Bivariate and Multivariate Functions

For functions depending on two variables, we automatically construct tri6 angulations subject to the condition that the continuous, piecewise linear approxi7 mation, underor overestimation never deviates more than a given δ -tolerance from 8 the original function over a given domain. This tolerance is ensured by solving sub9 problems over each triangle to global optimality. The continuous, piecewise linear 10 approximators, underand overestimators involve shift variables at the vertices of the 11 triangles leading to a small number of triangles while still ensuring continuity over 12 the entire domain. For functions depending on more than two variables, we provide 13 appropriate transformations and substitutions, which allow the use of oneor two14 S. Rebennack (B) Division of Economics and Business, Colorado School of Mines, Golden, Co, USA E-mail: [email protected] J. Kallrath Department of Astronomy, University of Florida, Gainesville, FL, USA E-mail: [email protected] 2 Steffen Rebennack, Josef Kallrath dimensional δ -approximators. We address the problem of error propagation when 15 using these dimensionality reduction routines. We discuss and analyze the trade-off 16 between one-dimensional and two-dimensional approaches and we demonstrate the 17 numerical behavior of our approach on nine bivariate functions for five different δ 18 tolerances. 19