5,959 research outputs found
Exact asymptotics of the optimal Lp-error of asymmetric linear spline approximation
In this paper we study the best asymmetric (sometimes also called penalized
or sign-sensitive) approximation in the metrics of the space , , of functions with nonnegative
Hessian by piecewise linear splines , generated by given
triangulations with elements. We find the exact asymptotic
behavior of optimal (over triangulations and splines error of such approximation as
Apollonian structure in the Abelian sandpile
The Abelian sandpile process evolves configurations of chips on the integer
lattice by toppling any vertex with at least 4 chips, distributing one of its
chips to each of its 4 neighbors. When begun from a large stack of chips, the
terminal state of the sandpile has a curious fractal structure which has
remained unexplained. Using a characterization of the quadratic growths
attainable by integer-superharmonic functions, we prove that the sandpile PDE
recently shown to characterize the scaling limit of the sandpile admits certain
fractal solutions, giving a precise mathematical perspective on the fractal
nature of the sandpile.Comment: 27 Pages, 7 Figure
Singular Continuation: Generating Piece-wise Linear Approximations to Pareto Sets via Global Analysis
We propose a strategy for approximating Pareto optimal sets based on the
global analysis framework proposed by Smale (Dynamical systems, New York, 1973,
pp. 531-544). The method highlights and exploits the underlying manifold
structure of the Pareto sets, approximating Pareto optima by means of
simplicial complexes. The method distinguishes the hierarchy between singular
set, Pareto critical set and stable Pareto critical set, and can handle the
problem of superposition of local Pareto fronts, occurring in the general
nonconvex case. Furthermore, a quadratic convergence result in a suitable
set-wise sense is proven and tested in a number of numerical examples.Comment: 29 pages, 12 figure
Interpolation and scattered data fitting on manifolds using projected PowellāSabin splines
We present methods for either interpolating data or for fitting scattered data on a two-dimensional smooth manifold. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of a family of charts {(UĪ¾ , Ī¾)}Ī¾ā satisfying certain conditions of smooth dependence on Ī¾. If is a C2-manifold embedded into R3, then projections into tangent planes can be employed. The data fitting method is a two-stage method. We prove that the resulting function on the manifold is continuously differentiable, and establish error bounds for both methods for the case when the data are generated by a smooth function
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