Abstract
The differential evolution algorithm is applied to solve the optimization problem to reconstruct the production function (inverse problem) for the spatial Solow mathematical model using additional measurements of the gross domestic product for the fixed points. Since the inverse problem is ill-posed the regularized differential evolution is applied. For getting the optimized solution of the inverse problem the differential evolution algorithm is paralleled to 32 kernels. Numerical results for different technological levels and errors in measured data are presented and discussed.
Funding source: Russian Science Foundation
Award Identifier / Grant number: 18-71-10044
Funding source: Ministry of Education and Science of the Republic of Kazakhstan
Award Identifier / Grant number: AP05134121
Funding statement: This work is supported by the Russian Science Foundation (grant No. 18-71-10044), i.e. numerical investigation of spatial Solow mathematical model, and by the Ministry of Education and Science of the Republic of Kazakhstan (grant No. AP05134121), i.e. inverse problem statement (Section 2).
Acknowledgements
The authors thank to Professor Daniyar Nurseitov for problem statement and fruitfull discussions for concerning Solow mathematical model and thank to Dr. Igor Chernykh for help in parallel realization of DE algorithm.
References
[1] S. Acikgoz and M. Mert, A short note on the fallacy of identification of technological progress in models of economic growth, SAGE Open 5 (2015), 1–5. 10.1177/2158244015579942Search in Google Scholar
[2] S. Aniţa, V. Capasso, H. Kunze and D. La Torre, Dynamics and control of an integro-differential system of geographical economics, Ann. Acad. Rom. Sci. Ser. Math. Appl. 7 (2015), no. 1, 8–26. Search in Google Scholar
[3] Ç. Balkaya, An implementation of differential evolution algorithm for inversion of geoelectrical data, J. Appl. Geophys. 98 (2013), 160–175. 10.1016/j.jappgeo.2013.08.019Search in Google Scholar
[4] D. Bayanjargal R. Enkhbat and A. Griewank, Global optimization approach to the Solow growth theory, Adv. Model. Optimiz. 12 (2010), 133–140. Search in Google Scholar
[5] M. Burger, L. Caffarelli and P. A. Markowich, Partial differential equation models in the socio-economic sciences, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014), no. 2028, Article ID 20130406. 10.1098/rsta.2013.0406Search in Google Scholar PubMed PubMed Central
[6] C. Camacho, B. Zou and M. Briani, On the dynamics of capital accumulation across space, European J. Oper. Res. 186 (2008), no. 2, 451–465. 10.1016/j.ejor.2007.02.031Search in Google Scholar
[7] V. Capasso, R. Engbers and D. La Torre, On a spatial Solow model with technological diffusion and nonconcave production function, Nonlinear Anal. Real World Appl. 11 (2010), no. 5, 3858–3876. 10.1016/j.nonrwa.2010.01.016Search in Google Scholar
[8] S. Dasgupta, S. Das, A. Biswas and A. Abraham, On stability and convergence of the population-dynamics in differential evolution, AI Commun. 22 (2009), no. 1, 1–20. 10.3233/AIC-2009-0440Search in Google Scholar
[9] J. Davalos, R. Munguia and S. Urzua, Estimation of the Solow–Cobb–Douglas economic growth model with a Kalman filter: An observability-based approach, Heliyon 5 (2019), e01959. 10.1016/j.heliyon.2019.e01959Search in Google Scholar PubMed PubMed Central
[10] R. Engbers, Spatial Structures in Geographical Economics, Westfalische Wilhelms-Universität Münster, Münster, 2009. Search in Google Scholar
[11] R. Engbers, M. Burger and V. Capasso, Inverse problems in geographical economics: Parameter identification in the spatial Solow model, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014), no. 2028, Article ID 20130402. 10.1098/rsta.2013.0402Search in Google Scholar PubMed
[12] M. Gumpert, Regional economic disparities under the Solow model, Qual. Quant. (2019), 10.1007/s11135-019-00836-2. 10.1007/s11135-019-00836-2Search in Google Scholar
[13] W. Hahn, Theory and Application of Liapunov’s Direct Method, Prentice-Hall, Englewood Cliffs, 1963. Search in Google Scholar
[14] T. L. Hill, Statistical Mechanics: Principles and Selected Applications, McGraw-Hill, New York, 1956. Search in Google Scholar
[15] R. Horst, P. M. Pardalos and N. V. Thoai, Introduction to Global Optimization, Nonconvex Optim. Appl. 3, Kluwer Academic, Dordrecht, 1995. 10.1007/978-1-4615-2025-2Search in Google Scholar
[16] J. P. Juchem Neto, J. C. R. Claeyssen and S. S. Pôrto Júnior, Returns to scale in a spatial Solow–Swan economic growth model, Phys. A 533 (2019), Article ID 122055. 10.1016/j.physa.2019.122055Search in Google Scholar
[17] S. I. Kabanikhin, Definitions and examples of inverse and ill-posed problems, J. Inverse Ill-Posed Probl. 16 (2008), no. 4, 317–357. 10.1515/JIIP.2008.019Search in Google Scholar
[18] S. Kabanikhin, O. Krivorotko, S. Zhang and V. Kashtanova, Inverse problems for parabolic equations arising in social networks, 2019. Search in Google Scholar
[19] J. Kennedy and R. Eberhart, Particle swarm optimisation, IEEE International Conference on Neural Networks, IEEE Press, Piscataway (1995), 1942–1948. Search in Google Scholar
[20] S. Kirkpatrick, C. D. Gelatt, Jr. and M. P. Vecchi, Optimization by simulated annealing, Science 220 (1983), no. 4598, 671–680. 10.1126/science.220.4598.671Search in Google Scholar
[21] P. Krugman, Increasing returns and economic geography, J. Political Econ. 99 (1991), 483–499. 10.3386/w3275Search in Google Scholar
[22] S. Kuznets, National income, Technical Report 73d Congress, 2d Session, Senate Document no. 124, U.S. Government Printing Office, Washington, 1934. Search in Google Scholar
[23] S. Kuznets, National Income. 1919-1938, National Bureau of Economic Research, Washington, 1941. Search in Google Scholar
[24] P. Mossay, Increasing returns and heterogeneity in a spatial economy, Regional Sci. Urban Econ. 33 (2003), 419–444. 10.1016/S0166-0462(02)00041-8Search in Google Scholar
[25] K. Prettner, V. Kufenko and V. Geloso, Divergence, convergence, and the history-augmented Solow model, Struct. Change Econ. Dynam. 53 (2020), 62–76. 10.1016/j.strueco.2019.12.008Search in Google Scholar
[26] K. V. Price, R. M. Storn and J. A. Lampinen, Differential Evolution, Nat. Comput. Ser., Springer, Berlin, 2005. Search in Google Scholar
[27] A. Qing, Differential Evolution: Fundamentals and Applications in Electrical Engineering, John Wiley & Sons, New York, 2009. 10.1002/9780470823941Search in Google Scholar
[28] R. Sato, The impact of technical change on the holotheticity of production functions, Rev. Econ. Stud. 47 (1980), 767–776. 10.2307/2296942Search in Google Scholar
[29] R. G. Smirnov and K. Wang, In search of a new economic model determined by logistic growth, European J. Appl. Math. 31 (2020), no. 2, 339–368. 10.1017/S0956792519000081Search in Google Scholar
[30] R. M. Solow, A contribution to the theory of economic growth, Quart. J. Econ. 70 (1956), 65–94. 10.2307/1884513Search in Google Scholar
[31] R. Storn, Differential evolution research – Trends and open questions, Advances in Differential Evolution, Stud. Comput. Intell. 143, Springer, Berlin (2008), 1–31. 10.1007/978-3-540-68830-3_1Search in Google Scholar
[32] R. Storn and K. Price, Differential evolution – A simple and efficient adaptive scheme for global optimization over continuous spaces, Report no. TR-95-012, International Computer Science Institute, Berkeley, 1995. Search in Google Scholar
[33] R. Storn and K. Price, Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces, J. Global Optim. 11 (1997), no. 4, 341–359. 10.1023/A:1008202821328Search in Google Scholar
[34] B. Varszegi, On the convergence and the steady state in a delayed Solow model, IFAC-PapersOnLine 51 (2018), 94–99. 10.1016/j.ifacol.2018.07.205Search in Google Scholar
[35] S. J. Wright and J. Nocedal, Numerical Optimization, Springer Ser. Oper. Res. Financ. Eng., Springer, New York, 1999. 10.1007/b98874Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
