1. AN ITERATIVE METHOD FOR MODEL PARAMETER IDENTIFICATION 4. DIFFERENTIAL EQUATION MODELS E.Dimitrova, Chr. Boyadjiev E.Dimitrova, Chr. Boyadjiev BULGARIAN ACADEMY OF SCIENCES INSTITUTE OF CHEMICAL ENGINEERING 9 TH Workshop on “Transport Phenomena in Two-Phase Flow”

2. INTRODUCTION Mathematical modeling relevant to heat and mass transfer operation is based on development of adequate mathematical structures (differential equations) employing corresponding physical mechanisms. The model build-up requires values of coefficients that can be obtained through processing of experimental data. Such processing is based on solution of inverse problems and especially inverse identification problems. Very often this problem is incorrect (ill-posed) that implies a solution sensibility due to errors of the experimental data used. An iterative method for inverse problem solution was used for model parameter identification of different models. In this paper the method will be applied for differential equations models. 9 TH Workshop on “Transport Phenomena in Two-Phase Flow”

3. PROBLEM FORMULATION Let’s consider two-parameter differential equation model where and are the exact values of the parameter. The parameter identification problem will be solved by help of artificial experimental data provided by a random number generator: where A n are random numbers within the interval [0, 1]. 9 TH Workshop on “Transport Phenomena in Two-Phase Flow”

4. Fig.1 Mathematical model and “experimental” data [] - - “experimental” data with a maximal relative error  5% [──] - - model for b 1 =1 ; b 2 =5. x The values of objective function y n are obtained from the model when x changes in the interval [0;1] ( x=0.01n (n=0,…,99)). The maximal relative errors of these “experimental” data are  5%. The mathematical model and “experimental” data are shown on Fig. 1. These plots show that when 0<x<0.3 the inverse identification problem is correct, while in the case of 0.31<x<0.65 it is incorrect. The problem becomes essentially incorrect when 0.66<x<1. The parameters in the model can be determined through processing of N experimental values of the objective function ( n=1,…,N ). This requires a least square function to be used: 9 TH Workshop on “Transport Phenomena in Two-Phase Flow” PROBLEM FORMULATION Model:

5. Every method for solving of incorrect problems should solve also correct ones. Therefore, the first solution of the inverse problem considered here corresponds to the interval 0<x<0.3. Let’s consider two-parameter differential equation model. Figure 2 shows models for exact parameter values ( ), calculated parameter values ( ) and “experimental” data. x Fig.2 Model and “experimental” data [  ] - - “experimental” data with a maximal relative error  5% [──] - - model for [-----] - - model for 9 TH Workshop on “Transport Phenomena in Two-Phase Flow” CORRECT PROBLEM SOLUTION Model:

6. The results of the identification problem solution are summarized in Table 1, where and are calculated values of the parameters, i – iteration number, γ - regularization parameter. 55 0.99555.0415150 Table 1. Correct problem solution (0<x<0.3, γ=1.5 ) 9 TH Workshop on “Transport Phenomena in Two-Phase Flow” CORRECT PROBLEM SOLUTION

7. 0.56.00.99114.9442255 0.76.00.99214.9555235 0.96.00.99215.0012190 1.16.00.99555.0415150 1.36.00.99234.9575229 1.56.00.99925.0311164 The efficiency of every iterative method for function minimization depends on the initial approximation. Parameter values obtained under conditions imposed by different initial approximations are summarized in Table 2. Table 2. Effect of the initial approximation (0<x<0.3, γ=1.5) 9 TH Workshop on “Transport Phenomena in Two-Phase Flow” CORRECT PROBLEM SOLUTION

8. The iteration number depends on the regularization parameter value γ and the efficiency of the minimization increases when the value of γ is increased. This effect is demonstrated through data summarized on Table 3. 10.99444.9832183 1.50.99555.0415150 20.99705.0616138 Table 3. Effect of γ 9 TH Workshop on “Transport Phenomena in Two-Phase Flow” CORRECT PROBLEM SOLUTION

9. As commented before, if “experimental” data are obtained under conditions (regimes), corresponding to the interval 0.31<x<0.65, the parameter identification problem will be ill-posed. The problem incorrectness is due to solution sensibility with respect to “experimental” errors of the objective function. Let’s consider a solution of the parameter identification problem through minimization of the least square function, when the inverse identification problem is incorrect (with x=0.01n, n=31,…,65, i.e. 0.31<x<0.65. The solution ( ) ) is shown in Table 4. INCORRECT PROBLEM SOLUTION Table 4. Incorrect problem solution (0.31<x<0.65, γ=0.5 ) 55 1.13445.3709 203 9 TH Workshop on “Transport Phenomena in Two-Phase Flow”

10. Comparisons between obtained model, exact model, and “experimental” data in the case of correct and incorrect problem are illustrated by plots on Fig.2 and Fig.3. These plots indicate very small differences between obtained model and the exact model. x Fig.2 Model and “experimental” data (0<x<0.3, γ=1.5 ) [  ] - - “experimental” data with a maximal relative error  5% [──] - - model for [-----] - - model for x Fig.3 Model and “experimental” data (0.31<x<0.65, γ=0.5 ) [  ] - - “experimental” data with a maximal relative error  5% [──] - - model for [-----] - - model for 9 TH Workshop on “Transport Phenomena in Two-Phase Flow”

11. The iteration numbers depend on the initial approximations of the iterative procedure. The results for 0.31<x<0.65 are summarized in Table 5. Table 5. Effect of the initial approximation (0.31<x<0.65, γ=0.5) 0.56.01.23875.617867 0.76.01.16795.4573115 0.96.01.17035.4584112 1.16.01.13445.3709203 1.56.01.17115.4591103 2.06.01.17085.4585158 3.06.01.16035.4338333 9 TH Workshop on “Transport Phenomena in Two-Phase Flow” INCORRECT PROBLEM SOLUTION

12. The effect of the regularization parameter γ on the iteration numbers are summarized in Table 6. When the regularization parameter increase, then the iteration number is decrease. 0.051.16075.4344516 0.51.13445.3709203 1.21.25785.657232 Table 6. Effect of γ 9 TH Workshop on “Transport Phenomena in Two-Phase Flow” INCORRECT PROBLEM SOLUTION

13. The results presented on Fig. 3 demonstrate that the differences between the exact model and the models derived through parameter identifications are very small. On the other hand, the results in Table 5 show that the differences between the exact and the obtained values of the parameters are significant. The correctness of the parameter identification will be tested below through a criterion of model adequacy. x Fig.3 Model and “experimental” data (0.31<x<0.65, γ=0.5 ) [  ] - - “experimental” data with a maximal relative error  5% [──] - - model for [-----] - - model for Table 5. Effect of the initial approximation 0.56.01.23875.617867 0.76.01.16795.4573115 0.96.01.17035.4584112 1.16.01.13445.3709203 1.56.01.17115.4591103 2.06.01.17085.4585158 3.06.01.16035.4338333

14. The result of the parameter identification of the two-parameter model with initial approximations and where the inverse problem is essentially incorrect is shown in the Table 7. ESSENTIALLY INCORRECT PROBLEM SOLUTION Table 7. Essentially incorrect problem solution (0.65<x<1, γ=5 ) 55 1.92016.012142 9 TH Workshop on “Transport Phenomena in Two-Phase Flow”

15. The comparison with “experimental” data is shown on Fig. 4. Both the data in Table 7 and plots on Fig. 4 indicate that the differences between obtained and exact parameter values are very large. On the other hand, the differences between obtained and exact models exhibit just the opposite behavior. x Fig.4 Model and “experimental” data (0.31<x<0.65, γ=0.5 ) [  ] - - “experimental” data with a maximal relative error  5% [──] - - model for [-----] - - model for Table 7. Essentially incorrect problem solution (0.65<x<1, γ=5 ) 55 1.92016.012142 9 TH Workshop on “Transport Phenomena in Two-Phase Flow”

16. The effects of the initial approximation and the number of regularization parameter γ also were investigated. The results are summarized in the Tables 8 and 9. Table 8. Effect of the initial approximation 0.56.02.27976.258596 0.76.02.08506.090655 0.96.01.77145.872129 1.16.01.92016.012142 1.56.02.29056.264978 2.06.01.79245.917417 3.06.03.07866.650275 0.54.08277.0680902 14.07887.0664901 22.27606.256290 32.20626.195462 51.92016.012142 103.42876.7541233 Table 9. Effect of γ (0.65<x<1, γ=5) 9 TH Workshop on “Transport Phenomena in Two-Phase Flow” ESSENTIALLY INCORRECT PROBLEM SOLUTION Model adequacy test as a criterion for correctness of results of parameter identification is applied below.

17. Statistical Analysis of Model Adequacy The model adequacy is defined by the variance ratio: where S and S ε are model and experimental error variance. The value of F is compared to the tabulated values ( F J ) of the Fisher’s distribution. The condition of the model adequacy is where ν=N - J, ν ε =K – 1,  =0.01 9 TH Workshop on “Transport Phenomena in Two-Phase Flow”

18. The statistical analysis of the model adequacy was performed when the inverse problem is correct, incorrect and essentially incorrect and the results are presented in Table 10. The results confirm the adequacy of the model. When the inverse problem is incorrect the model is adequate despite the large differences between the calculated and the exact values of the model parameters. In case of essential incorrectness of the inverse problem the model employed in this paper is adequate irrespective of the large differences between calculated and exact values of model parameters. 9 TH Workshop on “Transport Phenomena in Two-Phase Flow” Statistical Analysis of Model Adequacy 55 0.99555.04151.51.79411.79200.99762.25 55 1.13445.37090.52.60502.35290.81572.20 55 1.92016.012152.78542.61990.88472.20 Table 10. Statistical analysis of the model adequacy

19. Conclusions An iterative method and algorithm for differential equations model parameters identification in cases of incorrect inverse problems are proposed. Solutions of model parameters identification problems through least square function minimizations show large differences between exact and calculated parameter values. This difference cannot be explained by experimental data size only, but they come also from the inverse problem incorrectness that is mainly due to parameter sensibility with respect to the experimental data errors. Thus, a minimization of least square function cannot be assumed as a solution of parameter identification problem. Additional condition for inverse problem regularization is introduced in the procedure proposed. This condition permits the least square function minimization to be employed for solutions of model parameter identification problem. The model adequacy was tested through a statistical analysis. The latter can be assumed as a criterion of applicability of the iterative method proposed. The solution procedure of the essential incorrect inverse problem for model parameter identification shows, that the iterative procedure convergences when the parameter values are very sensible with respect to the errors of the experimental data. 9 TH Workshop on “Transport Phenomena in Two-Phase Flow”