Recovery of a time-dependent bottom topography function from the shallow water equations via an adjoint approach. (English) Zbl 1523.65079
Summary: 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 of including 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.
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 65M30 | Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65K10 | Numerical optimization and variational techniques |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35L60 | First-order nonlinear hyperbolic equations |
| 35L65 | Hyperbolic conservation laws |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35R30 | Inverse problems for PDEs |
| 35R25 | Ill-posed problems for PDEs |
Keywords:
inverse problems; adjoint methods; discontinuous Galerkin methods; hyberbolic balance laws; shallow water equations; \(L^1\) and \(H^1\) regularizationSoftware:
TAFReferences:
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