Estimating two time-varying reaction coefficients in a water quality model from inexact initial and boundary data

Dinh Nho Hào; Nguyen Trung Thành; Nguyen Van Duc; Nguyen Van Thang

doi:10.3934/eect.2024053

Article Contents

2025, Volume 14, Issue 2: 246-274. Doi: 10.3934/eect.2024053

This issue Previous Article Asymptotic behavior of global solutions to the complex Ginzburg–Landau type equation in the super Fujita-critical case Next Article Qualitative behavior of solutions for a chemotaxis-haptotaxis model with flux limitation

Estimating two time-varying reaction coefficients in a water quality model from inexact initial and boundary data

1.
Hanoi Institute of Mathematics, VAST, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam
2.
Department of Mathematics, Rowan University, 201 Mullica Hill Rd, Glassboro, NJ 08043, USA
3.
Department of Mathematics, Vinh University, 182 Le Duan, Vinh City, Vietnam
4.
Quan Hanh Secondary School, Quan Hanh Town, Nghi Loc District, Nghe An Province, Vietnam

^*Corresponding author: Nguyen Trung Thành
^*Corresponding author: Nguyen Trung Thành

Received: July 2024

Revised: September 2024

Early access: September 2024

Published: April 2025

This research was supported by the Vingroup Innovation Foundation under grant number VINIF.2020.DA16.

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Abstract

An inverse problem of identifying two time-varying reaction coefficients in a system of one-dimensional advection-reaction equations is investigated. The system is used for modeling the transportation of pollutants in rivers or streams. The coefficients to be identified represent the deoxygenation and reaeration rates. All the initial and boundary data in the model are assumed to contain noise. Lipschitz-type stability estimates of the coefficients are obtained. The proofs of the stability estimates are based on a Carleman estimate for a first-order transport operator. For numerical computation, the coefficient identification problem is reformulated as an optimization problem using the least-squares method coupled with the adjoint equation method for computing the gradient of the objective functional. Error estimates are derived and numerical examples are provided for demonstrating the performance of the proposed algorithm.

Keywords:

Mathematics Subject Classification: Primary: 35R30; Secondary: 35Q49.

Citation:

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Figure 1. Exact and reconstructed coefficients $ k_d $ and $ k_r $ in Example 1. (a)-(b): Using the $ H^1 $ data; (c)-(d): Using the $ L^2 $ data

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Figure 2. Noisy data and the solution of the forward problem at $ x = L $ in Example 1. (a)-(b): $ H^1 $ data; (c)-(d): $ L^2 $ data

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Figure 3. Exact and reconstructed coefficients $ k_d $ and $ k_r $ in Example 2 using the $ H^1 $ data

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Figure 4. Exact and reconstructed coefficients $ k_d $ and $ k_r $ in Example 2 using the $ L^2 $ data

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Figure 5. Exact and reconstructed coefficients $ k_d $ and $ k_r $ in Example 1 using $ H^1 $ data in $ t\in [0.3465, 2] $

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Figure 6. Exact and reconstructed coefficients $ k_d $ and $ k_r $ in Example 1 using $ L^2 $ data in $ t\in [0.3465, 2] $

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Figure 7. Exact and reconstructed coefficients $ k_d $ and $ k_r $ in Example 2 using $ H^1 $ data in $ t\in [0.3465, 2] $

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Figure 8. Exact and reconstructed coefficients $ k_d $ and $ k_r $ in Example 2 using $ L^2 $ data in $ t\in [0.3465, 2] $

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