Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Gino Montecinos, Rodrigo Lecaros, Juan López-Ríos, Enrique Zuazua

TL;DR

This work addresses the inverse problem of recovering a time-dependent bathymetry from surface measurements in a 1D shallow-water context modeled by the BP system. It establishes local well-posedness for BP with a moving bottom, formulates a PDE-constrained optimization to identify the bottom, and proves the existence of a global minimizer, coupled with an adjoint-based gradient. A universal FORCE-$\alpha$ finite-volume scheme is developed to discretize both the forward BP problem and its adjoint, enabling stable, low-dissipation solutions for the inverse problem. Numerical tests on smooth, discontinuous, and large-gradient bottoms demonstrate accurate bottom recovery and highlight the superiority of the FORCE-$\alpha$+CSF scheme over conventional discretizations. The results have implications for tsunami-genesis studies and ocean engineering, and point to future work on richer models and controllability analyses.

Abstract

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 PDEconstrained 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-α 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.

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

TL;DR

This work addresses the inverse problem of recovering a time-dependent bathymetry from surface measurements in a 1D shallow-water context modeled by the BP system. It establishes local well-posedness for BP with a moving bottom, formulates a PDE-constrained optimization to identify the bottom, and proves the existence of a global minimizer, coupled with an adjoint-based gradient. A universal FORCE- finite-volume scheme is developed to discretize both the forward BP problem and its adjoint, enabling stable, low-dissipation solutions for the inverse problem. Numerical tests on smooth, discontinuous, and large-gradient bottoms demonstrate accurate bottom recovery and highlight the superiority of the FORCE-+CSF scheme over conventional discretizations. The results have implications for tsunami-genesis studies and ocean engineering, and point to future work on richer models and controllability analyses.

Abstract

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 PDEconstrained 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-α 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.
Paper Structure (13 sections, 7 theorems, 95 equations, 13 figures, 1 algorithm)

This paper contains 13 sections, 7 theorems, 95 equations, 13 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The mathematical framework
  3. The optimal control approach
  4. Numerical reconstruction of a time dependent bathymetry
  5. Numerical discretization
  6. Numerical results
  7. Smooth bottom profile
  8. Discontinuous bottom profile
  9. Smooth bottom profile with large gradients
  10. Conclusions
  11. Existence and uniqueness of solutions
  12. First order necessary condition and optimality system
  13. The reference scheme Rusanov+FD

Key Result

Theorem 1

Let $s>3/2$ and $(\zeta_0,V_0)\in \mathbf{X}^s$. Let $b\in W^{2,\infty}([0,\infty);H^{s+1})$ and assume ES34 is valid. Then, there exists $T_{BP}>0$, uniformly bounded from below with respect to $\epsilon$, such that system ES32 admits a unique solution $(\zeta,V)^T\in C([0,T_{BP}/\epsilon];\mathbf{

Figures (13)

  • Figure 1: Sketch of the physical domain and main notation
  • Figure 2: Smooth bottom profile (\ref{['eq:b-smooth']}): The $L_\infty$ norm of $\nabla J$ for $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells at $t = 1$, $\alpha_F = 2$. (Dot line) Rusanov+FD scheme. (Dash line) FORCE-$\alpha$+FD scheme. (Full line) FORCE-$\alpha$+CSF scheme.
  • Figure 3: Smooth bottom profile (\ref{['eq:b-smooth']}): Result for the reconstruction procedure resulting from Rusanov+FD. Parameters $\Delta t = 0.01$, $\alpha_F = 2$, $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells. Feft:$t=0.25$, centered$t=0.5$, right:$t=0.75$.
  • Figure 4: Smooth bottom profile (\ref{['eq:b-smooth']}): Result for the reconstruction procedure resulting from non-conservative FORCE-$\alpha$+FD. Parameters $\Delta t = 0.01$, $\alpha_F = 2$, $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells. Feft:$t=0.25$, centered$t=0.5$, right:$t=0.75$.
  • Figure 5: Smooth bottom profile (\ref{['eq:b-smooth']}): Result for the reconstruction procedure resulting from FORCE-$\alpha$+CSF. Parameters $\Delta t = 0.01$, $\alpha_F = 2$, $\varepsilon = 0.001$, $\lambda_b = 0.71$, $100$ cells. Feft:$t=0.25$, centered$t=0.5$, right:$t=0.75$.
  • ...and 8 more figures

Theorems & Definitions (13)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Remark 3
  • ...and 3 more