Cases where the derivative of a boundary value problem does not exist or is constantly changing, traditional derivative can easily get stuck in the local optima or does not factually represent a constantly changing solution. Hence the need for evolutionary algorithms becomes evident. However, evolutionary algorithms are compute-intensive since they scan the entire solution space for an optimal solution. Larger populations and smaller step sizes allow for improved quality solution but results in an increase in the complexity of the optimization process. In this research a population-distributed implementation for differential evolution algorithm is presented for solving systems of 2^{nd}-order, 2-point boundary value problems (BVPs). In this technique, the system is formulated as an optimization problem by the direct minimization of the overall individual residual error subject to the given constraint boundary conditions and is then solved using differential evolution in the sense that each of the derivatives is replaced by an appropriate difference quotient approximation. Four benchmark BVPs are solved using the proposed parallel framework for differential evolution to observe the speedup in the execution time. Meanwhile, the statistical analysis is provided to discover the effect of parametric changes such as an increase in population individuals and nodes representing features on the quality and behavior of the solutions found by differential evolution. The numerical results demonstrate that the algorithm is quite accurate and efficient for solving 2^{nd}-order, 2-point BVPs.
[1] Gong, Y.J., Chen, W.N., Zhan, Z.H., Zhang, J., Li, Y., Zhang, Q. and Li, J.J., 2015, Distributed evolutionary algorithms and their models: A survey of the state-of-the-art, Applied Soft Computing, 34, pp. 286-300. DOI: 10.1016/j.asoc.2015.04.061
[2] Zelinka, I., 2015, A survey on evolutionary algorithms dynamics and its complexity–Mutual relations, past, present and future, Swarm and Evolutionary Computation, 25, pp. 2-14. DOI: 10.1016/j.swevo.2015.06.002
[3] Price, K., Storn, R.M. and Lampinen, J.A., 2006, Differential evolution: a practical approach to global optimization, Springer Science Business Media, ISBN: 978-3-540-20950-8
[4] Storn, R. and Price, K., 1997, Differential Evolution–a simple and efficient heuristic for global optimization over continuous spaces, Journal of global optimization, 11(4), pp. 341-359. DOI: 10.1023/A:1008202821328
[5] Charles, A.J. and Parks, G.T., 2017, Mixed Oxide LWR Assembly Design Optimization Using Differential Evolution Algorithms, 2017 25th International Conference on Nuclear Engineering, Shanghai, China, 9, pp. V009T15A065. DOI: 10.1115/ICONE25-67936
[6] Zaharie, D. and Petcu, D., 2005, Parallel implementation of multi-population differential evolution, Proc. of the NATO Advanced Research Workshop on Concurrent information processing and computing, Nicolau, A. and Grigoras, D., eds., Sinaia, Romania, pp. 223-232.
[7] Ge, Y.F., Yu, W.J. and Zhang, J., 2016, Diversity-Based Multi-Population Differential Evolution for Large-Scale Optimization, Proc. of the 2016 on Genetic and Evolutionary Computation Conference Companion, Denver, Colorado, USA, pp. 31-32. DOI: 10.1145/2908961.2908995
[8] Cheng, J., Zhang, G., Caraffini, F. and Neri, F., 2015, Multicriteria adaptive differential evolution for global numerical optimization, Integrated Computer-Aided Engineering, 22(2), pp. 103-107. DOI: 10.3233/ICA-150481
[9] Lobato, F.S., Steffen Jr, V. and Silva Neto, A.J., 2010, A comparative study of the application of differential evolution and simulated annealing in radiative transfer problems, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 32(SPE), pp. 518-526. DOI: 10.1590/S1678-58782010000500012
[10] Hartfield, R.J., Jenkins, R.M. and Burkhalter, J.E., 2007, Ramjet powered missile design using a genetic algorithm, Journal of Computing and Information Science in Engineering, 7(2), pp. 167-173. DOI: 0.1115/1.2738722
[11] Penas, D.R., Banga, J.R., González, P. and Doallo, R., 2015, Enhanced parallel differential evolution algorithm for problems in computational systems biology, Applied Soft Computing, 33, pp. 86-99. DOI: 10.1016/j.asoc.2015.04.025
[12] González-Álvarez, D.L., Vega-Rodríguez, M.A. and Rubio-Largo, Á., 2014, Parallelizing and optimizing a hybrid differential evolution with Pareto tournaments for discovering motifs in DNA sequences, The Journal of Supercomputing, 70(2), pp. 880-905. DOI: 10.1007/s11227-014-1266-y
[13] Kozlov, K. and Samsonov, A., 2011, DEEP—differential evolution entirely parallel method for gene regulatory networks, The Journal of Supercomputing, 57(2), pp. 172-178. DOI: 10.1007/s11227-010-0390-6
[14] Maciejewski, Ł., 2007, Application of differential evolution algorithm for identification of experimantal data, Archive of Mechanical Engineering, 54(4), pp. 327-337.
[15] Nayak, N., Routray, S.K. and Rout, P.K., 2016, Design of Takagi-Sugeno fuzzy controller for VSCHVDC parallel AC transmission system using differential evolution algorithm, International Journal of Computer Aided Engineering and Technology, 8(3), pp. 277-294. DOI: 10.1504/IJCAET.2016.077605
[16] Mokhtari, H. and Salmasnia, A., 2015, A Monte Carlo simulation based chaotic differential evolution algorithm for scheduling a stochastic parallel processor system, Expert Systems with Applications, 42(20), pp. 7132-7147. DOI: 10.1016/j.eswa.2015.05.015
[17] Acebrón, J.A. and Spigler, R., 2007, Supercomputing applications to the numerical modeling of industrial and applied mathematics problems, The Journal of Supercomputing, 40(1), pp. 67-80. DOI: 10.1007/s11227-006-0014-3
[18] Tardivo, M.L., Caymes-Scutari, P., Mendez-Garabetti, M. and Bianchini, G., 2013, Two models for parallel differential evolution, Proc. of HPCLatAm, C. Garcia Garino and M. Printista, eds., Mendoza, Argentina, pp. 25-36.
[19] Ntipteni, M.S., Valakos, I.M. and Nikolos, I.K., 2006, An asynchronous parallel differential evolution algorithm, Proc. of the ERCOFTAC conference on design optimisation: methods and application.
[20] Fateh, M.F., Zameer, A., Mirza, N.M., Mirza, S.M. and Raja, M.A.Z., 2017, Biologically inspired computing framework for solving two-point boundary value problems using differential evolution, Neural Computing and Applications, 28(8), pp. 2165-2179. DOI: 10.1007/s00521-016-2185-z
[21] Tasoulis, D.K., Pavlidis, N.G., Plagianakos, V.P. and Vrahatis, M.N., 2004, Parallel differential evolution, Proc. of the 2004 Congress on Evolutionary Computation, Portland, Oregon, USA, pp. 2023-2029. DOI: 10.1109/CEC.2004.1331145
[22] Abo-Hammour, Z.S., Yusuf, M., Mirza, N.M., Mirza, S.M., Arif, M. and Khurshid, J., 2004, Numerical solution of second-order, two-point boundary value problems using continuous genetic algorithms, International Journal for Numerical Methods in Engineering, 61(8), pp. 1219-1242. DOI: 10.1002/nme.1108
[23] Tat, C.K., Majid, Z.A., Suleiman, M. and Senu, N., 2012, Solving Linear Two-Point Boundary Value, Applied Mathematical Sciences, 6(99), pp. 4921-4929.
[24] Zurita, N.F.S., Colby, M.K., Tumer, I.Y., Hoyle, C. and Tumer, K., 2018, Design of Complex Engineered Systems Using Multi-Agent Coordination, Journal of Computing and Information Science in Engineering, 18(1), pp. 011003. DOI: 10.1115/1.4038158