# American Institute of Mathematical Sciences

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
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##### References:
Flow chart of parameter estimation with NCG method
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
The evolution of the cost functional as a function of the number of minimization iterations for the case of initial guess value $(1, 1)$
Retrieval process of $K_1$ and $(\mu_{\ast})_0$ with iteration in different cases: $(a).~ \sigma=0, \gamma=0$; $(b).~\sigma=0, \gamma\neq{0}$; $(c).~\sigma\neq{0}, \gamma=0$; $(d).~\sigma\neq{0}, \gamma\neq{0}$
Change of $(\mu_{\ast})_0$ magnified partially: $(a)$. in Fig.5 (c) without regularization; $(b)$. in Fig.5 (d) with regularization
Comparison between the methods of NCG, ncg, lbfgs and tn with different initial guess. The true parameters are (0.38, 50)
 NCG ncg lbfgs tn Iter. Result Iter. Result Iter. Result Iter. 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]
 NCG ncg lbfgs tn Iter. Result Iter. Result Iter. Result Iter. 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]
Results obtained with several sets of experiments
 Guessed value Error level $\sigma$ Iters $J$ Estimated Value Regularization.Para.$\gamma$ (1, 1) 0.00 12 2.339E-7 (49.4799, 0.3974) 0 (1, 1) 0.00 13 1.225E-6 (49.5557, 0.3734) 3.5E-10 (1, 1) 0.02 15 3.020E-6 (50.4796, 0.2683) 0 (1, 1) 0.02 14 4.900E-6 (50.2095, 0.3864) 3.5E-10 (20, 20) 0.02 5 4.1237E-6 (50.4308, 0.3902) 3.0E-10 (20, 20) 0.02 5 3.1178E-6 (50.0751, 0.4118) 0 (20, 20) 0.02 50 2.9000E-6 (49.1653, 0.4669) 0
 Guessed value Error level $\sigma$ Iters $J$ Estimated Value Regularization.Para.$\gamma$ (1, 1) 0.00 12 2.339E-7 (49.4799, 0.3974) 0 (1, 1) 0.00 13 1.225E-6 (49.5557, 0.3734) 3.5E-10 (1, 1) 0.02 15 3.020E-6 (50.4796, 0.2683) 0 (1, 1) 0.02 14 4.900E-6 (50.2095, 0.3864) 3.5E-10 (20, 20) 0.02 5 4.1237E-6 (50.4308, 0.3902) 3.0E-10 (20, 20) 0.02 5 3.1178E-6 (50.0751, 0.4118) 0 (20, 20) 0.02 50 2.9000E-6 (49.1653, 0.4669) 0

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