Bashkir State Univerity The Chair of Mathematical Modeling , Ufa, Zaki Validi str. 32 Phone: , Institute of Petrochemistry and Catalysis Russia Academy of Science The Laboratory of Mathematical Chemistry , Ufa, October Avenue 141 SEMYEN I. SPIVAK INFORMATIVITY OF AN EXPERIMENT AND UNCERTAINTY REGIONS OF MODEL PARAMETERS

2 k+k+ k-k- - number of stages in the reaction mechanism, - the characters involved in the reactions of substances; - total number of substances; and stoichiometric coefficients of the initial reactants and reaction products, respectively; and - rate constants of stages in the forward and reverse directions respectively. Numbers and characterize the reaction order COMPLEX CHEMICAL REACTIONS MECHANISM THE SET OF ELEMENTARY STAGES

3 The Low of Mass Action: – concentrations of substances REACTION RATE

4 The general formula for the element of the stoichiometric matrix Г The chemical composition of substances reflect the molecular matrix A. Element number of atoms of q-th element contained in the molecule of the j-th agent. ГА = 0 The most important condition - the existence of linear integrals (linear relations between the concentrations)/ The number of independence integrals are the number of different types of chemical elements that make up the system. THE SYSTEMS OF DIFFERENTIAL EQUATIONS OF CHEMICAL KINETICS – transpose Г.

5 X – vector of the measured substances, precursors and reaction products; У – vector of unmeasured substances, intermediates (radicals, catalysts and their complexes, substances on the surface of the catalyst, enzymes, etc.). Consequence of the lack of experimental data – ambiguity of the solution the inverse problem identification of the reaction mechanism. PROBLEM ABSENSE OF MEASUREMENTS PART OF SUBSTANCES

6 Identify which when substituted in the system of differential equations of chemical kinetics, reproduce the experimentally measured X Informativity of the experiment -the number of independent parametric functions of the constants that allow unambiguous estimation based on the available information. What kind of individual constants, or their non-linear parametric combinations? 1. Nonstationary 2. Quasistationary Bodenshtein – Semenov Quasistationary Principle Choriuti – Temkin stationary reactions theory 3. Equilibrium Zeldovich function STRUCTURE OF THE EXPERIMENT INVERSE PROBLEMS OF CHEMICAL KINETICS

7 The number of independent columns equals the number of unknown constants s. The investigation noninformayivity of measurements The problem of non-uniqueness of solutions of principle. The main problem - informativity analysis of the measurements. Establishment of appropriate methods and algorithms mathematical software. UNIQUENESS CRITERION (Calman and oth.)

8 Consequence of Jacobi matrix degeneration dimensionis existence The system of independence nonlinear parametric functions is existence - the independence parameters of model The equations system for independence parameters The system of linear partial differential equations of the first order with variable coefficients

9 The main theorem There is the dimension of the matrix Instead of the Jacobian matrix is sufficient to consider the matrix U, structure is completely determined by the type of right-hand sides of the original system of equations, therefore, the reaction mechanism We prove the solvability of analytic systems of differential equations to determine the independent parametric functions. A constructive procedure to determine them. Implemented as a system of analytical calculations QUASISTIONARITY

10 Non-uniqueness of solutions There is a transformation invariant with respect to X. Transformations allowed by the original system, form a group of transformations. The number of generators of the group - number of independent parameters of the model. The explicit form of generators - expressions for the independent parameters. For quasi-stationary approximationproved to be equivalent to the approach based on an analysis of the matrix U. NONSTAIONARY

11 Derivation of explicit expressions for the independent model parameters. The system of design algorithms for exceptions based on the methods of computer algebra. The software is developed. Direct exclusion of Y by moving to systems of differential equations high dimension of X. EXCEPTION UNMEASURED CONCENTRATIONS

12 N – number of measurements Normal distribution of measurement error – the sum of squared deviations (the minimal squares method) Laplace distribution of measurement error - sum of absolute deviations The uniform law of distribution of measurement error - the module maximum deviation (Chebyshev method for alignment) "Everyone believes in the normal distribution in its own way a mathematician thinks he should be out of the experiment, the experimenter believes that it is strictly proved mathematic”. A. Poincaré CRITERIA FOR A CALCULATION OF MEASUREMENTS

13 Leonid V. Kantorovich. On some new approaches to numerical methods and observations processing // Siberian Mathematical Journal, 1962, v.3, №5, p The basic idea - Correlation of the measurement error with errors of parameters of mathematical models. The main result - No hypothesis on the distribution of the measurement error. Compute the domain within which each point describes a measurement conforming measurement errors. No minimization of the criterion of error measurements conformity. Nowadays, this is also knows as set-membership approach

14 Calculation - experiment: - vector of the maximum permissible error of measurement We determine under the conditions of compliance We solve 2s mathematical programming problems: For a fixed s, compute No information provided by a constant. The problem - Planning of measurements in order to reduce the uncertainty. The possibility of using solutions dual tasks. UNCERTAINTY INTERVALS FOR THE PARAMETERS

15 The methods and algorithms for calculating the domains of uncertainty of kinetic parameters in constructing mathematical models of complex chemical reactions are developed and implemented. SCHEME OF STUDY Analytical analysis of information content. The ideology of the Gauss relation of the parameters kinetic models of complex chemical reactions with the characteristics of the measurements. Kantorovich approach (aka ‘set-membership approach’). CONCLUSION

16 Thank You for Your Attention!