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L 2 -error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes

Thomas Apel, Dieter Sirch (2011)

Applications of Mathematics

An L 2 -estimate of the finite element error is proved for a Dirichlet and a Neumann boundary value problem on a three-dimensional, prismatic and non-convex domain that is discretized by an anisotropic tetrahedral mesh. To this end, an approximation error estimate for an interpolation operator that is preserving the Dirichlet boundary conditions is given. The challenge for the Neumann problem is the proof of a local interpolation error estimate for functions from a weighted Sobolev space.

La différentiation automatique et son utilisation en optimisation

Jean-Pierre Dussault (2008)

RAIRO - Operations Research

In this work, we present an introduction to automatic differentiation, its use in optimization software, and some new potential usages. We focus on the potential of this technique in optimization. We do not dive deeply in the intricacies of automatic differentiation, but put forward its key ideas. We sketch a survey, as of today, of automatic differentiation software, but warn the reader that the situation with respect to software evolves rapidly. In the last part of the paper, we present some...

Lanczos-like algorithm for the time-ordered exponential: The * -inverse problem

Pierre-Louis Giscard, Stefano Pozza (2020)

Applications of Mathematics

The time-ordered exponential of a time-dependent matrix 𝖠 ( t ) is defined as the function of 𝖠 ( t ) that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in 𝖠 ( t ) . The authors have recently proposed the first Lanczos-like algorithm capable of evaluating this function. This algorithm relies on inverses of time-dependent functions with respect to a non-commutative convolution-like product, denoted by * . Yet, the existence of such inverses, crucial to...

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