEs). We focus on the stability of general linear methods with piecewise linear interpolation. The new concepts of G S ( p ) -stability, G A S ( p ) -stability and weak G A S ( p ) -stability are introduced. These stability properties for ( k , p , 0 ) -algebraically stable general linear methods (GLMs) are further investigated. Some extant results are unified.

Goal-oriented adaptive mesh refinement for the quasicontinuum approximation of a Frenkel-Kontorova model

The quasicontinuum approximation [E.B. Tadmor, M. Ortiz, R. Phillips, Quasicontinuum analysis of defects in solids, Philos. Mag. A 73(6) (1996) 1529–1563] is a method to reduce the atomistic degrees of freedom of a crystalline solid by piecewise linear interpolation from representative atoms that are nodes for a finite element triangulation. In regions of the crystal with a highly nonuniform deformation such as around defects, every atom must be a representative atom to obtain sufficient accuracy, but the mesh can be coarsened away from such regions to remove atomistic degrees of freedom while retaining sufficient accuracy. We present an error estimator and a related adaptive mesh refinement algorithm for the quasicontinuum approximation of a generalized Frenkel-Kontorova model that enables a quantity of interest to be efficiently computed to a predetermined accuracy.

Stable, linear spline wavelets on nonuniform knots with vanishing moments

In wavelet analysis on nonuniform grids it is desirable that the wavelet scheme is stable in some norm independently of the grid spacing (grid stability). It is known that this kind of stability is difficult to achieve for spline wavelets based on orthogonal complements, with stability measured in the L 2 -norm. On the other hand, a wavelet scheme based on piecewise linear interpolation (Faber decomposition) is known to be grid stable in the L ∞ norm. In this paper we show that Faber decomposition can be extended with preservation of moments, without sacrificing grid stability in the L ∞ norm.

Asymptotic stability of multistep methods for nonlinear delay differential equations

This paper is concerned with the numerical solution of delay differential equations. The emphasis is on the nonlinear stability of multistep methods. It is shown that every A-stable linear multistep method with piecewise constant or linear interpolation can preserve the asymptotic stability of a class of nonlinear systems, which is an extension of the well known GP-stability result to the nonlinear case. As a comparison, it is also proved that strict stability at infinity is necessary to the asymptotic stability of one-leg methods for non-autonomous equations.

Asteroseismic constraints on the OPAL opacity interpolation

AbstractThere is a difference of a few Kelvins in the effective temperature between a model used only two-point interpolation of opacity and a model used piecewise linear interpolation of opacity. However the frequency difference between the models is of the order of several microHertz at a certain stage, which is almost 10 times worse than the observational precision of p-modes of solar-like stars. Therefore, the two-point interpolation of opacity is unsuitable in modelling of solar-like stars with element diffusion.

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have “nicely shaped” triangles that optimize several criteria, and they are easy to construct and update. The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained regular triangulations. CDTs and weighted CDTs are powerful and practical models of geometric domains, especially in two and three dimensions. The main contributions are rigorous, theory-tested definitions of CDTs and piecewise linear complexes (geometric domains that incorporate nonconvex faces with “internal” boundaries), a characterization of the combinatorial properties of CDTs and weighted CDTs (including a generalization of the Delaunay Lemma), the proof of several optimality properties of CDTs when they are used for piecewise linear interpolation, and a simple and useful condition that guarantees that a domain has a CDT. These results provide foundations for reasoning about CDTs and proving the correctness of algorithms. Later articles in this series discuss algorithms for constructing and updating CDTs.

Adaptive placement method on pricing arithmetic average options

Since there is no analytic solution for arithmetic average options until present, developing an efficient numerical algorithm becomes a promising alternative. One of the most famous numerical algorithms is introduced by Hull and White (J Deriv 1:21–31, 1993). Motivated by the common idea of reducing the nonlinearity error in the adaptive mesh model in Figlewski and Gao (J Financ Econ 53:313–351, 1999) and the adaptive quadrature method, we propose an adaptive placement method to replace the logarithmically equally-spaced placement rule in the Hull and White’s model by placing more representative average prices in the highly nonlinear area of the option value as the function of the arithmetic average stock price. The basic idea of this method is to design a recursive algorithm to limit the error of the linear interpolation between each pair of adjacent representative average prices. Numerical experiments verify the superior performance of this method for reducing the interpolation error and hence improving the convergence rate. To show that the adaptive placement method can improve any numerical algorithm with the techniques of augmented state variables and the piece-wise linear interpolation approximation, we also demonstrate how to integrate the adaptive placement method into the GARCH option pricing algorithm in Ritchken and Trevor (J Finance 54:377–402, 1999). Similarly great improvement of the convergence rate suggests the potential applications of this novel method to a broad class of numerical pricing algorithms for exotic options and complex underlying processes.

BV-norm convergence of interpolation approximations for Frobenius-Perron operators

Let S be a chaotic transformation from an interval into itself and let P be the corresponding Frobenius-Perron operator associated with S. In this paper we prove the convergence under the variation norm for a piecewise linear interpolation method that was recently proposed by the authors for computing a stationary density of P.

Uniform convergence of the multigrid V‐cycle on graded meshes for corner singularities

AbstractThis paper analyzes a multigrid (MG) V‐cycle scheme for solving the discretized 2D Poisson equation with corner singularities. Using weighted Sobolev spaces Kma(Ω) and a space decomposition based on elliptic projections, we prove that the MG V‐cycle with standard smoothers (Richardson, weighted Jacobi, Gauss–Seidel, etc.) and piecewise linear interpolation converges uniformly for the linear systems obtained by finite element discretization of the Poisson equation on graded meshes. In addition, we provide numerical experiments to demonstrate the optimality of the proposed approach. Copyright © 2008 John Wiley & Sons, Ltd.

Balance of datum land prices among cities based on the city gravitation model and stochastic diffusion equation

A balance of urban datum land prices is achieved to harmonize regional land prices and make the prices truly reflect different economic development levels and land prices among cities. The current piecewise linear interpolation balance method widely used has two drawbacks that always lead to an unsatisfactory balance among some cities. When the excess of land price in the central city to the surrounding zone reaches a certain degree, land price in the circumjacent city is not only consistent with the local land grade and land use level, but also influenced by the diffusion of land price in the central city. Thus, a new balanced scheme of datum land prices based on the city gravitation model and stochastic diffusion equation is brought forward. Finally, the new method is examined in the practice of datum land price balance in Hubei Province, China.

Dynamic transport simulation code including plasma rotation and radial electric field

A new one-dimensional transport code named TASK/TX, which is able to describe dynamic behavior of tokamak plasmas, has been developed. It solves simultaneously a set of flux-surface averaged equations composed of Maxwell’s equations, continuity equations, equations of motion, heat transport equations, fast-particle slowing-down equations and two-group neutral diffusion equations. The set of equations describes plasma rotations in both toroidal and poloidal directions through momentum transfer and evaluates the radial electric field self-consistently. The finite element method with a piecewise linear interpolation function is employed with a fine radial mesh near the plasma surface. The Streamline Upwind Petrov–Galerkin method is also used for robust calculation. We have confirmed that the neoclassical properties are well described by the poloidal neoclassical viscous force. The modification of density profile during neutral beam injection is presented. In the presence of ion orbit loss, the generation of the inward radial electric field and torque due to radial current is self-consistently calculated.

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have “nicely shaped” triangles that optimize several criteria, and they are easy to construct and update. The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained regular triangulations. CDTs and weighted CDTs are powerful and practical models of geometric domains, especially in two and three dimensions. The main contributions are rigorous, theory-tested definitions of CDTs and piecewise linear complexes (geometric domains that incorporate nonconvex faces with “internal” boundaries), a characterization of the combinatorial properties of CDTs and weighted CDTs (including a generalization of the Delaunay Lemma), the proof of several optimality properties of CDTs when they are used for piecewise linear interpolation, and a simple and useful condition that guarantees that a domain has a CDT. These results provide foundations for reasoning about CDTs and proving the correctness of algorithms. Later articles in this series discuss algorithms for constructing and updating CDTs.

Numerical meshfree path integration method for non-linear dynamic systems

We present a new numerical meshfree path integration (MPI) method for non-linear dynamic systems. The obtained MPI method can be performed in the irregular computational domain and the probability density values of the random nodes in the domain can be calculated via the MPI method and the ordinary differential equations for the first and second-order moments on the basis of Gaussian closure method. The piecewise linear interpolation based on adaptive least squares is utilized as a post processor to approximate the probability density values on arbitrary positions. The good performance of the resulting method is finally shown in the numerical examples by using three specific non-linear dynamic systems: Duffing oscillator subjected to both harmonic and stochastic excitations, Duffing–Rayleigh oscillator subjected to both harmonic and stochastic excitations, and CHEN system driven by three different Gaussian white noises.