Ensemble-based data assimilation and the localisation problem

The “butterfly effect” is a popularly known paradigm; commonly it is said that when a butterfly flaps its wings in Brazil, it may cause a tornado in Texas. This essentially describes how weather forecasts can be extremely senstive to small changes in the given atmospheric data, or initial conditions, used in computer model simulations. In 1961 Edward Lorenz found, when running a weather model, that small changes in the initial conditions given to the model can, over time, lead to entriely different forecasts (Lorenz, 1963). This discovery highlights one of the major challenges in modern weather forecasting; that is to provide the computer model with the most accurately specified initial conditions possible. A process known as data assimilation seeks to minimize the errors in the given initial conditions and was, in 1911, described by Bjerkness as “the ultimate problem in meteorology” (Bjerkness, 1911).

A wavenumber independent boundary element method for an acoustic scattering problem

In this paper we consider the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data, a problem which models, for example, outdoor sound propagation over inhomogeneous. at terrain. To achieve good approximation at high frequencies with a relatively low number of degrees of freedom, we propose a novel Galerkin boundary element method, using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that, on each element, the approximation space contains polynomials ( of degree.) multiplied by traces of plane waves on the boundary. We prove stability and convergence and show that the error in computing the total acoustic field is O( N-(v+1) log(1/2) N), where the number of degrees of freedom is proportional to N logN. This error estimate is independent of the wavenumber, and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity.

Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test

In this article, we use the no-response test idea, introduced in Luke and Potthast (2003) and Potthast (Preprint) and the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient gamma of the equation del (.) gamma(x)del + c(x) with piecewise regular gamma and bounded function c(x). We use infinitely many Cauchy data as measurement and give a reconstructive method to localize the interface. We will base this multiwave version of the no-response test on two different proofs. The first one contains a pointwise estimate as used by the singular sources method. The second one is built on an energy (or an integral) estimate which is the basis of the probe method. As a conclusion of this, the probe and the singular sources methods are equivalent regarding their convergence and the no-response test can be seen as a unified framework for these methods. As a further contribution, we provide a formula to reconstruct the values of the jump of gamma(x), x is an element of partial derivative D at the boundary. A second consequence of this formula is that the blow-up rate of the indicator functions of the probe and singular sources methods at the interface is given by the order of the singularity of the fundamental solution.

A high-wavenumber boundary-element method for an acoustic scattering problem

In this paper we show stability and convergence for a novel Galerkin boundary element method approach to the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data. This problem models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve a good approximation with a relatively low number of degrees of freedom we employ a graded mesh with smaller elements adjacent to discontinuities in impedance, and a special set of basis functions for the Galerkin method so that, on each element, the approximation space consists of polynomials (of degree $\nu$) multiplied by traces of plane waves on the boundary. In the case where the impedance is constant outside an interval $[a,b]$, which only requires the discretization of $[a,b]$, we show theoretically and experimentally that the $L_2$ error in computing the acoustic field on $[a,b]$ is ${\cal O}(\log^{\nu+3/2}|k(b-a)| M^{-(\nu+1)})$, where $M$ is the number of degrees of freedom and $k$ is the wavenumber. This indicates that the proposed method is especially commendable for large intervals or a high wavenumber. In a final section we sketch how the same methodology extends to more general scattering problems.

Efficient tree construction for the multicast problem

A new heuristic for the Steiner Minimal Tree problem is presented here. The method described is based on the detection of particular sets of nodes in networks, the “Hot Spot” sets, which are used to obtain better approximations of the optimal solutions. An algorithm is also proposed which is capable of improving the solutions obtained by classical heuristics, by means of a stirring process of the nodes in solution trees. Classical heuristics and an enumerative method are used CIS comparison terms in the experimental analysis which demonstrates the goodness of the heuristic discussed in this paper.

Efficient tree construction for the multicast problem

A new heuristic for the Steiner minimal tree problem is presented. The method described is based on the detection of particular sets of nodes in networks, the “hot spot” sets, which are used to obtain better approximations of the optimal solutions. An algorithm is also proposed which is capable of improving the solutions obtained by classical heuristics, by means of a stirring process of the nodes in solution trees. Classical heuristics and an enumerative method are used as comparison terms in the experimental analysis which demonstrates the capability of the heuristic discussed

A fuzzy approach for the network congestion problem

In the recent years, the unpredictable growth of the Internet has moreover pointed out the congestion problem, one of the problems that historicallyha ve affected the network. This paper deals with the design and the evaluation of a congestion control algorithm which adopts a FuzzyCon troller. The analogyb etween Proportional Integral (PI) regulators and Fuzzycon trollers is discussed and a method to determine the scaling factors of the Fuzzycon troller is presented. It is shown that the Fuzzycon troller outperforms the PI under traffic conditions which are different from those related to the operating point considered in the design.