A feasible set for chemical speciation problems

Numerical methods applied to solve chemical speciation problems often do no t converge at the desired global minimum. The risk of failure can be reduce d by starting calculations on the residual map in a region where convergenc e is assured. This is possible because the topography of the residual map f or all ideal chemical equilibrium problems is predictable and contains thre e important regions: a region of mostly steep gradient where the simulation assumes an excess of mass in the system; another region of mass deficit wi th areas of low to zero gradients and, in between, a channel where total ma ss is conserved. The channel constitutes a feasible set and always contains the global minimum. Although apparently an advantage in guiding the descen t towards the solution, the feasible set channel causes a major problem for convergence by hiding the global minimum from a starting point in either e xcess or deficit regions. In addition, the steep slopes of the channel caus e most line search algorithms to overshoot several times before converging. The best approach to maximize both convergence and speed is to start in th e mass excess region with a Levenberg-Marquardt method using a cubic polyno mial line search and, in the small number of cases where convergence fails, to use the slower and robust simplex method. (C) 2000 Published by Elsevie r Science Ltd. All rights reserved.