A geometric programming approach for bivariate cubic \(L_{1}\) splines (Q2485428)

This paper deals with a geometric programming approach to get bivariate cubic \(L_1\) splines. Recently this new class of polynomial splines has been introduced and developed in the univariate case. In fact, while conventional polynomial splines have excellent approximatation power and ensure efficient evaluation, but they do not ``preserve shape well'', cubic \(L_1\) splines provide \(C^1\)-smooth, shape-preserving interpolation of data, including those with abrupt changes in spacing and magnitude. The authors of the present paper extend the geometric programming framework from univariate to bivariate case. The coefficients of \(L_1\) splines are obtained by minimizing the \(L_1\) norm of second partial derivatives of candidate \(C^1\)-smooth piecewise cubic surfaces. This process is equivalent to solving a non linear programming problem with a convex feasible region and a nondifferentiable convex objective function. The nondifferentiability of the objective function makes it difficult to directly characterize the corresponding optimality condition and traditional nonlinear programming techniques can hardly be applied directly to this nonsmooth optimization problem. The use of geometric programming can alleviate this problem by transforming it into a differentiable convex dual programming problem with a linear objective function and convex cubic constraints. It is now possible to apply any general-purpose nonlinear solver to find a dual solution. A primal optimal solution is then obtained from the dual one by satisfying suitable optimality conditions. Since the dual problem is ``simpler'' than the primal one, theoretical investigations of shape-preserving properties of bivariate cubic \(L_1\) splines can be better carried out using the dual solution. Computational experiments are also presented.