Continuous Piecewise Linear Delta-Approximations for Bivariate and Multivariate Functions

Abstract

For functions depending on two variables, we automatically construct triangulations subject to the condition that the continuous, piecewise linear approximation, under-, or overestimation, never deviates more than a given $\delta$-tolerance from the original function over a given domain. This tolerance is ensured by solving subproblems over each triangle to global optimality. The continuous, piecewise linear approximators, under-, and overestimators, involve shift variables at the vertices of the triangles leading to a small number of triangles while still ensuring continuity over the entire domain. For functions depending on more than two variables, we provide appropriate transformations and substitutions, which allow the use of one- or two-dimensional $\delta$-approximators. We address the problem of error propagation when using these dimensionality reduction routines. We discuss and analyze the trade-off between one-dimensional (1D) and two-dimensional (2D) approaches, and we demonstrate the numerical behavior of our approach on nine bivariate functions for five different $\delta$-tolerances.