Parallel mesh methods for tension splines

This paper addresses the problem of shape preserving spline interpolation formulated as a differential multipoint boundary value problem (DMBVP for short). Its discretization by mesh method yields a five-diagonal linear system which can be ill-conditioned for unequally spaced data. Using the superposition principle we split this system in a set of tridiagonal linear systems with a diagonal dominance. The latter ones can be stably solved either by direct (Gaussian elimination) or iterative methods (SOR method and finite-difference schemes in fractional steps) and admit effective parallelization. Numerical examples illustrate the main features of this approach.

[1]  Tom Lyche,et al.  Interpolation with Exponential B-Splines in Tension , 1993, Geometric Modelling.

[2]  Miljenko Marušić,et al.  Sharp error bounds for interpolating splines in tension , 1995 .

[3]  D. Schweikert.  An Interpolation Curve Using a Spline in Tension , 1966 .

[4]  Carla Manni,et al.  On discrete hyperbolic tension splines , 1999, Adv. Comput. Math..

[5]  Boris I. Kvasov.  Approximation by Discrete GB-Splines , 2004, Numerical Algorithms.

[6]  Richard Franke,et al.  Thin plate splines with tension , 1985, Comput. Aided Geom. Des..

[7]  Helmuth Späth,et al.  Two-dimensional spline interpolation algorithms , 1993 .

[8]  Nira Dyn,et al.  Exponentials Reproducing Subdivision Schemes , 2003, Found. Comput. Math..

[9]  Gene H. Golub,et al.  Matrix computations , 1983 .

[10]  A. Le Méhauté,et al.  Spline curves and surfaces with tension , 1994 .

[11]  B. McCartin.  Theory of exponential splines , 1991 .

[12]  Weiyin Ma,et al.  A generalized curve subdivision scheme of arbitrary order with a tension parameter , 2010, Comput. Aided Geom. Des..

[13]  Hiroshi Akima,et al.  A New Method of Interpolation and Smooth Curve Fitting Based on Local Procedures , 1970, JACM.

[14]  N. N. Yanenko,et al.  The Method of Fractional Steps , 1971 .

[15]  Jean Duchon,et al.  Splines minimizing rotation-invariant semi-norms in Sobolev spaces , 1976, Constructive Theory of Functions of Several Variables.

[16]  A. Samarskii.  The Theory of Difference Schemes , 2001 .

[17]  Panagiotis D. Kaklis,et al.  An algorithm for constructing convexity and monotonicity-preserving splines in tension , 1988, Comput. Aided Geom. Des..

[18]  Robert J. Renka,et al.  Interpolatory tension splines with automatic selection of tension factors , 1987 .

[19]  P. Rentrop.  An algorithm for the computation of the exponential spline , 1980 .

[20]  Joe D. Warren,et al.  A subdivision scheme for surfaces of revolution , 2001, Comput. Aided Geom. Des..

[21]  Sanja Singer,et al.  Conditions of matrices in discrete tension spline approximations of DMBVP , 2007 .

[22]  Boris I. Kvasov,et al.  Methods of Shape-Preserving Spline Approximation , 2000 .

[23]  Josef Hoschek,et al.  Fundamentals of computer aided geometric design , 1996 .