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Unit# MATH - Gestion

Institute

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Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathema

Finite element method

The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the tr

Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.
The function is often thought of as

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Assyr Abdulle, Giacomo Rosilho De Souza

Stabilized explicit methods are particularly efficient, for large systems of stiff stochastic differential equations (SDEs) due to their extended stability domain. However, they lose their efficiency when a severe stiffness is induced by very few "fast" degrees of freedom, as the stiff and nonstiff terms are evaluated concurrently. Therefore, inspired by [A. Abdulle, M. J. Grote, and G. Rosilho de Souza, Explicit stabilized multirate method for stiff differential equations, Math. Comp., in press, 2022], we introduce a stochastic modified equation whose stiffness depends solely on vi the "slow" terms. By integrating this modified equation with a stabilized explicit scheme, we devise a multirate method which overcomes the bottleneck caused by a few severely still terms and recovers the efficiency of stabilized schemes for large systems of nonlinear SDEs. The scheme is not based on any scale separation assumption of the SDE. Therefore, it is employable for problems stemming from the spatial discretization of stochastic parabolic partial differential equations on locally refined grids. The multirate scheme has strong Order 1/2, weak order 1, and its stability is proved on a model problem. Numerical experiments confirm the efficiency and accuracy of the scheme.

We discuss criteria for a stable map of genus two and degree 4 to the projective plane to be smoothable, as an application of our modular desingularisation of (M) over bar (2,n)(P-r, d)(main) via logarithmic geometry and Gorenstein singularities.

We consider expected performances based on max-stable random fields and we are interested in their derivatives with respect to the spatial dependence parameters of those fields. Max-stable fields, such as the Brown-Resnick and Smith fields, are very popular in spatial extremes. We focus on the two most popular unbiased stochastic derivative estimation approaches: the likelihood ratio method (LRM) and the infinitesimal perturbation analysis (IPA). LRM requires the multivariate density of the max-stable field to be explicit, and IPA necessitates the computation of the derivative with respect to the parameters for each simulated value. We propose convenient and tractable conditions ensuring the validity of LRM and IPA in the cases of the Brown-Resnick and Smith field, respectively. Obtaining such conditions is intricate owing to the very structure of max-stable fields. Then we focus on risk and dependence measures, which constitute one of the several frameworks where our theoretical results can be useful. We perform a simulation study which shows that both LRM and IPA perform well in various configurations, and provide a real case study that is valuable for the insurance industry. (C) 2021 The Authors. Published by Elsevier B.V.