There is growing awareness that errors in the model equations cannot be ignored data assimilation methods such as four-dimensional variational (4D-Var). If allowed for, more information can extracted from observations, longer time windows are possible, and minimisation process easier, at least principle. Weak-constraint 4D-Var estimates error minimises a series of quadratic cost functions, whic...

The discontinuous Galerkin (DG) discretization relies on an integral (weak) formulation of the hyperbolic conservation law, which leads to the evaluation of several surface and line integrals for multidimensional problems. An alternative formulation of the DG method is possible under the flux reconstruction (FR) framework, where the equations are solved in differential form and the discretizati...

We investigate the ultra weak variational formulation for simulating time-harmonic Maxwell problems. This study has two main goals. First, we introduce a novel derivation of the UWVF method which shows that the UWVF is an unusual version of the standard upwind discontinuous Galerkin (DG) method with a special choice of basis functions. Second, we discuss the practical implementation of an elect...

An explicit formulation of Mach’s principle is given, yielding a finite gravitational background. Its value appears to be the same as the local acceleration in systems of various levels of hierarchy, which indicates that Machian forces control the formation of local structure. The large-scale chain structure can be interpreted in the same way. The level and scale at which the transition from th...

We provide general formulation of weak identification in semiparametric models and an efficiency concept. Weak occurs when a parameter is weakly regular, that is, it locally homogeneous degree zero. When this happens, consistent or equivariant estimation shown to be impossible. then show there exists underlying regular fully characterizes the parameter. While not unique, concepts sufficiency mi...

In this paper we investigate the feasibility of using the ultra weak variational formulation (UWVF) to solve the timeharmonic 3D elastic wave propagation problem. The UWVF is a non-polynomial volume based method that uses plane waves as basis functions which reduces the computational burden. More general, the UWVF is a special form of the discontinuous Galerkin method. As a model problem we con...

The ultra weak variational formulation (UWVF) approach has been proposed as an effective method for solving Helmholtz problems with high wave numbers. However, for coarse meshes the method can suffer from instability. In this paper we consider computational aspects of the ultra weak variational formulation for the inhomogeneous Helmholtz problem. We introduce a method to improve the UWVF scheme...