Absolute convergence

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_{n=0}^\infty a_n is said to converge absolutely if \textstyle\sum_{n=0}^\infty \left|a_n\right| = L for some real number \textstyle L. Similarly, an improper integral of a function, \textstyle\int_0^\infty f(x),dx, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if \textstyle\int_0^\infty |f(x)|dx = L. Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess – a convergent series that is not absolutely convergent is called conditionally convergent, while absolutely convergent series behave "nicely". For instance, rear

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A decoupled and unconditionally convergent linear FEM integrator for the Landau-Lifshitz-Gilbert equation with magnetostriction

Jonathan Rochat

To describe and simulate dynamic micromagnetic phenomena, we consider a coupled system of the non-linear Landau-Lifshitz-Gilbert equation and the conservation of momentum equation. This coupling allows one to include magnetostrictive effects into the simulations. Existence of weak solutions has recently been shown in Carbou et al. (2011) (Global weak solutions for the Landau-Lifschitz equation with magnetostriction. Math. Meth. Appl. Sci., 34, 1274-1288). In our contribution, we give an alternate proof which additionally provides an effective numerical integrator. The latter is based on linear finite elements (FEs) in space and a linear-implicit Euler time-stepping. Despite the nonlinearity, only two linear systems have to be solved per timestep, and the integrator fully decouples both equations. Finally, we prove unconditional convergence-at least of a subsequence-towards, and hence existence of, a weak solution of the coupled system, as timestep size and spatial mesh size tend to zero. We conclude the work with numerical experiments, which study the discrete blow-up of the LLG equation as well as the influence of the magnetostrictive term on the discrete blow-up.

Oxford University Press2014

A Continuation Multi Level Monte Carlo (C-MLMC) method for uncertainty quantification in compressible inviscid aerodynamics

Pénélope Leyland, Fabio Nobile, Michele Pisaroni

In this work we apply the Continuation Multi-Level Monte Carlo (C-MLMC) algorithm proposed in Collier et al. (2014) to efficiently propagate operating and geometric uncertainties in inviscid compressible aerodynamics numerical simulations. The key idea of MLMC is that one can draw MC samples simultaneously and independently on several approximations of the problem under investigations on a hierarchy of nested computational grids (levels). The expectation of an output quantity is computed as a sample average using the coarsest solutions and corrected by averages of the differences of the solutions of two consecutive grids in the hierarchy. By this way, most of the computational effort is transported from the finest level (as in a standard Monte Carlo approach) to the coarsest one. The continuation algorithm (C-MLMC) is a robust and self-tuning version that estimates on the fly the optimal number of level and realizations per level. In this work we describe in detail how C-MLMC can be adapted to perform uncertainty quantification analysis in compressible aerodynamics and we apply it to the relevant test cases of a quasi 1D convergent–divergent Laval nozzle and the 2D transonic RAE-2822 airfoil.

Elsevier2017

Series (mathematics)

In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus

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In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and fun

Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used

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