A numerical procedure and coupled system formulation for the adjoint approach in hyperbolic PDE-constrained optimization problems

Montecinos, Gino I.; López-Ríos, J.; Ortega, Juan‐Pablo; Lecaros, Rodrigo

doi:10.1093/imamat/hxy067

Cited by 3 publications

(2 citation statements)

References 45 publications

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“…The Manning roughness's coefficient has also been identified in the context of a complex channel network in [Ding et al, 2004, Ding and Wang, 2005, Ding and Wang, 2012a. Furthermore, a general framework to deal with hyperbolic PDE˙constrained optimization problems was presented in [Montecinos et al, 2019]. A coupled system of the PDE-constrained problem and the adjoint formulation was presented, and conditions are provided to guarantee existence of an optimal solution.…”

Section: Introductionmentioning

confidence: 99%

Bathymetry and friction estimation from transient velocity data for 1D shallow water flows in open channels with varying width

Moreles¹,

Hernández-Dueñas²,

González–Casanova³

2021

Preprint

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The shallow water equations (SWE) model a variety of geophysical flows. Flows in channels with rectangular cross sections may be modelled with a simplified one-dimensional SWE with varying width.Among other model parameters, information about the bathymetry and friction coefficient is needed for the correct and precise prediction of the flow. Although synthetic values of the model parameters may suffice for testing numerical schemes, approximations of the bathymetry and other parameters may be required for applications. Estimations may be obtained by experimental methods but some of those techniques may be expensive, time consuming, and not always available. In this work, we propose to solve the inverse problem to estimate the bathymetry and the Manning's friction coefficient from transient velocity data. This is done with the aid of a cost functional which includes the SWE through Lagrange multipliers. The solution is obtained by solving the constrained optimization problem by a continuous descent method. The direct and the adjoint problems are both solved numerically using a second-order accurate Roe-type upwind scheme. Numerical tests are included to show the merits of the algorithm.

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Section: Introductionmentioning

confidence: 99%

Bathymetry and friction estimation from transient velocity data for 1D shallow water flows in open channels with varying width

Moreles¹,

Hernández-Dueñas²,

González–Casanova³

2021

Preprint

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“…The flux function of a scalar conservation law was reconstructed using the information from the shock that forms in the work by Kang and Tanuma [34]. In a more general setting for balance laws, Montecinos et al [42] derived a unified scheme for solving the forward and adjoint problems simultaneously. Methodology for the scalar Burger's equation was presented by Lellouche et al [38] in which the authors aimed to find the best approximation for the measured data by means of boundary control and an adjoint approach.…”

mentioning

confidence: 99%

Recovery of a Time-Dependent Bottom Topography Function from the Shallow Water Equations via an Adjoint Approach

Britton¹,

Chow²,

Chen³

et al. 2020

Preprint

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We develop an adjoint approach for recovering the topographical function included in the source term of one-dimensional hyperbolic balance laws. We focus on a specific system, namely the shallow water equations, in an effort to recover the riverbed topography. The novelty of this work is the ability to robustly recover the bottom topography using only noisy boundary data from one measurement event and the inclusion of two regularization terms in the iterative update scheme. The adjoint scheme is determined from a linearization of the forward system and is used to compute the gradient of a cost function. The bottom topography function is recovered through an iterative process given by a three-operator splitting method which allows the feasibility to include two regularization terms. Numerous numerical tests demonstrate the robustness of the method regardless of the choice of initial guess and in the presence of discontinuities in the solution of the forward problem.

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Optimal control approach for moving bottom detection in one‐dimensional shallow waters by surface measurements

Lecaros,

López‐Ríos,

Montecinos

et al. 2024

Math Methods in App Sciences

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We consider the Boussinesq‐Peregrine (BP) system as described by Lannes [Lannes, D. (2013). The water waves problem: mathematical analysis and asymptotics (Vol. 188). American Mathematical Soc.], within the shallow water regime, and study the inverse problem of determining the time and space variations of the channel bottom profile, from measurements of the wave profile and its velocity on the free surface. A well‐posedness result within a Sobolev framework for (BP), considering a time dependent bottom, is presented. Then, the inverse problem is reformulated as a nonlinear PDE‐constrained optimization one. An existence result of the minimum, under constraints on the admissible set of bottoms, is presented. Moreover, an implementation of the gradient descent approach, via the adjoint method, is considered. For solving numerically both, the forward (BP) and its adjoint system, we derive a universal and low‐dissipation scheme, which contains non‐conservative products. The scheme is based on the FORCE‐ method proposed in [Toro, E. F., Saggiorato, B., Tokareva, S., and Hidalgo, A. (2020). Low‐dissipation centred schemes for hyperbolic equations in conservative and non‐conservative form. Journal of Computational Physics, 416, 109545]. Finally, we implement this methodology to recover three different bottom profiles; a smooth bottom, a discontinuous one, and a continuous profile with a large gradient. We compare with two classical discretizations for (BP) and the adjoint system. These results corroborate the effectiveness of the proposed methodology to recover bottom profiles.

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A numerical procedure and coupled system formulation for the adjoint approach in hyperbolic PDE-constrained optimization problems

Cited by 3 publications

References 45 publications

Bathymetry and friction estimation from transient velocity data for 1D shallow water flows in open channels with varying width

Bathymetry and friction estimation from transient velocity data for 1D shallow water flows in open channels with varying width

Recovery of a Time-Dependent Bottom Topography Function from the Shallow Water Equations via an Adjoint Approach

Optimal control approach for moving bottom detection in one‐dimensional shallow waters by surface measurements

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