A feasible set for chemical speciation problems
Numerical methods applied to solve chemical speciation problems often do not converge at the desired global minimum. The risk of failure can be reduced by starting calculations on the residual map in a region where convergence is assured. This is possible because the topography of the residual map for all ideal chemical equilibrium problems is predictable and contains three important regions: a region of mostly steep gradient where the simulation assumes an excess of mass in the system; another region of mass deficit with areas of low to zero gradients and, in between, a channel where total mass is conserved. The channel constitutes a feasible set and always contains the global minimum. Although apparently an advantage in guiding the descent towards the solution, the feasible set channel causes a major problem for convergence by hiding the global minimum from a starting point in either excess or deficit regions. In addition, the steep slopes of the channel cause most line search algorithms to overshoot several times before converging. The best approach to maximize both convergence and speed is to start in the mass excess region with a LevenbergMarquardt method using a cubic polynomial line search and, in the small number of cases where convergence fails, to use the slower and robust simplex method.
convergence algorithm simplex chemical-equilibrium optimization