Inverse Problem and Specialized Minimization Methods

Inverse problem can also be considered as an optimization problem. If one has an analytical expression for the component concentration C(k, k2, k , t) and a set of experimental values Cexp, corresponding to time t, it is possible to find the unknown constants by minimization of a sum of squared differences between the experimental and calculated values. 

This is a typical minimization problem for a function of n variables that can be solved using a Mathcad built-in function MINIMIZE. The latter implements gradient search algorithms to find the local minimum. The SSq function in this case is called the target function, and the unknown kinetic constants are the optimization parameters. When there are no additional limitations for the values of optimization parameters or the sought function, we have a case of the so called unconstrained optimization. Likewise, if the unknown parameters or the target function itself are mathematically constrained with some equalities or inequalities, then one deals with the constrained optimization. Such additional constrains are usually set on the basis on the physical nature of the problems (e.g. rate constants must be positive, a ratio of the direct reaction rate to that of the inverse one must equal the equilibrium constant, etc.) The constraints are sometimes added in order to speed up the computations (for example, the value of target function in the found minimum should not exceed some number TOL). 

Usage of the Mathcad MINIMIZE function requires following three main rules  

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