Existence of piecewise affine Lyapunov functions in two dimensions

In Marinosson (2002) [10], a method to compute Lyapunov functions for systems with asymptotically stable equilibria was presented. The method uses finite differences on finite elements to generate a linear programming problem for the system in question, of which every feasible solution parameterises a piecewise affine Lyapunov function. In Hafstein (2004) [2] it was proved that the method always succeeds in generating a Lyapunov function for systems with an exponentially stable equilibrium. However, the proof could not guarantee that the generated function has negative orbital derivative locally in a small neighbourhood of the equilibrium. In this article we give an example of a system, where no piecewise affine Lyapunov function with the proposed triangulation scheme exists. This failure is due to the triangulation of the method being too coarse at the equilibrium, and we suggest a fan-like triangulation around the equilibrium. We show that for any two-dimensional system with an exponentially stable equilibrium there is a local triangulation scheme such that the system possesses a piecewise affine Lyapunov function. Hence, the method might eventually be equipped with an improved triangulation scheme that does not have deficits locally at the equilibrium.     

Estimation of the Bezout number for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function.In this paper.a coniecture on trianguation is confirmed The relation between the piecewise linear algebraiccurve and four-color conjecture is also presented.By Morgan-Scott triangulation, we will show the instabilityof Bezout number of piecewise algebraic curves. By using the combinatorial optimization method,an upper     

The Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at mnT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m + n − 2 containing all but one point of them also contains the last point.     

Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems

In this paper, we present a new approach for computing Lyapunov functions for nonlinear discrete-time systems with an asymptotically stable equilibrium at the origin. Given a suitable triangulation of a compact neighbourhood of the origin, a continuous and piecewise affine function can be parameterized by the values at the vertices of the triangulation. If these vertex values satisfy system-dependent linear inequalities, the parameterized function is a Lyapunov function for the system. We propose calculating these vertex values using constructions from two classical converse Lyapunov theorems originally due to Yoshizawa and Massera. Numerical examples are presented to illustrate the proposed approach.     

The Delaunay triangulation for multidimensional surfaces and its approximative properties

We define the Delaunay triangulation for surfaces and prove an analog of the G. Voronoi empty sphere theorem. We also prove a convergence theorem for gradients of piecewise linear approximations constructed on the Delaunay triangulation for functions differentiable on smooth surfaces.     

Analysis on a Finite Volume Element Method for Stokes Problems

Finite volume element method for the Stokes problem is considered. We use a conforming piecewise linear function on a fine grid for velocity and piecewise constant element on a coarse grid for pressure. For general triangulation we prove the equivalence of the finite volume element method and a saddle-point problem, the inf-sup condition and the uniqueness of the approximation solution. We also give the optimal order H^1 norm error estimate. For two widely used dual meshes we give the L^2 norm error estimates, which is optimal in one case and quasi-optimal in another ease. Finally we give a numerical example.     

Bivariate interpolatory rational splines

Smoothing conditions in terms of Bézier coefficients of piecewise rational functions on an arbitrary triangulation are derived. This facilitates the solution of the problem of bivariate rational spline interpolation, with or without convexity constraints, particularly on the three and four-directional meshes. For such a triangulation, we also derive the conformality condition that a bivariate rationale spline function must satisfy, and we demonstrate the interpolation scheme with a low-degree example.The research of this author was supported by NSF Grant # DMS-92-06928.     

Nöther-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve. Nöther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space. In this paper, using the properties of bivariate splines, the Nöther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented.     

N(o)ther-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve.N(o)ther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space. In this paper, using the properties of bivariate splines, the N(o)ther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented.     

The Bezout number of piecewise algebraic curves

Based on the discussion of the number of roots of univariate spline and the common zero points of two piecewise algebraic curves, a lower upbound of Bezout number of two piecewise algebraic curves on any given non-obtuse-angled triangulation is found. Bezout number of two piecewise algebraic curves on two different partitions is also discussed in this paper.     

Construction of basis functions in the finite element method

In many applications of the finite element method, the explicit form of the basis functions is not known. A well-known exception is that of piecewise linear approximation over a triangulation of the plane, where the basis functions are pyramid functions. In the present paper, the basis functions are displayed in closed form for piecewise polynomial approximation of degreen over a triangulation of the plane. These basis functions are expressed simply in terms of the pyramid functions for linear approximation.     

Scattered data interpolation using minimum energy Powell-Sabin elements and data dependent triangulations

A popular approach for obtaining surfaces interpolating to scattered data is to define the interpolant in a piecewise manner over a triangulation with vertices at the data points. In most cases, the interpolant cannot be uniquely determined from the prescribed function values since it belongs to a space of functions of dimension greater than the number of data points. Thus, additional parameters are needed to define an interpolant and have to be estimated somehow from the available data. It is intuitively clear that the quality of approximation by the interpolant depends on the choice of the triangulation and on the method used to provide the additional parameters. In this paper we suggest basing the selection of the triangulation and the computation of the additional parameters on the idea of minimizing a given cost functional measuring the quality of the interpolant. We present a scheme that iteratively updates the triangulation and computes values of the additional parameters so that the quality of the interpolant, as measured by the cost functional, improves from iteration to iteration. This method is discussed and tested numerically using an energy functional and Powell-Sabin twelve split interpolants.     

On the dimension of spaces of pp functions with boundary conditions

In this paper we study the dimension of spaces of pp (piecewise polynomial) functions satisfying boundary conditions on a three direction mesh (or type-1 triangulation) and on a four direction mesh (or type-2 triangulation). The work in this area was initiated by Chui, Schumaker and Wang in [3, 4, 5].     

Data Dependent Triangulations for Piecewise Linear Interpolation

Given a set of data points in R² and corresponding data values,it is clear that the quality of a piecewise linear interpolationover triangles depends on the specific triangulation of thedata points. While conventional triangulation methods dependonly on the distribution of the data points in R² in this paperwe suggest that the triangulation should depend on the datavalues as well. Several data dependent criteria for definingthe triangulation are discussed and efficient algorithms forcomputing these triangulations are presented. It is shown fora variety of test cases that data dependent triangulations canimprove significantly the quality of approximation and thatlong and thin triangles, which are traditionally avoided, aresometimes very suitable.     

Piecewise linear interpolants to Lagrange and Hermite convex scattered data

This paper concerns two fundamental interpolants to convex bivariate scattered data. The first,u, is the supremum over all convex Lagrange interpolants and is piecewise linear on a triangulation. The other,l, is the infimum over all convex Hermite interpolants and is piecewise linear on a tessellation. We discuss the existence, uniqueness, and numerical computation ofu andl and the associated triangulation and tessellation. We also describe how to generate convex Hermite data from convex Lagrange data.Research partially supported by the EU Project FAIRSHAPE, CHRX-CT94-0522. The first author was also partially supported by DGICYT PB93-0310 Research Grant.     

Equivariant perturbation in Gomory and Johnson’s infinite group problem—III: foundations for the k-dimensional case with applications to $$ $$

We develop foundational tools for classifying the extreme valid functions for the k-dimensional infinite group problem. In particular, we present the general regular solution to Cauchy’s additive functional equation on restricted lower-dimensional convex domains. This provides a k-dimensional generalization of the so-called Interval Lemma, allowing us to deduce affine properties of the function from certain additivity relations. Next, we study the discrete geometry of additivity domains of piecewise linear functions, providing a framework for finite tests of minimality and extremality. We then give a theory of non-extremality certificates in the form of perturbation functions. We apply these tools in the context of minimal valid functions for the two-dimensional infinite group problem that are piecewise linear on a standard triangulation of the plane, under a regularity condition called diagonal constrainedness. We show that the extremality of a minimal valid function is equivalent to the extremality of its restriction to a certain finite two-dimensional group problem. This gives an algorithm for testing the extremality of a given minimal valid function.     

Construction of a CPA contraction metric for periodic orbits using semidefinite optimization

A Riemannian metric with a local contraction property can be used to prove existence and uniqueness of a periodic orbit and determine a subset of its basin of attraction. While the existence of such a contraction metric is equivalent to the existence of an exponentially stable periodic orbit, the explicit construction of the metric is a difficult problem.In this paper, the construction of such a contraction metric is achieved by formulating it as an equivalent problem, namely a feasibility problem in semidefinite optimization. The contraction metric, a matrix-valued function, is constructed as a continuous piecewise affine (CPA) function, which is affine on each simplex of a triangulation of the phase space. The contraction conditions are formulated as conditions on the values at the vertices.The paper states a semidefinite optimization problem. We prove on the one hand that a feasible solution of the optimization problem determines a CPA contraction metric and on the other hand that the optimization problem is always feasible if the system has an exponentially stable periodic orbit and the triangulation is fine enough. An objective function can be used to obtain a bound on the largest Floquet exponent of the periodic orbit.     

Asymptotic Expansion for the Derivative of Finite Elements

It is proved in this paper that there exists an expansion for the derivative of the linear finite element approximation to a model Dirichlet problem in a polygonal domain with a piecewise uniform triangulation.     

Solving discrete systems of nonlinear equations

We study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice Zⁿ of the n-dimensional Euclidean space Rⁿ. It is assumed that the set is integrally convex, which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of Zⁿ and each simplex of the triangulation lies in an n-dimensional cube of size one. With respect to this triangulation we assume that the function satisfies some property that replaces continuity. Under this property and some boundary condition the function has a zero point. To prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations. The standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied, because the ‘continuity property’ is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself. We apply the main existence result to prove the existence of a pure Cournot-Nash equilibrium in a Cournot oligopoly model. We further obtain a discrete analogue of the well-known Borsuk-Ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem.     

Regge?s Einstein-Hilbert functional on the double tetrahedron

The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, constant scalar curvature metrics, and the Yamabe problem on the double tetrahedron, with some reference to the possibilities on a general piecewise flat manifold. The main tool is analysis of Regge?s Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar curvature) functional on Riemannian manifolds. We study the Einstein-Hilbert-Regge functional on the space of metrics and on discrete conformal classes of metrics.