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 non-differentiable minimization method can be defined as a generalization of the gradient-based unconstrained methods, such as the preconditioned conjugate-gradient and quasi-Newton 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 gradient-based form. The non-differentiable aspect of the MLEF algorithm is illustrated in an example with one-dimensional Burgers model and simulated observations. The MLEF algorithm has a robust performance, producing satisfactory results for tested non-differentiable observation operators.

Abstract not found

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.

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.

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 homepage: www.elsevier.com/locate/jcp Non-linear Petrov–Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods

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 con-12 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.

We investigate the issue of variational and sequential data assimilation with nonlinear and non-smooth observation operators using a two-dimensional limitedarea shallow-water equation model and its adjoint. The performance of the “Four-Dimensional ” Variational Approach (4D-Var, 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 non-smooth 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 (L-BFGS) quasi-Newton algorithm originally intended for smooth optimization, the well-known non-linear conjugate gradient method also originally intended for smooth optimization, and the Limited-Memory Bundle Method algorithm (LMBM) (Karmitsa, 2007) specifically designed to address large-scale non-smooth minimization problems. Numerical results obtained for the MLEF method show that the LMBM algorithm gives superior results to the CG method. Results for 4D-Var suggest that L-BFGS performs well when the non-smoothness is not extreme, but fails for non-smooth functions with large Lipschitz constants. The LMBM method is found to be suitable choice for large-scale non-smooth optimization, although additional work is needed to improve its numerical stability. Finally, the results and methodologies of 4D-Var and MLEF are compared and contrasted. Copyright c ○ 0000 Royal Meteorological Society

A new Petrov-Galerkin 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 space-time and the direction of the gradient of the solution. The stabilization of the proper orthogonal decomposition (POD) model using the new Petrov-Galerkin approach is demonstrated in 1D and 2D advection and 1D shock wave cases. Error estimation is carried out for evaluating the accuracy of the Petrov-Galerkin POD model. Numerical results show the new nonlinear Petrov-Galerkin method is a promising approach for stablisation of reduced order modelling. Keywords: Finite Element, Petrov-Galerkin, Proper orthogonal decomposition, Reduced order modelling, Shock wave.