Restricted gene expression programming: a new approach for parameter identification inverse problems of partial differential equation Methodologies and Application First Online: 09 December 2015 DOI :
10.1007/s00500-015-1965-1

Cite this article as: Chen, Y., Li, K., Chen, Z. et al. Soft Comput (2017) 21: 2651. doi:10.1007/s00500-015-1965-1
Abstract Traditionally, solving the parameter identification inverse problems of partial differential equation encountered many difficulties and insufficiency. In this article, we proposed a restricted gene expression programming (GEP) for parameter identification inverse problems of partial differential equation based on the self-adaption, self-organization, and self-learning characters of GEP. The algorithm simulates parametric function itself of partial differential equation directly through the observed values by taking effect to inverse results caused by noise of the measured value into full consideration. Modeling is unnecessary to add regularization in modeling process aiming at special problems again. The experiment results also showed that the algorithm has good noise-immunity. In case there is no noise or noise is very low, the identified parametric function is almost the same as the original accurate value; when noise is very high, accurate result can still be obtained, which successfully realizes automation of parameter modeling process for partial differential equation.

Keywords Gene expression programming Partial differential equation Inverse problems Thomas algorithm

References Ahmadi MB, Kiani NA, Mikaeilvand N (2014) Laplace transform formula on fuzzy nth-order derivative and its application in fuzzy ordinary differential equations[J]. Soft Comput 18(12):2461–2469

CrossRef MATH Google Scholar Ali A, Soheil S, Seng CC (2015) A Runge-Kutta method with reduced number of function evaluations to solve hybrid fuzzy differential equations[J]. Soft Comput 19(4):1051–1062

CrossRef MATH Google Scholar Allahviranloo T, Abbasbandy S, Behzadi SS (2014) Solving nonlinear fuzzy differential equations by using fuzzy variational iteration method. Soft Comput 18(1):2191–2200

CrossRef MATH Google Scholar Allahviranloo T, Behzadi SS (2014) The use of airfoil and Chebyshev polynomials methods for solving fuzzy Fredholm integro-differential equations with Cauchy kernel[J]. Soft Comput 18(10):1885–1897

CrossRef MATH Google Scholar Allahviranloo T, Chehlabi M (2015) Solving fuzzy differential equations based on the length function properties. Soft Comput 19(2):307–320

CrossRef MATH Google Scholar Aoki Y, Hayami K, De Sterck H et al (2014) Cluster Newton method for sampling multiple solutions of underdetermined inverse problems: application to a parameter identification problem in pharmacokinetics. SIAM J Sci Comput 36(1):B14–B44

MathSciNet CrossRef MATH Google Scholar Bao W, Song Y (2014) Multiquadric quasi-interpolation methods for solving partial differential algebraic equations. Numer Methods Partial Differ Equ 30(1):95–119

MathSciNet CrossRef MATH Google Scholar Behzadi SS (2015) A new study on first-order fuzzy Fredholm-Volterra integro-differential equations by Jacobi polynomials and collocation methods. Soft Comput 19(2):421–429

CrossRef MATH Google Scholar Bui-Thanh T (2015) From Godunov to a unified hybridized discontinuous Galerkin framework for partial differential equations[J]. J Comput Phys 295:114–146

MathSciNet CrossRef MATH Google Scholar Che H, Zhou Z, Jiang Z et al (2013) H1-Galerkin expanded mixed finite element methods for nonlinear pseudo-parabolic integro-differential equations. Numer Methods Partial Differ Equ 29(3):799–817

CrossRef MATH Google Scholar Chen C, Jiang Y-L (2015) Lie group analysis method for two classes of fractional partial differential equations. Commun Nonlinear Sci Numer Simul 26(1—-3):24–35

MathSciNet CrossRef Google Scholar Chiappino S, Morerio P, Marcenaro L, Regazzoni CS (2015) Bio-inspired relevant interaction modelling in cognitive crowd management. J Ambient Intell Humaniz Comput 6(2):171–192

Churbanov DV, Shcheglov AY (2013) An iterative method for solving an inverse problem for a first-order nonlinear partial differential equation with estimates of guaranteed accuracy and the number of steps[J]. Comput Math Math Phys 53(2):215–220

MathSciNet CrossRef MATH Google Scholar Denisov AM (2014) Inverse problem for a quasilinear system of partial differential equations with a nonlocal boundary condition[J]. Comput Math Math Phys 54(10):1513–1521

MathSciNet CrossRef MATH Google Scholar Engbers R, Burger M, Capasso V (2028) Inverse problems in geographical economics: parameter identification in the spatial Solow model. Philos Trans R Soc A 2014:372

MATH Google Scholar Fenu G, Pau PL (2015) Evaluating complex network indices for vulnerability analysis of a territorial power grid. J Ambient Intell Humaniz Comput 6(3):297–306

CrossRef Google Scholar Ferreira C (2001) Gene expression programming: a new adaptive algorithm for solving problems. Complex Syst 13(2):87–129

MathSciNet MATH Google Scholar Groetsch CW, Groetsch CW (1993) Inverse problems in the mathematical sciences. Springer, New York

CrossRef MATH Google Scholar Harker M, O’Leary P (2015) Sylvester equations and the numerical solution of partial fractional differential equations. J Comput Phys 293:370–384

MathSciNet CrossRef MATH Google Scholar Herrera I, de la Cruz LM, Rosas-Medina A (2014) Nonoverlapping discretization methods for partial differential equations. Numer Methods Partial Differ Equ 30(5):1427–1454

MathSciNet CrossRef MATH Google Scholar Isakov V (2006) Inverse problems for partial differential equations. Springer, Berlin

MATH Google Scholar Kwon K (2015) Uniqueness and nonuniqueness in inverse problems for elliptic partial differential equations and related medical imaging. Adv Math Phys 2015(2015):908251

Lee C-C, Chang P-C (2006) Integrated traffic modeling and frame skipping for pre-stored streaming videos over cellular networks[J]. J High Speed Netw 15(4):329–340

MathSciNet Google Scholar Li K, Chen Z, Li Y, Zhou A (2005) An application of genetic programming to economic forecasting. In: Proceedings of the international conference on high performance computing and applications (HPCA2004) (Lecture notes in computational science and engineering), vol 8. Springer, Heidelberg, pp 71–80

Li K, Li Y, Chen Z, Wu Z (2005) A new dynamic evolutionary algorithm based on particle transportation theory. In: Proceedings of the international conference on high performance computing and applications (HPCA2004) (Lecture notes in computational science and engineering), vol 8. Springer, Heidelberg, pp 81–92

Liu H, Wu Z, Li H, Wang Z (2015) Parameter identification of hyperbolic equation based on differential evolution algorithm. J Wuhan Univ 61(2):117–123

MathSciNet MATH Google Scholar Marin L, Karageorghis A, Lesnic D (2015) A numerical study of the SVD-MFS solution of inverse boundary value problems in two-dimensional steady-state linear thermoelasticity. Numer Methods Partial Differ Equ 31(1):168–201

MathSciNet CrossRef MATH Google Scholar Naz R, Mahomed FM (2015) A complex Noether approach for variational partial differential equations. Commun Nonlinear Sci Numer Simul 27(1–3):120–135

MathSciNet CrossRef Google Scholar Nieto JJ (2012) Periodic boundary value problems for first-order impulsive ordinary differential equations. Nonlinear Anal 51(7):1223–1232

MathSciNet CrossRef MATH Google Scholar Ozkum G, Demir A, Erman S et al. (2013) On the inverse problem of the fractional heat-like partial differential equations: determination of the source function. Adv Math Phys 2013(2013):476154

Pao CV, Chang YH, Jau G-C (2013) Numerical methods for a coupled system of differential equations arising from a thermal ignition problem[J]. Numer Methods Partial Differ Equ 29(1):251–279

MathSciNet CrossRef MATH Google Scholar Romanov VG (1987) Inverse problems of mathematical physics. BRILL, Leiden

Google Scholar Siddiqi SS, Arshed S (2015) Numerical solution of time-fractional fourth-order partial differential equations. Int J Comput Math 92(7):1496–1518

MathSciNet CrossRef MATH Google Scholar Srivastava V, Tripathi BK, Pathak VK (2014) Biometric recognition by hybridization of evolutionary fuzzy clustering with functional neural networks. J Ambient Intell Humaniz Comput 5(4):525–537

CrossRef Google Scholar Vieira FHT, Bianchi GR, Lee LL (2008) A network traffic prediction approach based on multifractal modeling. J High Speed Netw 17(2):83–96

Google Scholar Wang Z, Wang H, Qiu S (2015) A new method for numerical differentiation based on direct and inverse problems of partial differential equations[J]. Appl Math Lett 43:61–67

MathSciNet CrossRef MATH Google Scholar Wang Q, Wen J (2014) Analytical and numerical stability of partial differential equations with piecewise constant arguments. Numer Methods Partial Differ Equ 30(1):1–6

MathSciNet CrossRef MATH Google Scholar Watanabe Y, Kinoshita T, Nakao MT (2013) A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations. Math Comput 82(283):1543–1557

Wazwaz A (2012) A new method for solving singular initial value problems in the second-order ordinary differential equations[J]. Appl Math Comput 128(1):45–57

MathSciNet CrossRef MATH Google Scholar Wu Z (2004) Evolutionary optimization and its applications in inverse problems of differential equations. Wuhan University, Wuhan

Google Scholar © Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations 1. College of Mathematics and Informatics South China Agricultural University Guangzhou China 2. Department of Chemical and Petroleum Engineering University of Calgary Calgary Canada