510 Mathematik
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The efficient time integration of the dynamic core equations for numerical weather prediction (NWP) remains a key challenge. One of the most popular methods is currently provided by implementations of the semi-implicit semi-Lagrangian (SISL) method, originally proposed by Robert (J. Meteorol. Soc. Jpn., 1982). Practical implementations of the SISL method are, however, not without certain shortcomings with regard to accuracy, conservation properties and stability. Based on recent work by Gottwald, Frank and Reich (LNCSE, Springer, 2002), Frank, Reich, Staniforth, White and Wood (Atm. Sci. Lett., 2005) and Wood, Staniforth and Reich (Atm. Sci. Lett., 2006) we propose an alternative semi-Lagrangian implementation based on a set of regularized equations and the popular Stormer-Verlet time stepping method in the context of the shallow-water equations (SWEs). Ultimately, the goal is to develop practical implementations for the 3D Euler equations that overcome some or all shortcomings of current SISL implementations.
The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean–Vlasov equations as the starting point to derive ensemble Kalman–Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems.
In this study we present iterative regularization methods using rational approximations, in particular, Pade approximants, which work well for ill-posed problems. We prove that the (k,j)-Pade method is a convergent and order optimal iterative regularization method in using the discrepancy principle of Morozov. Furthermore, we present a hybrid Pade method, compare it with other well-known methods and found that it is faster than the Landweber method. It is worth mentioning that this study is a completion of the paper [A. Kirsche, C. Bockmann, Rational approximations for ill-conditioned equation systems, Appl. Math. Comput. 171 (2005) 385-397] where this method was treated to solve ill-conditioned equation systems. (c) 2006 Elsevier Inc. All rights reserved.
A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations is proposed and analysed. Application of regularization to the geopotential field used in the momentum equations leads to an unconditionally stable scheme. The analysis, together with a fully nonlinear example application, suggests that this approach is a promising, efficient, and accurate alternative to traditional schemes.
We study mixed boundary value problems for an elliptic operator A on a manifold X with boundary Y, i.e., Au = f in int X, T (+/-) u = g(+/-) on int Y+/-, where Y is subdivided into subsets Y+/- with an interface Z and boundary conditions T+/- on Y+/- that are Shapiro-Lopatinskij elliptic up to Z from the respective sides. We assume that Z subset of Y is a manifold with conical singularity v. As an example we consider the Zaremba problem, where A is the Laplacian and T- Dirichlet, T+ Neumann conditions. The problem is treated as a corner boundary value problem near v which is the new point and the main difficulty in this paper. Outside v the problem belongs to the edge calculus as is shown in Bull. Sci. Math. ( to appear). With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along Z {v} of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.
We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as bundles, connections and spin structures. Using right Kan extensions, we can assign to any such theory an ordinary quantum field theory defined on the category of spacetimes and we shall clarify under which conditions it satisfies the axioms of locally covariant quantum field theory. The same constructions can be performed in a homotopy theoretic framework by using homotopy right Kan extensions, which allows us to obtain first toy-models of homotopical quantum field theories resembling some aspects of gauge theories.
In a recent paper, the Lefschetz number for endomorphisms (modulo trace class operators) of sequences of trace class curvature was introduced. We show that this is a well defined, canonical extension of the classical Lefschetz number and establish the homotopy invariance of this number. Moreover, we apply the results to show that the Lefschetz fixed point formula holds for geometric quasiendomorphisms of elliptic quasicomplexes.
This paper is concerned with localization properties of coherent states. Instead of classical uncertainty relations we consider "generalized" localization quantities. This is done by introducing measures on the reproducing kernel. In this context we may prove the existence of optimally localized states. Moreover, we provide a numerical scheme for deriving them.
The aim of this paper is to express the Conley-Zehnder index of a symplectic path in terms of an index due to Leray and which has been studied by one of us in a previous work. This will allow us to prove a formula for the Conley-Zehnder index of the product of two symplectic paths in terms of a symplectic Cayley transform. We apply our results to a rigorous study of the Weyl representation of metaplectic operators, which plays a crucial role in the understanding of semiclassical quantization of Hamiltonian systems exhibiting chaotic behavior.
