1,133 research outputs found
Discrete embeddings for Lagrangian and Hamiltonian systems
The general topic of the present paper is to study the conservation for some
structural property of a given problem when discretising this problem.
Precisely we are interested with Lagrangian or Hamiltonian structures and thus
with variational problems attached to a least action principle. Considering a
partial differential equation (PDE) deriving from such a variational principle,
a natural question is to know whether this structure at the continuous level is
preserved at the discrete level when discretising the PDE. To address this
question a concept of \textit{coherence} is introduced. Both the differential
equation (the PDE translating the least action principle) and the variational
structure can be embedded at the discrete level. This provides two discrete
embeddings for the original problem. In case these procedures finally provide
the same discrete problem we will say that the discretisation is
\textit{coherent}. Our purpose is illustrated with the Poisson problem.
Coherence for discrete embeddings of Lagrangian structures is studied for
various classical discretisations (finite elements, finite differences and
finite volumes). Hamiltonian structures are shown to provide coherence between
a discrete Hamiltonian structure and the discretisation of the mixed
formulation of the PDE, both for mixed finite elements and mimetic finite
differences methods.Comment: Acta Mathematica Vietnamica, Springer Singapore, A Para{\^i}tr
Raviart-Thomas finite elements of Petrov-Galerkin type
The mixed finite element method for the Poisson problem with the
Raviart-Thomas elements of low-level can be interpreted as a finite volume
method with a non-local gradient. In this contribution, we propose a variant of
Petrov-Galerkin type for this problem to ensure a local computation of the
gradient at the interfaces of the elements. The shape functions are the
Raviart-Thomas finite elements. Our goal is to define test functions that are
in duality with these shape functions: Precisely, the shape and test functions
will be asked to satisfy a L2-orthogonality property. The general theory of
Babu\v{s}ka brings necessary and sufficient stability conditions for a
Petrov-Galerkin mixed problem to be convergent. We propose specific constraints
for the dual test functions in order to ensure stability. With this choice, we
prove that the mixed Petrov-Galerkin scheme is identical to the four point
finite volumes scheme of Herbin, and to the mass lumping approach developed by
Baranger, Maitre and Oudin. Finally, we construct a family of dual test
functions that satisfy the stability conditions. Convergence is proven with the
usual techniques of mixed finite elements
Neural Expectation Maximization
Many real world tasks such as reasoning and physical interaction require
identification and manipulation of conceptual entities. A first step towards
solving these tasks is the automated discovery of distributed symbol-like
representations. In this paper, we explicitly formalize this problem as
inference in a spatial mixture model where each component is parametrized by a
neural network. Based on the Expectation Maximization framework we then derive
a differentiable clustering method that simultaneously learns how to group and
represent individual entities. We evaluate our method on the (sequential)
perceptual grouping task and find that it is able to accurately recover the
constituent objects. We demonstrate that the learned representations are useful
for next-step prediction.Comment: Accepted to NIPS 201
Application of direct inverse analogy method (DIVA) and viscous design optimization techniques
A direct-inverse approach to the transonic design problem was presented in its initial state at the First International Conference on Inverse Design Concepts and Optimization in Engineering Sciences (ICIDES-1). Further applications of the direct inverse analogy (DIVA) method to the design of airfoils and incremental wing improvements and experimental verification are reported. First results of a new viscous design code also from the residual correction type with semi-inverse boundary layer coupling are compared with DIVA which may enhance the accuracy of trailing edge design for highly loaded airfoils. Finally, the capabilities of an optimization routine coupled with the two viscous full potential solvers are investigated in comparison to the inverse method
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