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We address multiscale elliptic problems with random coefficients that are a perturbation
of multiscale deterministic problems. Our approach consists in taking benefit of the
perturbative context to suitably modify the classical Finite Element basis into a
deterministic multiscale Finite Element basis. The latter essentially shares the same
approximation properties as a multiscale Finite Element basis directly generated on the
random problem. The specific reference method that we use is the Multiscale Finite Element
Method. Using numerical experiments, we demonstrate the efficiency of our approach and the
computational speed-up with respect to a more standard approach. In the stationary
setting, we provide a complete analysis of the approach, extending that available for the
deterministic periodic setting.
The parareal in time algorithm allows for efficient parallel numerical simulations of
time-dependent problems. It is based on a decomposition of the time interval into
subintervals, and on a predictor-corrector strategy, where the propagations over each
subinterval for the corrector stage are concurrently performed on the different processors
that are available. In this article, we are concerned with the long time integration of
Hamiltonian systems. Geometric, structure-preserving integrators are preferably employed
for such systems because they show interesting numerical properties, in particular
excellent preservation of the total energy of the system. Using a symmetrization procedure
and/or a (possibly also symmetric) projection step, we introduce here several variants of
the original plain parareal in time algorithm [L. Baffico, et al. Phys. Rev. E
66 (2002) 057701; G. Bal and Y. Maday, A parareal time
discretization for nonlinear PDE’s with application to the pricing of an American put, in
Recent developments in domain decomposition methods, Lect.
Notes Comput. Sci. Eng. 23 (2002) 189–202; J.-L. Lions, Y. Maday
and G. Turinici, C. R. Acad. Sci. Paris, Série I 332 (2001)
661–668.] that are better adapted to the Hamiltonian context. These variants are
compatible with the geometric structure of the exact dynamics, and are easy to implement.
Numerical tests on several model systems illustrate the remarkable properties of the
proposed parareal integrators over long integration times. Some formal elements of
understanding are also provided.
This paper considers the inversion problem related to the
manipulation of quantum
systems using laser-matter interactions. The focus
is on the identification of the field free Hamiltonian and/or
the dipole moment of a
quantum system. The evolution of the system is given by the Schrödinger
equation. The available data are observations as a function of time
corresponding to dynamics generated by electric fields. The
well-posedness of the problem is proved, mainly focusing on the uniqueness of
the solution. A numerical approach is also introduced with an
illustration of its efficiency on a test problem.
The present article is an overview of some mathematical results, which
provide elements of rigorous basis for some multiscale
computations in materials science. The emphasis is laid upon atomistic
to continuum limits for crystalline materials. Various mathematical
approaches are addressed. The
setting is stationary. The relation to existing techniques used in the engineering
literature is investigated.
In order to describe a solid which deforms smoothly in some region, but
non smoothly in some other region, many multiscale methods have recently
been proposed. They aim at coupling an atomistic model (discrete
mechanics) with a macroscopic model
We provide here a theoretical ground for such a coupling in a
one-dimensional setting. We briefly study the general case of a convex
energy, and next concentrate on
a specific example of a nonconvex energy, the Lennard-Jones case. In the
latter situation, we prove that the discretization needs to account in
an adequate way for the coexistence of a discrete model and a continuous
one. Otherwise, spurious discretization effects may appear.
We provide a numerical analysis of the approach.
We present the field of computational chemistry from the standpoint of numerical analysis. We introduce the most commonly used models and comment on their applicability. We briefly outline the results of mathematical analysis and then mostly concentrate on the main issues raised by numerical simulations. A special emphasis is laid on recent results in numerical analysis, recent developments of new methods and challenging open issues.
The present work is a mathematical analysis of two algorithms, namely
the Roothaan and the level-shifting algorithms, commonly used in
practice to solve the Hartree-Fock equations. The level-shifting
algorithm is proved to be well-posed and to converge provided the shift
parameter is large enough. On the contrary, cases when the Roothaan
algorithm is not well defined or fails in converging are
exhibited. These mathematical results are confronted to numerical
experiments performed by chemists.
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