April  2018, 14(2): 817-831. doi: 10.3934/jimo.2017077

Parameter identification techniques applied to an environmental pollution model

1. 

School of Mathematics and Statistics, Nanjing University of Information Science and Technology (NUIST), Nanjing, China, 210044

2. 

Department of Scientific Computing, Florida State University, Tallahassee, FL 32306, USA

* Corresponding author: I. Michael Navon

* Corresponding author: Yuepeng Wang

Received  January 2016 Revised  August 2017 Published  September 2017

Fund Project: The first author is supported by NSFC (41375115,61572015) and (ICT1600262)

The retrieval of parameters related to an environmental model is explored. We address computational challenges occurring due to a significant numerical difference of up to two orders of magnitude between the two model parameters we aim to retrieve. First, the corresponding optimization problem is poorly scaled, causing minimization algorithms to perform poorly (see Gill et al., practical optimization, AP, 1981,401pp). This issue is addressed by proper rescaling. Difficulties also arise from the presence of strong nonlinearity and ill-posedness which means that the parameters do not converge to a single deterministic set of values, but rather there exists a range of parameter combinations that produce the same model behavior. We address these computational issues by the addition of a regularization term in the cost function. All these computational approaches are addressed in the framework of variational adjoint data assimilation. The used observational data are derived from numerical simulation results located at only two spatial points. The effect of different initial guess values of parameters on retrieval results is also considered. As indicated by results of numerical experiments, the method presented in this paper achieves a near perfect parameter identification, and overcomes the indefiniteness that may occur in inversion process even in the case of noisy input data.

Citation: Yuepeng Wang, Yue Cheng, I. Michael Navon, Yuanhong Guan. Parameter identification techniques applied to an environmental pollution model. Journal of Industrial & Management Optimization, 2018, 14 (2) : 817-831. doi: 10.3934/jimo.2017077
References:
[1]

Q. ChaiR. LoxtonK. L. Teo and C. Yang, A unified parameter identification method for nonlinear time delay systems, Journal of Industrial and Management Optimization, 9 (2013), 471-486. doi: 10.3934/jimo.2013.9.471. Google Scholar

[2]

T. Chen and C. Xu, Computational optimal control of the Saint-Venant PDE model using the time-scaling technique, Asia-Pacific Journal of Chemical Engineering, 11 (2016), 70-80. doi: 10.1002/apj.1944. Google Scholar

[3]

J. DuI. M. NavonJ. ZhuF. Fang and A. K. Alekseev, Reduced order modeling based on POD of a parabolized Navier-Stokes equations model Ⅱ: trust region POD 4DVAR data assimilation, Comput. Math. Appl., 65 (2013), 380-394. doi: 10.1016/j.camwa.2012.06.001. Google Scholar

[4]

D. M. Dunlavy, T. G. Kolda and E. Acar, Poblano v1. 0: A Matlab Toolbox for Gradient-Based Optimization, SANDIA Report, 2010. doi: 10.2172/989350. Google Scholar

[5]

H. W. Engl, Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates, J. Optim. Theory Appl., 52 (1987), 209-215. doi: 10.1007/BF00941281. Google Scholar

[6]

C. FarhatR. Tezaur and R. Djellouli, On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method, Inverse Problems, 18 (2002), 1229-1246. doi: 10.1088/0266-5611/18/5/302. Google Scholar

[7]

R. Giering, Tangent linear and adjoint biogeochemical models, in Inverse Methods in Global Biogeochemical Cycles (eds. P. Kasibhatla, M. Heimann, P. Rayner, N. Mahowald, R. G. Prinn and D. E. Hartley), American Geophysical Union: Washington DC, 2000, 33-47.Google Scholar

[8]

P. E. Gill, W. Murray and M. H. Wright, Practical Optimization, Academic Press, 1981. Google Scholar

[9]

M. Gunzburger, Adjoint equation-based methods for control problems in incompressible, viscous flows, Flow Turbulence Combust., 65 (2000), 249-272. doi: 10.1023/A:1011455900396. Google Scholar

[10]

P. C. Hansen, Rank-Deficient and Discrete Ill-posed Problems, Practical optimization, SIAM, 1998. doi: 10.1137/1.9780898719697. Google Scholar

[11]

M. J. HossenI. M. Navon and D. N. Daescu, Effect of random perturbations on adaptive observation techniques, International Journal For Numerical Methods in Fluids, 69 (2012), 110-123. doi: 10.1002/fld.2545. Google Scholar

[12]

S. X. HuangW. Han and R. S. Wu, Theoretical analysis and numerical experiments of variational assimilation for one-dimensional ocean temperature model with techniques in inverse problems, Science in China D, 47 (2004), 630-638. Google Scholar

[13]

D. Krawczyk-stando and M. Rudnicki, Regularization parameter selection in discrete ill-posed problems{The use of the U-curve, Int. J. Appl. Math. Comput. Sci., 17 (2007), 157-164. doi: 10.2478/v10006-007-0014-3. Google Scholar

[14]

Q. LinR. LoxtonC. Xu and K. L. Teo, Parameter estimation for nonlinear time-delay systems with noisy output measurements, Automatica, 60 (2015), 48-56. doi: 10.1016/j.automatica.2015.06.028. Google Scholar

[15]

W. J. Liu, G. Li and L. H. Hong, General decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping, Journal of Function Spaces, 2 (2014), article ID 284809, 21pp. doi: 10.1155/2014/284809. Google Scholar

[16]

R. LoxtonK. L. Teo and V. Rehbock, An optimization approach to state-delay identification, IEEE Transactions on Automatic Control, 55 (2010), 2113-2119. doi: 10.1109/TAC.2010.2050710. Google Scholar

[17]

B. Malengier and R. V. Keer, Parameter estimation in convection dominated nonlinear convection-diffusion problems by the relaxation method and the adjoint equation, Journal of Computational and Applied Mathematics, 215 (2008), 477-483. doi: 10.1016/j.cam.2006.03.050. Google Scholar

[18]

I. M. Navon, Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography, Dynam. Atmos. Oceans, 27 (1998), 55-79. doi: 10.1016/S0377-0265(97)00032-8. Google Scholar

[19]

I. M. Navon and D. M. Legler, Conjugate-gradient methods for large-scale minimization in meteorology, Monthly Weather Review, 115 (1987), 1479-1502. doi: 10.1175/1520-0493(1987)115<1479:CGMFLS>2.0.CO;2. Google Scholar

[20]

I. M. NavonX. ZouJ. Derber and J. Sela, Variational data assimilation with an adiabatic version of the nmc spectral model, Monthly Weather Review, 120 (1992), 1433-1446. doi: 10.1175/1520-0493(1992)120<1433:VDAWAA>2.0.CO;2. Google Scholar

[21]

T. NieminenJ. Kangas and L. Kettunen, Use of Tikhonov regularization to improve the accuracy of position estimates in inertial navigation, International Journal of Navigation and Observation, 2011 (2011), 1-10. doi: 10.1155/2011/450269. Google Scholar

[22]

J. Nocedal and S. J. Wright, Numerical Optimization, Springer, 1999. doi: 10.1007/b98874. Google Scholar

[23]

D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math., 3 (1955), 28-41. doi: 10.1137/0103003. Google Scholar

[24]

B. ProtasT. R. Bewley and G. Hagen, A computational framework for the regularization of adjoint analysis in multiscale PDE systems, Journal of Computational Physics, 195 (2004), 49-89. doi: 10.1016/j.jcp.2003.08.031. Google Scholar

[25]

R. B. StorchL. C. G. Pimentel and H. R. B. Orlande, Identification of atmospheric boundary layer parameters by inverse problem, Atmospheric Environment, 41 (2007), 1417-1425. doi: 10.1016/j.atmosenv.2006.10.014. Google Scholar

[26]

Y. P. WangI. M. NavonX. Y. Wang and Y. Cheng, 2D Burgers equation with large Reynolds number using POD/DEIM and calibration, International Journal For Numerical Methods in Fluids, 82 (2016), 909-931. doi: 10.1002/fld.4249. Google Scholar

[27]

Y. P. Wang and S. L. Tao, Application of regularization technique to variational adjoint method: A case for nonlinear convection-diffusion problem, Applied Mathematics and Computation, 218 (2011), 4475-4482. doi: 10.1016/j.amc.2011.10.028. Google Scholar

[28]

Y. W. Wen and A. M. Yip, Adaptive Parameter Selection for Total Variation Image Deconvolution, Numer. Math. Theor. Meth. Appl., 2 (2009), 427-438. Google Scholar

[29]

J. C.-F. Wong and J.-L. Xie, Inverse determination of a heat source from natural convection in a porous cavity, Computers and Fluids, 52 (2011), 1-14. doi: 10.1016/j.compfluid.2011.07.013. Google Scholar

[30]

Y. Zhu and I. M. Navon, Impact of parameter estimation on the performance of the FSU global spectral model using its full-physics adjoint, Monthly Weather Review, 127 (1999), 1497-1517. doi: 10.1175/1520-0493(1999)127<1497:IOPEOT>2.0.CO;2. Google Scholar

[31]

X. ZouI. M. Navon and F. X. LeDimet, An optimal nudging data assimilation scheme using parameter estimation, Quarterly Journal of the Royal Meteorological Society, 118 (1992), 1163-1186. doi: 10.1002/qj.49711850808. Google Scholar

show all references

References:
[1]

Q. ChaiR. LoxtonK. L. Teo and C. Yang, A unified parameter identification method for nonlinear time delay systems, Journal of Industrial and Management Optimization, 9 (2013), 471-486. doi: 10.3934/jimo.2013.9.471. Google Scholar

[2]

T. Chen and C. Xu, Computational optimal control of the Saint-Venant PDE model using the time-scaling technique, Asia-Pacific Journal of Chemical Engineering, 11 (2016), 70-80. doi: 10.1002/apj.1944. Google Scholar

[3]

J. DuI. M. NavonJ. ZhuF. Fang and A. K. Alekseev, Reduced order modeling based on POD of a parabolized Navier-Stokes equations model Ⅱ: trust region POD 4DVAR data assimilation, Comput. Math. Appl., 65 (2013), 380-394. doi: 10.1016/j.camwa.2012.06.001. Google Scholar

[4]

D. M. Dunlavy, T. G. Kolda and E. Acar, Poblano v1. 0: A Matlab Toolbox for Gradient-Based Optimization, SANDIA Report, 2010. doi: 10.2172/989350. Google Scholar

[5]

H. W. Engl, Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates, J. Optim. Theory Appl., 52 (1987), 209-215. doi: 10.1007/BF00941281. Google Scholar

[6]

C. FarhatR. Tezaur and R. Djellouli, On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method, Inverse Problems, 18 (2002), 1229-1246. doi: 10.1088/0266-5611/18/5/302. Google Scholar

[7]

R. Giering, Tangent linear and adjoint biogeochemical models, in Inverse Methods in Global Biogeochemical Cycles (eds. P. Kasibhatla, M. Heimann, P. Rayner, N. Mahowald, R. G. Prinn and D. E. Hartley), American Geophysical Union: Washington DC, 2000, 33-47.Google Scholar

[8]

P. E. Gill, W. Murray and M. H. Wright, Practical Optimization, Academic Press, 1981. Google Scholar

[9]

M. Gunzburger, Adjoint equation-based methods for control problems in incompressible, viscous flows, Flow Turbulence Combust., 65 (2000), 249-272. doi: 10.1023/A:1011455900396. Google Scholar

[10]

P. C. Hansen, Rank-Deficient and Discrete Ill-posed Problems, Practical optimization, SIAM, 1998. doi: 10.1137/1.9780898719697. Google Scholar

[11]

M. J. HossenI. M. Navon and D. N. Daescu, Effect of random perturbations on adaptive observation techniques, International Journal For Numerical Methods in Fluids, 69 (2012), 110-123. doi: 10.1002/fld.2545. Google Scholar

[12]

S. X. HuangW. Han and R. S. Wu, Theoretical analysis and numerical experiments of variational assimilation for one-dimensional ocean temperature model with techniques in inverse problems, Science in China D, 47 (2004), 630-638. Google Scholar

[13]

D. Krawczyk-stando and M. Rudnicki, Regularization parameter selection in discrete ill-posed problems{The use of the U-curve, Int. J. Appl. Math. Comput. Sci., 17 (2007), 157-164. doi: 10.2478/v10006-007-0014-3. Google Scholar

[14]

Q. LinR. LoxtonC. Xu and K. L. Teo, Parameter estimation for nonlinear time-delay systems with noisy output measurements, Automatica, 60 (2015), 48-56. doi: 10.1016/j.automatica.2015.06.028. Google Scholar

[15]

W. J. Liu, G. Li and L. H. Hong, General decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping, Journal of Function Spaces, 2 (2014), article ID 284809, 21pp. doi: 10.1155/2014/284809. Google Scholar

[16]

R. LoxtonK. L. Teo and V. Rehbock, An optimization approach to state-delay identification, IEEE Transactions on Automatic Control, 55 (2010), 2113-2119. doi: 10.1109/TAC.2010.2050710. Google Scholar

[17]

B. Malengier and R. V. Keer, Parameter estimation in convection dominated nonlinear convection-diffusion problems by the relaxation method and the adjoint equation, Journal of Computational and Applied Mathematics, 215 (2008), 477-483. doi: 10.1016/j.cam.2006.03.050. Google Scholar

[18]

I. M. Navon, Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography, Dynam. Atmos. Oceans, 27 (1998), 55-79. doi: 10.1016/S0377-0265(97)00032-8. Google Scholar

[19]

I. M. Navon and D. M. Legler, Conjugate-gradient methods for large-scale minimization in meteorology, Monthly Weather Review, 115 (1987), 1479-1502. doi: 10.1175/1520-0493(1987)115<1479:CGMFLS>2.0.CO;2. Google Scholar

[20]

I. M. NavonX. ZouJ. Derber and J. Sela, Variational data assimilation with an adiabatic version of the nmc spectral model, Monthly Weather Review, 120 (1992), 1433-1446. doi: 10.1175/1520-0493(1992)120<1433:VDAWAA>2.0.CO;2. Google Scholar

[21]

T. NieminenJ. Kangas and L. Kettunen, Use of Tikhonov regularization to improve the accuracy of position estimates in inertial navigation, International Journal of Navigation and Observation, 2011 (2011), 1-10. doi: 10.1155/2011/450269. Google Scholar

[22]

J. Nocedal and S. J. Wright, Numerical Optimization, Springer, 1999. doi: 10.1007/b98874. Google Scholar

[23]

D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math., 3 (1955), 28-41. doi: 10.1137/0103003. Google Scholar

[24]

B. ProtasT. R. Bewley and G. Hagen, A computational framework for the regularization of adjoint analysis in multiscale PDE systems, Journal of Computational Physics, 195 (2004), 49-89. doi: 10.1016/j.jcp.2003.08.031. Google Scholar

[25]

R. B. StorchL. C. G. Pimentel and H. R. B. Orlande, Identification of atmospheric boundary layer parameters by inverse problem, Atmospheric Environment, 41 (2007), 1417-1425. doi: 10.1016/j.atmosenv.2006.10.014. Google Scholar

[26]

Y. P. WangI. M. NavonX. Y. Wang and Y. Cheng, 2D Burgers equation with large Reynolds number using POD/DEIM and calibration, International Journal For Numerical Methods in Fluids, 82 (2016), 909-931. doi: 10.1002/fld.4249. Google Scholar

[27]

Y. P. Wang and S. L. Tao, Application of regularization technique to variational adjoint method: A case for nonlinear convection-diffusion problem, Applied Mathematics and Computation, 218 (2011), 4475-4482. doi: 10.1016/j.amc.2011.10.028. Google Scholar

[28]

Y. W. Wen and A. M. Yip, Adaptive Parameter Selection for Total Variation Image Deconvolution, Numer. Math. Theor. Meth. Appl., 2 (2009), 427-438. Google Scholar

[29]

J. C.-F. Wong and J.-L. Xie, Inverse determination of a heat source from natural convection in a porous cavity, Computers and Fluids, 52 (2011), 1-14. doi: 10.1016/j.compfluid.2011.07.013. Google Scholar

[30]

Y. Zhu and I. M. Navon, Impact of parameter estimation on the performance of the FSU global spectral model using its full-physics adjoint, Monthly Weather Review, 127 (1999), 1497-1517. doi: 10.1175/1520-0493(1999)127<1497:IOPEOT>2.0.CO;2. Google Scholar

[31]

X. ZouI. M. Navon and F. X. LeDimet, An optimal nudging data assimilation scheme using parameter estimation, Quarterly Journal of the Royal Meteorological Society, 118 (1992), 1163-1186. doi: 10.1002/qj.49711850808. Google Scholar

Figure 1.  Gradient test
Figure 2.  Flow chart of parameter estimation with NCG method
Figure 3.  Evolution of the value of parameters with iterations for the initial guess values (1, 1)(solid line -), (10, 1)(dashed line --), (30, 10)(dotted line :), (20, 3)(dotted-dashed line -.) and (30, 1)(star dotted-dashed line *-.), respectively
Figure 4.  The evolution of the cost functional as a function of the number of minimization iterations for the case of initial guess value $(1, 1)$
Figure 5.  Retrieval process of $K_1$ and $(\mu_{\ast})_0$ with iteration in different cases: $(a).~ \sigma=0, \gamma=0$; $(b).~\sigma=0, \gamma\neq{0}$; $(c).~\sigma\neq{0}, \gamma=0$; $(d).~\sigma\neq{0}, \gamma\neq{0}$
Figure 6.  Change of $(\mu_{\ast})_0$ magnified partially: $(a)$. in Fig.5 (c) without regularization; $(b)$. in Fig.5 (d) with regularization
Table 1.  Comparison between the methods of NCG, ncg, lbfgs and tn with different initial guess. The true parameters are (0.38, 50)
NCGncg lbfgs tn
Iter.ResultIter.ResultIter.ResultIter.Result
[0.3;40]2[0.3751;
50.7179]
2[0.3759;
50.7398]
5[0.3752;
50.7179]
6[0.3753;
50.7524]
[0.15;18]3[0.3794;
50.0722]
2 [0.3962;
52.4113]
11 [0.3791;
50.1205]
4[0.3797;
50.0491]
[0.01;1.2] 3 [0.3771;
50.4490]
2 [0.6661;
89.1588]
10 [0.3773;
50.4361]
4 [0.3534;
54.4259]
NCGncg lbfgs tn
Iter.ResultIter.ResultIter.ResultIter.Result
[0.3;40]2[0.3751;
50.7179]
2[0.3759;
50.7398]
5[0.3752;
50.7179]
6[0.3753;
50.7524]
[0.15;18]3[0.3794;
50.0722]
2 [0.3962;
52.4113]
11 [0.3791;
50.1205]
4[0.3797;
50.0491]
[0.01;1.2] 3 [0.3771;
50.4490]
2 [0.6661;
89.1588]
10 [0.3773;
50.4361]
4 [0.3534;
54.4259]
Table 2.  Results obtained with several sets of experiments
Guessed valueError level $\sigma$Iters $J$Estimated ValueRegularization.Para.$\gamma$
(1, 1)0.00122.339E-7(49.4799, 0.3974)0
(1, 1)0.00131.225E-6(49.5557, 0.3734)3.5E-10
(1, 1)0.02153.020E-6(50.4796, 0.2683)0
(1, 1)0.02144.900E-6(50.2095, 0.3864)3.5E-10
(20, 20)0.0254.1237E-6(50.4308, 0.3902)3.0E-10
(20, 20)0.0253.1178E-6(50.0751, 0.4118)0
(20, 20)0.02502.9000E-6(49.1653, 0.4669)0
Guessed valueError level $\sigma$Iters $J$Estimated ValueRegularization.Para.$\gamma$
(1, 1)0.00122.339E-7(49.4799, 0.3974)0
(1, 1)0.00131.225E-6(49.5557, 0.3734)3.5E-10
(1, 1)0.02153.020E-6(50.4796, 0.2683)0
(1, 1)0.02144.900E-6(50.2095, 0.3864)3.5E-10
(20, 20)0.0254.1237E-6(50.4308, 0.3902)3.0E-10
(20, 20)0.0253.1178E-6(50.0751, 0.4118)0
(20, 20)0.02502.9000E-6(49.1653, 0.4669)0
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