Piecewise

About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid 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.

Piecewise isometries have zero topological entropy

Abstract: We show that piecewise isometries, i.e. non-necessarily invertible maps defined on a finite union of polytopes and coinciding with an isometry on the interior of each polytope, have zero topological entropy in any dimension. This had been conjectured by a number of authors. The proof is by an induction on the dimension and uses a device (the differential of a piecewise linear map) introduced by M. Tsujii.

Robust model predictive control

Abstract: This thesis deals with the topic of min-max formulations of robust model predictive control problems. The sets involved in guaranteeing robust feasibility of the min-max program in the presence of state constraints are of particular interest, and expanding the applicability of well understood solvers of linearly constrained quadratic min-max programs is the main focus. To this end, a generalisation for the set of uncertainty is considered: instead of fixed bounds on the uncertainty, state- and input-dependent bounds are used. To deal with state- and input dependent constraint sets a framework for a particular class of set-valued maps is utilised, namely parametrically convex set-valued maps. Relevant properties and operations are developed to accommodate parametrically convex set-valued maps in the context of robust model predictive control. A quintessential part of this work is the study of fundamental properties of piecewise polyhedral set-valued maps which are parametrically convex, we show that one particular property is that their combinatorial structure is constant. The study of polytopic maps with a rigid combinatorial structure allows the use of an optimisation based approach of robustifying constrained control problems with probabilistic constraints. Auxiliary polytopic constraint sets, used to replace probabilistic constraints by deterministic ones, can be optimised to minimise the conservatism introduced while guaranteeing constraint satisfaction of the original probabilistic constraint. We furthermore study the behaviour of the maximal robust positive invariant set for the case of scaled uncertainty and show that this set is continuously polytopic up to a critical scaling factor, which we can approximate a-priori with an arbitrary degree of accuracy. Relevant theoretical statements are developed, discussed and illustrated with examples.