Shock capturing data assimilation algorithm for 1D shallow water equations

Tirupathi, Seshu; Tchrakian, Tigran; Zhuk, Sergiy; McKenna, Sean Andrew

doi:10.1016/j.advwatres.2015.12.021

Cited by 7 publications

(11 citation statements)

References 30 publications

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“…The adjoint system (12) must be solved backwards in time from t = T tot = 0 to find the gradient which is defined in terms of its solution evaluated at t = 0, cf. (11). For convenience, we introduce the backwards time variable τ = T − t and define vectors…”

Section: Analytic Solution Of the Data Assimilation Problem In The LImentioning

confidence: 99%

“…This approach, often referred to as "4D-VAR", has received much attention in the literature [5], particularly for applications to operational weather forecasting [6] and climate modelling. Given the ubiquity of the SWE as a model in Earth sciences, variational data assimilation approaches for this system have already received significant attention [7,8,9,10,11].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the convergence of data assimilation for the one-dimensional shallow water equations with sparse observations

2019

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The shallow water equations (SWE) are a widely used model for the propagation of surface waves on the oceans. In particular, the SWE are used to model the propagation of tsunami waves in the open ocean. We consider the associated data assimilation problem of optimally determining the initial conditions for the one-dimensional SWE in an unbounded domain from a small set of observations of the sea surface height and focus on how the structure of the observation operator affects the convergence of the gradient approach employed to solve the data assimilation problem computationally. In the linear case we prove a theorem that gives sufficient conditions for convergence to the true initial conditions. It asserts that at least two observation points must be used and at least one pair of observation points must be spaced more closely than half the effective minimum wavelength of the energy spectrum of the initial conditions. Our analysis is confirmed by numerical experiments for both the linear and nonlinear SWE data assimilation problems which reveal a relation between the convergence rate of gradient iterations and the number and spacing of the observation points. More specifically, these results show that convergence rates improve with increasing numbers of observation points and that at least three observation points are required for the practically useful results. Better results are obtained for the nonlinear equations provided more than two observation points are used. This paper is a first step in understanding the conditions for observability of the SWE for small numbers of observation points in more physically realistic settings.

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Section: Analytic Solution Of the Data Assimilation Problem In The LImentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the convergence of data assimilation for the one-dimensional shallow water equations with sparse observations

2019

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“…In the numerical computations, we apply finite-element approximation spaces based on rectangular elements, and we opt to approximate the error in (40) by u h/2 − u h , where u h/2 is an approximation obtained from a Galerkin approximation in a finite-element space V h/2 0 in which each element is uniformly divided into 4 elements; see Figure 2. The refined finite-element space V h/2 0 moreover serves to construct an approximation to the dual solution in (40). In summary, considering an approximation u h ∈ V h 0 and a refined approximation u h/2 ∈ V h/2 0 , the worst-case multi-objective error estimate pertaining to u h is r(u h ), z h/2 with z h/2 according to Figure 1 plots the worst-case multi-objective error estimate, r(u h ), z h/2 , versus the dimension of the finite-element approximation space, dim V h 0 , for finite-element approximations with polynomial degrees p ∈ {1, 2, 3, 4} on uniform meshes with mesh width h = 2 −2 , 2 −3 , .…”

Section: Worst-case Multi-objective Error Estimation Without Data Incmentioning

confidence: 99%

“…This technique applies to general linear operators with non-trivial null-spaces and possibly non-closed ranges, and it is based on an extension of Young-Fenchel duality to closed unbounded linear operators [50,48]. It is noteworthy that this technique for estimating a linear functional of a solution of an operator equation, which we consider in this paper in the context of worst-case multi-objective error estimation, yields a natural generalization of adaptive-state-estimation or so-called dataassimilation procedures (such as online sequential filters [40,51,39] or offline variational approaches [18]) to a wide class of linear and nonlinear partial differential equations, enabling incorporation of a-posteriori knowledge (for instance, sensor readings) into the error estimate rendering it less conservative [51,49].…”

Section: Introductionmentioning

confidence: 99%

Worst-case multi-objective error estimation and adaptivity

Brummelen

Zhuk

Zwieten

2017

Computer Methods in Applied Mechanics and Engineering

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This paper introduces a new computational methodology for determining aposteriori multi-objective error estimates for finite-element approximations, and for constructing corresponding (quasi-)optimal adaptive refinements of finiteelement spaces. As opposed to the classical goal-oriented approaches, which consider only a single objective functional, the presented methodology applies to general closed convex subsets of the dual space and constructs a worst-case error estimate of the finite-element approximation error. This worst-case multiobjective error estimate conforms to a dual-weighted residual, in which the dual solution is associated with an approximate supporting functional of the objective set at the approximation error. We regard both standard approximation errors and data-incompatibility errors associated with incompatibility of boundary data with the trace of the finite-element space. Numerical experiments are presented to demonstrate the efficacy of applying the proposed worst-case multiobjective error in adaptive refinement procedures.

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“…Modeling of flood events are usually considered as boundary value problem, with boundary conditions often described by a time changing water level (hydrograph), guiding the overflow across the domain (Tirupathi et al, 2016). The ability to update the model state through time with real time data is changing our vision of flood prediction and the way we forecast the flood evolution in time and space.…”

Section: Introductionmentioning

confidence: 99%

Enhanced flood forecasting through ensemble data assimilation and joint state-parameter estimation

Ziliani

Ghostine

Ait‐El‐Fquih

et al. 2019

Journal of Hydrology

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Accurate water level forecasts during flood events are crucial to mitigate the loss of human lives and economic damages. However, the accuracy of flood models can be affected by various factors, including the complexity of the terrain geometry and bathymetry, imperfect physics as well as uncertainties in the inflows and parameters. This paper describes a practical implementation of an ensemble Kalman filter (EnKF) based data assimilation system that is aimed towards enhancing the forecasting skill of flood models.

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Shock capturing data assimilation algorithm for 1D shallow water equations

Cited by 7 publications

References 30 publications

On the convergence of data assimilation for the one-dimensional shallow water equations with sparse observations

On the convergence of data assimilation for the one-dimensional shallow water equations with sparse observations

Worst-case multi-objective error estimation and adaptivity

Enhanced flood forecasting through ensemble data assimilation and joint state-parameter estimation

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