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Progress in adjoint error correction for integral functionals
 COMPUTING AND VISUALIZATION IN SCIENCE
, 2004
"... When approximating the solutions of partial differential equations, it is a few key output integrals which are often of most concern. This paper shows how the accuracy of these values can be improved through a correction term which is an inner product of the residual error in the original p.d.e. an ..."
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When approximating the solutions of partial differential equations, it is a few key output integrals which are often of most concern. This paper shows how the accuracy of these values can be improved through a correction term which is an inner product of the residual error in the original p.d.e. and the solution of an appropriately defined adjoint p.d.e. A number of applications are presented and the challenges of smooth reconstruction on unstructured grids and error correction for shocks are discussed.
Sharp error estimates for a discretisation of the 1D convection/diffusion equation with Dirac initial data
, 2004
"... This paper derives sharp l and l 1 estimates of the error arising from an explicit approximation of the constant coe#cient 1D convection/di#usion equation with Dirac initial data. The analysis embeds the discrete equations within a semidiscrete system of equations which can be solved by Fouri ..."
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Cited by 2 (1 self)
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This paper derives sharp l and l 1 estimates of the error arising from an explicit approximation of the constant coe#cient 1D convection/di#usion equation with Dirac initial data. The analysis embeds the discrete equations within a semidiscrete system of equations which can be solved by Fourier analysis. The error estimates are then obtained though asymptotic approximation of the integrals resulting from the inverse Fourier transform. This research is motivated by the desire to prove convergence of approximations to adjoint partial di#erential equations
On a posteriori pointwise error estimation using adjoint temperature and Lagrange remainder
, 2005
"... ..."
2 Optimal control of flow with discontinuities
, 2003
"... 7 Optimal control of the 1D Riemann problem of Euler equations is studied, with the initial values for pressure and 8 density as control parameters. The leastsquares type cost functional employs either distributed observations in time or 9 observations calculated at the end of the assimilation win ..."
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7 Optimal control of the 1D Riemann problem of Euler equations is studied, with the initial values for pressure and 8 density as control parameters. The leastsquares type cost functional employs either distributed observations in time or 9 observations calculated at the end of the assimilation window. Existence of solutions for the optimal control problem is 10 proven. Smooth and nonsmooth optimization methods employ the numerical gradient (respectively, a subgradient) of 11 the cost functional, obtained from the adjoint of the discrete forward model. The numerical flow obtained with the 12 optimal initial conditions obtained from the nonsmooth minimization matches very well with the observations. The 13 algorithm for smooth minimization converges for the shorter time horizon but fails to perform satisfactorily for the 14 longer time horizon, except when the observations corresponding to shocks are detected and removed.
ARTICLE IN PRESS 2 On a posteriori pointwise error estimation using
, 2004
"... 10 The pointwise estimation of heat conduction solution as a function of truncation error of a finite difference scheme is 11 addressed. The truncation error is estimated using a Taylor series with the remainder in the Lagrange form. The con12 tribution of the local error to the total pointwise err ..."
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10 The pointwise estimation of heat conduction solution as a function of truncation error of a finite difference scheme is 11 addressed. The truncation error is estimated using a Taylor series with the remainder in the Lagrange form. The con12 tribution of the local error to the total pointwise error is estimated via an adjoint temperature. It is demonstrated that 13 the results of numerical calculation of the temperature at an observation point may thus be refined via adjoint error 14 correction and that an asymptotic error bound may be found.
www.elsevier.com/locate/jcp Optimal control of flow with discontinuities
, 2003
"... Optimal control of the 1D Riemann problem of Euler equations is studied, with the initial values for pressure and density as control parameters. The leastsquares type cost functional employs either distributed observations in time or observations calculated at the end of the assimilation window. E ..."
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Optimal control of the 1D Riemann problem of Euler equations is studied, with the initial values for pressure and density as control parameters. The leastsquares type cost functional employs either distributed observations in time or observations calculated at the end of the assimilation window. Existence of solutions for the optimal control problem is proven. Smooth and nonsmooth optimization methods employ the numerical gradient (respectively, a subgradient) of the cost functional, obtained from the adjoint of the discrete forward model. The numerical flow obtained with the optimal initial conditions obtained from the nonsmooth minimization matches very well with the observations. The algorithm for smooth minimization converges for the shorter time horizon but fails to perform satisfactorily for the longer time horizon, except when the observations corresponding to shocks are detected and removed.