Identification of Physical Dynamical Processes via Linear Structure Models (Part 1)

The article discusses methods for identifying parameters of partial differential equations. Identification problems are often called poorly conditioned. However, the reason is the ambiguity of the solution, even at the point of minimality of the criterion. In particular, the article discusses: (1) An analysis of singular values for identify the unambiguous solution. The basis of these methods is a singular value decomposition of the matrix of experimental data, what makes it possible to abandon the inversion of matrices and, as a consequence, translate the problem of ill-conditioned problems into the problem of ambiguity of the solution. (2) The issues of anomalous measurements and combination of various experiments. (3) A universal optimization method for identifying parameters by their complete simple enumeration. The method is based on fast calculation of points on a multidimensional sphere. (4) The issues of identifiability of linear structure models and construction of experiments guaranteeing identification. (5) A method for identifying parameters via projecting linear structure model elements onto the plane of guarantors. (6) An approach to constructing histograms of unknown parameters of dynamic systems before calculating them using any algorithm. The approach is based on linear structure models with parameters on a sphere and a rather unexpected application of singular value decomposition. (7) The methods are accompanied by examples of heat equations. The Appendix containsalgorithmsintheMATLABlanguageforallexamples. (8)The presented optimization, projection and statistical methods based on the concept of linear structure models allow solving the same identification problem in fundamentally different ways, which significantly increases the reliability of the results obtained.