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The Maximum Likelihood Ensemble Filter as a . . .
, 2008
"... The Maximum Likelihood Ensemble Filter (MLEF) equations are derived without the differentiability requirement for the prediction model and for the observation operators. Derivation reveals that a new nondifferentiable minimization method can be defined as a generalization of the gradientbased un ..."
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Cited by 24 (8 self)
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The Maximum Likelihood Ensemble Filter (MLEF) equations are derived without the differentiability requirement for the prediction model and for the observation operators. Derivation reveals that a new nondifferentiable minimization method can be defined as a generalization of the gradientbased unconstrained methods, such as the preconditioned conjugategradient and quasiNewton methods. In the new minimization algorithm the vector of first order increments of the cost function is defined as a generalized gradient, while the symmetric matrix of second order increments of the cost function is defined as a generalized Hessian matrix. In the case of differentiable observation operators, the minimization algorithm reduces to the standard gradientbased form. The nondifferentiable aspect of the MLEF algorithm is illustrated in an example with onedimensional Burgers model and simulated observations. The MLEF algorithm has a robust performance, producing satisfactory results for tested nondifferentiable observation operators.
On a posteriori pointwise error estimation using adjoint temperature and Lagrange remainder
, 2005
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Data Assimilation for Numerical Weather Prediction: A Review
"... Abstract During the last 20 years data assimilation has gradually reached a mature center stage position at both Numerical Weather Prediction centers as well as being at the center of activities at many federal research institutes as well as at many universities. The research encompasses now activit ..."
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Cited by 2 (0 self)
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Abstract During the last 20 years data assimilation has gradually reached a mature center stage position at both Numerical Weather Prediction centers as well as being at the center of activities at many federal research institutes as well as at many universities. The research encompasses now activities which involve, beside meteorologists and oceanographers at operational centers or federal research facilities, many in the applied and computational mathematical research communities. Data assimilation or 4D VAR extends now also to other geosciences fields such as hydrology and geology and results in the publication of an ever increasing number of books and monographs related to the topic. In this short survey article we provide a brief introduction providing some historical perspective and background, a survey of data assimilation prior to 4D VAR and basic concepts of data assimilation. I first proceed to outline the early 4D VAR stages (1980–1990) and addresses in a succinct manner the period of the 1990s that saw the major developments and the flourishing of all aspects of 4D VAR both at operational centers and at research Universities and Federal Laboratories. Computational aspects of 4D Var data assimilation addressing computational burdens as well as ways to alleviate them are briefly outlined. Brief interludes are provided for each period surveyed allowing the reader to have a better perspective A brief survey of different topics related to state of the art 4D Var today is then presented and we conclude with what we perceive to be main directions of research and the future of data assimilation and some open problems. We will strive to use the unified notation of Ide et al. (J Meteor Soc Japan 75:181–189,
On Adjoint Variables for Discontinuous Flow
"... The standard approach to the solution of the adjoint equations stresses the similarity of direct and adjoint equations and implies the use of similar methods for their solution. Nevertheless, the adjoint equations have significant peculiarities in comparison with the direct problem equations at leas ..."
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The standard approach to the solution of the adjoint equations stresses the similarity of direct and adjoint equations and implies the use of similar methods for their solution. Nevertheless, the adjoint equations have significant peculiarities in comparison with the direct problem equations at least for compressible flows. From a numerical viewpoint these features concern the existence of the conservative form of the equations, linearity and specific boundary conditions or sources. From the flow field structure viewpoint, there are also sizable differences, for example, the compression shock formation in adjoint variables field is impossible when the rarefaction shock is stable and exists. The latter effect poses some restrictions on the solution of inverse problems.
On Adjoint Variables for Discontinuous Flow
"... The standard approach to the solution of the adjoint equations stresses the similarity of direct and adjoint equations and implies the use of similar methods for their solution. Nevertheless, the adjoint equations have significant peculiarities in comparison with the direct problem equations at leas ..."
Abstract
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The standard approach to the solution of the adjoint equations stresses the similarity of direct and adjoint equations and implies the use of similar methods for their solution. Nevertheless, the adjoint equations have significant peculiarities in comparison with the direct problem equations at least for compressible flows. From a numerical viewpoint these features concern the existence of the conservative form of the equations, linearity and specific boundary conditions or sources. From the flow field structure viewpoint, there are also sizable differences, for example, the compression shock formation in adjoint variables field is impossible when the rarefaction shock is stable and exists. The latter effect poses some restrictions on the solution of inverse problems.
and
"... The standard approach to the solution of the adjoint equations stresses the similarity of direct and adjoint equations and implies the use of similar methods for their solution. Nevertheless, the adjoint equations have significant peculiarities in comparison with the direct problem equations at leas ..."
Abstract
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The standard approach to the solution of the adjoint equations stresses the similarity of direct and adjoint equations and implies the use of similar methods for their solution. Nevertheless, the adjoint equations have significant peculiarities in comparison with the direct problem equations at least for compressible flows. From a numerical viewpoint these features concern the existence of the conservative form of the equations, linearity and specific boundary conditions or sources. From the flow field structure viewpoint, there are also sizable differences, for example, the compression shock formation in adjoint variables field is impossible when the rarefaction shock is stable and exists. The latter effect poses some restrictions on the solution of inverse problems. 1.
Journal of Computational Physics xxx (2012) xxx–xxx Contents lists available at SciVerse ScienceDirect Journal of Computational Physics
"... journal homepage: www.elsevier.com/locate/jcp Nonlinear Petrov–Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods ..."
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journal homepage: www.elsevier.com/locate/jcp Nonlinear Petrov–Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods
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.
Impact of NonSmooth Observation Operators on Variational and Sequential Data Assimilation for a LimitedArea Shallow Water Equations Model
"... We investigate the issue of variational and sequential data assimilation with nonlinear and nonsmooth observation operators using a twodimensional limitedarea shallowwater equation model and its adjoint. The performance of the “FourDimensional ” Variational Approach (4DVar, here: two dimensions ..."
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We investigate the issue of variational and sequential data assimilation with nonlinear and nonsmooth observation operators using a twodimensional limitedarea shallowwater equation model and its adjoint. The performance of the “FourDimensional ” Variational Approach (4DVar, here: two dimensions plus time) compared to that of the Maximum Likelihood Ensemble Filter (MLEF), a hybrid ensemble/variational method, is tested in the presence of nonsmooth observation operators. Following the work of (Lewis & Overton, 2008) and (Karmitsa, 2007), we investigate minimization of the data assimilation cost functional using the Limited Memory BFGS (LBFGS) quasiNewton algorithm originally intended for smooth optimization, the wellknown nonlinear conjugate gradient method also originally intended for smooth optimization, and the LimitedMemory Bundle Method algorithm (LMBM) (Karmitsa, 2007) specifically designed to address largescale nonsmooth minimization problems. Numerical results obtained for the MLEF method show that the LMBM algorithm gives superior results to the CG method. Results for 4DVar suggest that LBFGS performs well when the nonsmoothness is not extreme, but fails for nonsmooth functions with large Lipschitz constants. The LMBM method is found to be suitable choice for largescale nonsmooth optimization, although additional work is needed to improve its numerical stability. Finally, the results and methodologies of 4DVar and MLEF are compared and contrasted. Copyright c ○ 0000 Royal Meteorological Society
NonLinear PetrovGalerkin Methods for Reduced Order Hyperbolic Equations and Discontinuous Finite Element Methods
"... A new PetrovGalerkin approach for dealing with sharp or abrupt field changes in discontinuous Galerkin (DG) reduced order modelling (ROM) is outlined in this paper. This method presents a natural and easy way to introduce a diffusion term into ROM without tuning/optimising and provides appropriate ..."
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A new PetrovGalerkin approach for dealing with sharp or abrupt field changes in discontinuous Galerkin (DG) reduced order modelling (ROM) is outlined in this paper. This method presents a natural and easy way to introduce a diffusion term into ROM without tuning/optimising and provides appropriate modeling and stabilisation for the numerical solution of high order nonlinear PDEs. The approach is based on the use of the cosine rule between the advection direction in Cartesian spacetime and the direction of the gradient of the solution. The stabilization of the proper orthogonal decomposition (POD) model using the new PetrovGalerkin approach is demonstrated in 1D and 2D advection and 1D shock wave cases. Error estimation is carried out for evaluating the accuracy of the PetrovGalerkin POD model. Numerical results show the new nonlinear PetrovGalerkin method is a promising approach for stablisation of reduced order modelling. Keywords: Finite Element, PetrovGalerkin, Proper orthogonal decomposition, Reduced order modelling, Shock wave.