We present a local Fourier analysis of multigrid methods for the two-dimensional curl-curl formulation of Maxwell's equations. Both the hybrid smoother proposed by Hiptmair and the overlapping block smoother proposed by Arnold, Falk and Winther are considered. The key to our approach is the identification of twodimensional eigenspaces of the discrete curl-curl problem by decoupling the Fourier modes for edges with different orientations. Our analysis allows to quantify the smoothing properties of the considered smoothers and the convergence behavior of the considered multigrid methods. Additionally, we identify the Helmholtz splitting in Fourier space. This allows to recover several well known properties in Fourier space, such as the commutation properties of the classical Nédélec prolongator and the equivalence of the curl-curl operator and the vector Laplacian for divergence-free vectors. We show how the approach used in this paper can be generalized to twoand three-dimensional problems in H(curl) and H(div) and to other types of regular meshes.
We consider the numerical solution of time-dependent partial differential equations (PDEs) with random coefficients. A spectral approach, called stochastic finite element method, is used to compute the statistical characteristics of the solution. This method transforms a stochastic PDE into a coupled system of deterministic equations by means of a Galerkin projection onto a generalized polynomial chaos. An algebraic multigrid (AMG) method is presented to solve the algebraic systems that result after discretization of this coupled system. High-order time integration schemes of an implicit Runge-Kutta type and spatial discretization on unstructured finite element meshes are considered. The convergence properties of the AMG method are demonstrated by a convergence analysis and by numerical tests.
This paper studies the set of Pareto optimal insurance contracts and the core of an insurance game. Our setting allows multiple insurers with translation invariant preferences. We characterise the Pareto optimal contracts, which determines the shape of the indemnities. Closed-form and numerical solutions are found for various preferences that the insurance players might have. Determining associated premiums with any given optimal Pareto contract is another problem for which economic-based arguments are further discussed. We also explain how one may link the recent fast growing literature on risk-based optimality criteria to the Pareto optimality criterion and we show that the latter is much more general than the former one, which according to our knowledge, has not been pointed out by now. Further, we extend some of our results when model risk is included, i.e. there is some uncertainty with the risk model and/or the insurance players make decisions based on divergent beliefs about the underlying risk. These robust optimal contracts are investigated and we show how one may find robust and Pareto efficient contracts, which is a key decision-making problem under uncertainty.
We analyse models for panel data that arise in risk allocation problems, when a given set of sources are the cause of an aggregate risk value. We focus on the modeling and forecasting of proportional contributions to risk. Compositional data methods are proposed and the regression is flexible to incorporate external information from other variables. We guarantee that projected proportional contributions add up to 100%, and we introduce a method to generate confidence regions with the same restriction. An illustration using data from the stock exchange is provided.
Optimal reinsurance indemnities have widely been studied in the literature, yet the bargaining for optimal prices has remained relatively unexplored. Therefore, the key objective of this paper is to analyze the price of reinsurance contracts. We use a novel way to model the bargaining powers of the insurer and reinsurer, which allows us to generalize the contracts according to the Nash bargaining solution, indifference pricing and the equilibrium contracts. We illustrate these pricing functions by means of inverse-S shaped distortion functions for the insurer and the Value-at-Risk for the reinsurer.
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