One-log call iterative solution of the Colebrook equation for flow friction based on Pade polynomials

The 80 year-old empirical Colebrook function zeta, widely used as an informal standard for hydraulic resistance, relates implicitly the unknown flow friction factor lambda, with the known Reynolds number Re and the known relative roughness of a pipe inner surface epsilon* ; lambda = zeta(Re, epsilon* ,lambda). It is based on logarithmic law in the form that captures the unknown flow friction factor l in a way that it cannot be extracted analytically. As an alternative to the explicit approximations or to the iterative procedures that require at least a few evaluations of computationally expensive logarithmic function or non-integer powers, this paper offers an accurate and computationally cheap iterative algorithm based on Pade polynomials with only one log-call in total for the whole procedure (expensive log-calls are substituted with Pade polynomials in each iteration with the exception of the first). The proposed modification is computationally less demanding compared with the standard approaches of engineering practice, but does not influence the accuracy or the number of iterations required to reach the final balanced solution.Web of Science117art. no. 182

Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright omega-function

The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor, f. To date, the captured flow friction factor, f, can be extracted from the logarithmic form analytically only in the term of the Lambert W-function. The purpose of this study is to find an accurate and computationally efficient solution based on the shifted Lambert W-function also known as the Wright omega-function. The Wright omega-function is more suitable because it overcomes the problem with the overflow error by switching the fast growing term, y = W (e(x)), of the Lambert W-function to series expansions that further can be easily evaluated in computers without causing overflow run-time errors. Although the Colebrook equation transformed through the Lambert W-function is identical to the original expression in terms of accuracy, a further evaluation of the Lambert W-function can be only approximate. Very accurate explicit approximations of the Colebrook equation that contain only one or two logarithms are shown. The final result is an accurate explicit approximation of the Colebrook equation with a relative error of no more than 0.0096%. The presented approximations are in a form suitable for everyday engineering use, and are both accurate and computationally efficient.Web of Science71art. no. 3

Approximations to ruin probablities in infinite time using a Lévy process

Mestrado em Ciências ActuariaisEsta dissertação aborda especificamente problemas da área da teoria da ruína, sub-área da teoria do risco para a atividade seguradora. Em particular, estudamos a probabilidade de ruína eventual. Adaptamos o modelo de risco coletivo de Cramér-Lundberg, estendendo para o modelo perturbado. Adicionamos ao modelo de Poisson composto uma componente representativa de um processo de Lévy (alfa estável). Esta componente adicional permite-nos incorporar incertezas decorrentes de, por exemplo, flutuações de taxas de juro, alterações no número de apólices na carteira, em quaisquer dos casos mantendo as hipóteses tradicionais. Com o objetivo de cálculo da probabilidade de ruína no modelo perturbado, apresentamos novas técnicas, recuperando e generalizando modelos de aproximação bem conhecidos, tais como os de DE VYLDER (1996), DUFRESNE AND GERBER (1989), POLLACZEK-KHINCHINE, PADÉ (ver AVRAM ET AL. (2001) e JOHNSON AND TAAFFE (1989)), obtidas ajustando um, dois, três ou quatro momentos ordinários da distribuição dos montantes das indemnizações. Para além disso, considerámos também importante que as aproximações ajustassem a transformada de Laplace (para a probabilidade de ruína), veja-se FURRER (1998). Avaliamos a qualidade das aproximações estudadas exemplificando para um conjunto de distribuições de cauda leve e de cauda pesada. Ilustramos com detalhe com alguns resultados numéricos.In this thesis, we work with prominence to a key area in actuarial science, namely ruin theory. The Cramér-Lundberg model of collective risk theory is adapted for the perturbed model, by adding a Lévy (α-stabled) process to the compound Poisson process, which allows us to consider uncertainty to the premium income, fluctuations of the interest rates, changes to the number of policyholders, without neglecting all other assumptions. On the way, we present new approximation techniques, built for the perturbed model in infinite time, and recall a remarkable family of well-known approximations by DE VYLDER (1996), DUFRESNE AND GERBER (1989), POLLACZEK-KHINCHINE and PADÉ (see AVRAM ET. AL (2001) and JOHNSON AND TAAFFE (1989)), obtained by fitting one, two, three or four (we also attempt five) ordinary moments of the claim amount distribution, and thus significantly generalising these approximations. Finding such approximation which fit the Laplace transform of the ruin probability would also be quite valuable, see FURRER (1998). We test the accuracy of the approximations using a mixture of light and heavy tailed distributions for the individual claim amount. We evaluate the ultimate ruin probability and illustrate in detail some numerical results.N/

Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics

Exit problems for one-dimensional Lévy processes are easier when jumps only occur in one direction. In the last few years, this intuition became more precise: we know now that a wide variety of identities for exit problems of spectrally-negative Lévy processes may be ergonomically expressed in terms of two q-harmonic functions (or scale functions or positive martingales) W and Z. The proofs typically require not much more than the strong Markov property, which hold, in principle, for the wider class of spectrally-negative strong Markov processes. This has been established already in particular cases, such as random walks, Markov additive processes, Lévy processes with omega-state-dependent killing, and certain Lévy processes with state dependent drift, and seems to be true for general strong Markov processes, subject to technical conditions. However, computing the functions W and Z is still an open problem outside the Lévy and diffusion classes, even for the simplest risk models with state-dependent parameters (say, Ornstein–Uhlenbeck or Feller branching diffusion with phase-type jumps)