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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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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
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CONVERGENCE OF LINEARIZED AND ADJOINT APPROXIMATIONS FOR DISCONTINUOUS SOLUTIONS OF CONSERVATION LAWS. PART 1: LINEARIZED APPROXIMATIONS AND LINEARIZED OUTPUT FUNCTIONALS∗
"... Abstract. This paper analyzes the convergence of discrete approximations to the linearized equations arising from an unsteady onedimensional hyperbolic equation with a convex flux function. A simple modified Lax–Friedrichs discretization is used on a uniform grid, and a key point is that the numer ..."
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Abstract. This paper analyzes the convergence of discrete approximations to the linearized equations arising from an unsteady onedimensional hyperbolic equation with a convex flux function. A simple modified Lax–Friedrichs discretization is used on a uniform grid, and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined. It is proved that this gives pointwise convergence almost everywhere for the solution to the linearized discrete equations with smooth initial data, and also convergence in the discrete approximation of linearized output functionals. In Part 2 [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 905–921] we extend the results to Dirac initial data for the linearized equation and will prove the pointwise convergence almost everywhere for the solution of the adjoint discrete equations. In addition, we present numerical results illustrating the asymptotic behavior which is analyzed.
Analysis of discrete adjoints for upwind numerical schemes
 Lect Notes Comput Sc 2005
"... Abstract. This paper discusses several aspects related to the consistency and stability of the discrete adjoints of upwind numerical schemes. First and third order upwind discretizations of the onedimensional advection equation are considered in both the finite difference and finite volume formul ..."
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Abstract. This paper discusses several aspects related to the consistency and stability of the discrete adjoints of upwind numerical schemes. First and third order upwind discretizations of the onedimensional advection equation are considered in both the finite difference and finite volume formulations. We show that the discrete adjoints may lose consistency and stability near the points where upwinding is changed, and near inflow boundaries where the numerical scheme is changed. The impact of adjoint inconsistency and instability on data assimilation is analyzed. 1
equation with Dirac
, 2004
"... Sharp error estimates for discretizations of the 1D convection–diffusion ..."
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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.
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.
Convergence of linearised and adjoint
"... approximations for discontinuous solutions of conservation laws ..."
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