1. T he inverse methods - Introduction The inverse methods are an useful tool in determination of the physical parameters of any mathematical model by comparing its prediction with experimental data. It allows calculating the optimal values of model parameters p1, …, pN by performing computer simulations which give the solution for these values. The idea is to define the proper goal function, Gf(p1, …, pN), depending on model parameters, which measures the difference between the results produced by a real world experiment and model results, and then to seek its global minimum. Let us denote by yModel(x, t; p1, …, pN) the solution of the mathematical model (Model) as a function of x R3 and time t > 0 for any given set of parameters p1, …, pN . The values obtained from experiment (Exp) shall be denoted as yExp(x, t). One possible measure of the difference is Gf(p1,..., pN) := |yModel(x, t; p1,...,pN) - yExp(x, t)|2dxdt (1) where Ω R3 is a domain in space where the process occurs and tend > 0 is duration of the process. In the case when we measure the results only after some time (i.e. at the particular time tend only) the goal function will be defined rather as Gf(p1,..., pN) := |yModel(x, tend; p1,...,pN) - yExp(x, tend)|2dx (2) The assumption that we know the experimental data yExp(x, t) in the whole domain, x Ω R3, is rather optimistic. In practice the function is known at selected points only, yExp(xk, t). In this case the goal function is usually considered as Gf(p1,..., pN) := |yModel(xk, tend; p1,...,pN) - yExp(xk, tend)|2 (3) A procedure based on the inverse method ultimately requires the optimization function (goal function) be passed to various minimum seeking algorithms such as Hierarchical Genetic Strategy, Sequential Quadratic Procedure, Conjugate Gradient methods, or Coordinate Search Optimization Method to find the global optimum. But this is purely a mathematical and numerical pro[...]
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