Computing minimal interpolants in $C^{1, 1}(\mathbb{R}^d)$

We consider the following interpolation problem. Suppose one is given a finite set E⊂Rd, a function f:E→R, and possibly the gradients of ff at the points of E. We want to interpolate the given information with a function F∈C1,1(Rd) with the minimum possible value of Lip(∇F). We present practical, efficient algorithms for constructing an F such that Lip(∇F) is minimal, or for less computational effort, within a small dimensionless constant of being minimal.

[1]  David P. Dobkin,et al.  The quickhull algorithm for convex hulls , 1996, TOMS.

[2]  Some results of the Lipschitz constant of 1-Field on R n , 2014 .

[3]  E. Gruyer.  Minimal Lipschitz Extensions to Differentiable Functions Defined on a Hilbert Space , 2009 .

[4]  Raimund Seidel,et al.  Constructing higher-dimensional convex hulls at logarithmic cost per face , 1986, STOC '86.

[5]  Y. Peres,et al.  Tug-of-war and the infinity Laplacian , 2006, math/0605002.

[6]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[7]  Jean-Michel Morel,et al.  An axiomatic approach to image interpolation , 1997, Proceedings of International Conference on Image Processing.

[8]  Differentiable functions on Banach spaces with Lipschitz derivatives , 1973 .

[9]  C. Fefferman,et al.  Smooth Interpolation of Data by Efficient Algorithms , 2013 .

[10]  J. Glass.  Smooth-curve interpolation: A generalized spline-fit procedure , 1966 .

[11]  Daniel A. Spielman,et al.  Algorithms for Lipschitz Learning on Graphs , 2015, COLT.

[12]  Timothy M. Chan,et al.  Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three , 1995, SODA '95.

[13]  Nancy M. Amato,et al.  On computing Voronoi diagrams by divide-prune-and-conquer , 1996, SCG '96.

[14]  R. Jensen.  Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient , 1993 .

[15]  G. Aronsson.  Extension of functions satisfying lipschitz conditions , 1967 .

[16]  Bernard Chazelle,et al.  An optimal convex hull algorithm in any fixed dimension , 1993, Discret. Comput. Geom..

[17]  H. Whitney.  Analytic Extensions of Differentiable Functions Defined in Closed Sets , 1934 .

[18]  Arie Israel,et al.  Fitting a Sobolev function to data I , 2014, 1411.1786.

[19]  Sup-Inf explicit formulas for minimal Lipschitz extensions for 1-fields on R^n , 2015 .

[20]  Raimund Seidel,et al.  Small-dimensional linear programming and convex hulls made easy , 1991, Discret. Comput. Geom..

[21]  Kungl. Svenska vetenskapsakademien..  Arkiv för matematik , 1949 .

[22]  M. D. Kirszbraun.  Über die zusammenziehende und Lipschitzsche Transformationen , 1934 .

[23]  Franz Aurenhammer,et al.  Power Diagrams: Properties, Algorithms and Applications , 1987, SIAM J. Comput..

[24]  C. Fefferman.  Fitting a Cm-smooth function to data, III , 2009 .

[25]  Some results of the Lipschitz constant of 1-Field on $\mathbb{R}^n$ , 2014, 1402.4276.

[26]  Colin N. Jones,et al.  Optimized decision trees for point location in polytopic data sets - application to explicit MPC , 2010, Proceedings of the 2010 American Control Conference.

[27]  John C. Holladay,et al.  A smoothest curve approximation , 1957 .

[28]  Bo'az Klartag,et al.  Fitting a $C^m$-Smooth Function to Data II , 2009 .

[29]  G. Aronsson.  Minimization problems for the functional supxF(x, f(x), f′(x)) , 1965 .

[30]  阿部 浩一,et al.  Fundamenta Mathematicae私抄 : 退任の辞に代えて , 1987 .

[31]  Eugène Rouché.  Sur l'interpolation , 1859 .

[32]  C. Morawetz.  The Courant Institute of Mathematical Sciences , 1988 .

[33]  Gunnar Aronsson,et al.  Fast/Slow Diffusion and Growing Sandpiles , 1996 .

[34]  D. Rose,et al.  Generalized nested dissection , 1977 .

[35]  G. Glaeser.  Prolongement extr?mal de fonctions diff?rentiables d'une variable , 1973 .

[36]  S. Rao Kosaraju,et al.  A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields , 1995, JACM.

[37]  A general theorem of existence of quasi absolutely minimal Lipschitz extensions , 2012, 1211.5700.

[38]  Sariel Har-Peled.  A replacement for Voronoi diagrams of near linear size , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[39]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[40]  Gunnar,et al.  Minimization problems for the functional supxF(x, f(x), f'(x)). (II) , 1966 .

[41]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[42]  Rex A. Dwyer,et al.  On the convex hull of random points in a polytope , 1988, Journal of Applied Probability.

[43]  Alberto Bemporad,et al.  Evaluation of piecewise affine control via binary search tree , 2003, Autom..

[44]  Yann Gousseau,et al.  Interpolation of digital elevation models using AMLE and related methods , 2002, IEEE Trans. Geosci. Remote. Sens..