A Posteriori Error Estimates for the Solution of Variational Inverse Problems

Rao, Vishwas; Sandu, Adrian

doi:10.1137/140990036

Cited by 22 publications

(15 citation statements)

References 22 publications

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“…The computational cost of the proposed sampling smoother can of course by be reduced by decreasing the ensemble size; however, this will result in higher sampling error. The impact of the sampling errors can be assessed by the techniques developed in .…”

Section: Numerical Experimentsmentioning

confidence: 99%

A Hybrid Monte‐Carlo sampling smoother for four‐dimensional data assimilation

Attia

Rao

Sandu

2016

Numerical Methods in Fluids

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Summary This paper constructs an ensemble‐based sampling smoother for four‐dimensional data assimilation using a Hybrid/Hamiltonian Monte‐Carlo approach. The smoother samples efficiently from the posterior probability density of the solution at the initial time. Unlike the well‐known ensemble Kalman smoother, which is optimal only in the linear Gaussian case, the proposed methodology naturally accommodates non‐Gaussian errors and nonlinear model dynamics and observation operators. Unlike the four‐dimensional variational method, which only finds a mode of the posterior distribution, the smoother provides an estimate of the posterior uncertainty. One can use the ensemble mean as the minimum variance estimate of the state or can use the ensemble in conjunction with the variational approach to estimate the background errors for subsequent assimilation windows. Numerical results demonstrate the advantages of the proposed method compared to the traditional variational and ensemble‐based smoothing methods. Copyright © 2016 John Wiley & Sons, Ltd.

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Section: Numerical Experimentsmentioning

confidence: 99%

A Hybrid Monte‐Carlo sampling smoother for four‐dimensional data assimilation

Attia

Rao

Sandu

2016

Numerical Methods in Fluids

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“…A more rigorous argument can be made using the a-posteriori error estimation methodology developed in [3,4,16,81]. The full order KKT equations (5) form a large system of nonlinear equations, written abstractly as (1) and adjoint model (5a) linearized across the trajectory x a .…”

Section: Estimation Of Ar Optimization Errormentioning

confidence: 99%

POD/DEIM reduced-order strategies for efficient four dimensional variational data assimilation

Ştefanescu

Sandu

Navon

2015

Journal of Computational Physics

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Discrete empirical interpolation method (DEIM) Reduced-order models (ROMs) Shallow water equations Finite difference methodsThis work studies reduced order modeling (ROM) approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models. It is shown that a key requirement for a successful reduced order solution is that reduced order Karush-Kuhn-Tucker conditions accurately represent their full order counterparts. In particular, accurate reduced order approximations are needed for the forward and adjoint dynamical models, as well as for the reduced gradient. New strategies to construct reduced order based are developed for proper orthogonal decomposition (POD) ROM data assimilation using both Galerkin and Petrov-Galerkin projections. For the first time POD, tensorial POD, and discrete empirical interpolation method (DEIM) are employed to develop reduced data assimilation systems for a geophysical flow model, namely, the two dimensional shallow water equations. Numerical experiments confirm the theoretical framework for Galerkin projection. In the case of Petrov-Galerkin projection, stabilization strategies must be considered for the reduced order models. The new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time.

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“…In this article we employ adjoint based a posteriori analysis to accurately quantify the error in a QoI computed from the numerical solution of the PBE. Adjoint based error estimation is widely used for a host of numerical methods including finite elements, finite difference, time integration, multi-scale simulations and inverse problems [30,29,32,1,7,8,10,37,13,16,22,57]. The error estimate weights computable residuals of the numerical solution with the solution of an adjoint problem to quantify the accumulation and propagation of error.…”

Section: Introductionmentioning

confidence: 99%

A Posteriori Analysis and Efficient Refinement Strategies for the Poisson--Boltzmann Equation

Chaudhry¹

2018

SIAM J. Sci. Comput.

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The Poisson-Boltzmann equation (PBE) models the electrostatic interactions of charged bodies such as molecules and proteins in an electrolyte solvent. The PBE is a challenging equation to solve numerically due to the presence of singularities, discontinuous coefficients and boundary conditions. Hence, there is often large error in the numerical solution of the PBE that needs to be quantified. In this work, we use adjoint based a posteriori analysis to accurately quantify the error in an important quantity of interest, the solvation free energy, for the finite element solution of the PBE. We identify various sources of error and propose novel refinement strategies based on a posteriori error estimates.

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A Posteriori Error Estimates for the Solution of Variational Inverse Problems

Cited by 22 publications

References 22 publications

A Hybrid Monte‐Carlo sampling smoother for four‐dimensional data assimilation

A Hybrid Monte‐Carlo sampling smoother for four‐dimensional data assimilation

POD/DEIM reduced-order strategies for efficient four dimensional variational data assimilation

A Posteriori Analysis and Efficient Refinement Strategies for the Poisson--Boltzmann Equation

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