The Experts below are selected from a list of 129 Experts worldwide ranked by ideXlab platform

Seddik M. Djouadi - One of the best experts on this subject based on the ideXlab platform.

  • Nonlinear model reduction for fluid flows
    Proceedings of the 2011 American Control Conference, 2011
    Co-Authors: Samir Sahyoun, Seddik M. Djouadi
    Abstract:

    Model Reduction is an essential tool that has been applied in many control applications such as control of fluid flows. Most model reduction algorithms assume linear models and fail when applied to nonlinear high dimensional systems, in particular, fluid flow problems with high Reynolds numbers. For example, proper orthogonal decomposition (POD) fails to capture the nonlinear degrees of freedom in these systems, since it assumes that data belong to a linear space and therefore relies on the Euclidean distance as the metric to minimize. However, snapshots generated by nonlinear partial differential equations (PDEs) belong to manifolds for which the geodesics do not correspond in general to the Euclidean distance. A geodesic is a curve that is locally the shortest path between points. In this paper, we propose a model reduction method which generalizes POD to nonlinear manifolds which have a Differentiable Structure at each of their points. Moreover, an optimal method in constructing reduced order models for the two-dimensional Burgers' equation subject to boundary control is presented and compared to the POD reduced models.

  • Nonlinear model reduction for fluid flows
    Proceedings of the 2011 American Control Conference, 2011
    Co-Authors: Samir Sahyoun, Seddik M. Djouadi
    Abstract:

    Model Reduction is an essential tool that has been applied in many control applications such as control of fluid flows. Most model reduction algorithms assume linear models and fail when applied to nonlinear high dimensional systems, in particular, fluid flow problems with high Reynolds numbers. For example, proper orthogonal decomposition (POD) fails to capture the nonlinear degrees of freedom in these systems, since it assumes that data belong to a linear space and therefore relies on the Euclidean distance as the metric to minimize. However, snapshots generated by nonlinear partial differential equations (PDEs) belong to manifolds for which the geodesies do not correspond in general to the Euclidean distance. A geodesic is a curve that is locally the shortest path between points. In this paper, we propose a model reduction method which generalizes POD to nonlinear manifolds which have a Differentiable Structure at each of their points. Moreover, an optimal method in constructing reduced order models for the two-dimensional Burgers' equation subject to boundary control is presented and compared to the POD reduced models.

Samir Sahyoun - One of the best experts on this subject based on the ideXlab platform.

  • Nonlinear model reduction for fluid flows
    Proceedings of the 2011 American Control Conference, 2011
    Co-Authors: Samir Sahyoun, Seddik M. Djouadi
    Abstract:

    Model Reduction is an essential tool that has been applied in many control applications such as control of fluid flows. Most model reduction algorithms assume linear models and fail when applied to nonlinear high dimensional systems, in particular, fluid flow problems with high Reynolds numbers. For example, proper orthogonal decomposition (POD) fails to capture the nonlinear degrees of freedom in these systems, since it assumes that data belong to a linear space and therefore relies on the Euclidean distance as the metric to minimize. However, snapshots generated by nonlinear partial differential equations (PDEs) belong to manifolds for which the geodesics do not correspond in general to the Euclidean distance. A geodesic is a curve that is locally the shortest path between points. In this paper, we propose a model reduction method which generalizes POD to nonlinear manifolds which have a Differentiable Structure at each of their points. Moreover, an optimal method in constructing reduced order models for the two-dimensional Burgers' equation subject to boundary control is presented and compared to the POD reduced models.

  • Nonlinear model reduction for fluid flows
    Proceedings of the 2011 American Control Conference, 2011
    Co-Authors: Samir Sahyoun, Seddik M. Djouadi
    Abstract:

    Model Reduction is an essential tool that has been applied in many control applications such as control of fluid flows. Most model reduction algorithms assume linear models and fail when applied to nonlinear high dimensional systems, in particular, fluid flow problems with high Reynolds numbers. For example, proper orthogonal decomposition (POD) fails to capture the nonlinear degrees of freedom in these systems, since it assumes that data belong to a linear space and therefore relies on the Euclidean distance as the metric to minimize. However, snapshots generated by nonlinear partial differential equations (PDEs) belong to manifolds for which the geodesies do not correspond in general to the Euclidean distance. A geodesic is a curve that is locally the shortest path between points. In this paper, we propose a model reduction method which generalizes POD to nonlinear manifolds which have a Differentiable Structure at each of their points. Moreover, an optimal method in constructing reduced order models for the two-dimensional Burgers' equation subject to boundary control is presented and compared to the POD reduced models.

José M. Isidro - One of the best experts on this subject based on the ideXlab platform.

  • A QUANTUM IS A COMPLEX Structure ON CLASSICAL PHASE SPACE
    International Journal of Geometric Methods in Modern Physics, 2020
    Co-Authors: José M. Isidro
    Abstract:

    Duality transformations within the quantum mechanics of a finite number of degrees of freedom can be regarded as the dependence of the notion of a quantum, i.e., an elementary excitation of the vacuum, on the observer on classical phase space. Under an observer we understand, as in general relativity, a local coordinate chart. While classical mechanics can be formulated using a symplectic Structure on classical phase space, quantum mechanics requires a complex-Differentiable Structure on that same space. Complex-Differentiable Structures on a given real manifold are often not unique. This article is devoted to analysing the dependence of the notion of a quantum on the complex-Differentiable Structure chosen on classical phase space. For that purpose we consider Kahler phase spaces, endowed with a dynamics whose Hamiltonian equals the local Kahler potential.

  • Complex Moduli of Physical Quanta
    Modern Physics Letters A, 2005
    Co-Authors: José M. Isidro
    Abstract:

    Classical mechanics can be formulated using a symplectic Structure on classical phase space, while quantum mechanics requires a complex-Differentiable Structure on that same space. Complex-Differentiable Structures on a given real manifold are often not unique. This paper is devoted to analysing the dependence of the notion of a quantum on the complex-Differentiable Structure chosen on classical phase space.

  • Moduli of Quanta
    arXiv: Quantum Physics, 2005
    Co-Authors: José M. Isidro
    Abstract:

    The classical phase of the matrix model of 11-dimensional M-theory is complex, infinite-dimensional Hilbert space. As a complex manifold, the latter admits a continuum of nonequivalent, complex-Differentiable Structures that can be placed in 1-to-1 correspondence with families of coherent states in the Hilbert space of quantum states. The moduli space of nonbiholomorphic complex Structures on classical phase space turns out to be an infinite-dimensional symmetric space. We argue that each choice of a complex Differentiable Structure gives rise to a physically different notion of an elementary quantum.

  • QUANTUM-MECHANICAL DUALITIES ON THE TORUS
    Modern Physics Letters A, 2004
    Co-Authors: José M. Isidro
    Abstract:

    On classical phase spaces admitting just one complex-Differentiable Structure, there is no indeterminacy in the choice of the creation operators that create quanta out of a given vacuum. In these cases the notion of a quantum is universal, i.e. independent of the observer on classical phase space. Such is the case in all standard applications of quantum mechanics. However, recent developments suggest that the notion of a quantum may not be universal. Transformations between observers that do not agree on the notion of an elementary quantum are called dualities. Classical phase spaces admitting more than one complex-Differentiable Structure thus provide a natural framework to study dualities in quantum mechanics. As an example we quantise a classical mechanics whose phase space is a torus and prove explicitly that it exhibits dualities.

Joseph A. Wolf - One of the best experts on this subject based on the ideXlab platform.

  • Differentiable Structure for direct limit groups
    Letters in Mathematical Physics, 1991
    Co-Authors: Loki Natarajan, Enriqueta Rodríguez-carrington, Joseph A. Wolf
    Abstract:

    A direct limit $$G = \mathop {\lim }\limits_ \to G_\alpha$$ of (finite-dimensional) Lie groups has Lie algebra $$\mathfrak{g} = \mathop {\lim }\limits_ \to \mathfrak{g}_\alpha$$ and exponential map exp_ G : g→ G . Both G and g carry natural topologies. G is a topological group, and g is a topological Lie algebra with a natural Structure of real analytic manifold. In this Letter, we show how a special growth condition, natural in certain physical settings and satisfied by the usual direct limits of classical groups, ensures that G carries an analytic group Structure such that exp_ G is a diffeomorphism from a certain open neighborhood of 0∈g onto an open neighborhood of 1_ G ∈ G . In the course of the argument, one sees that the Structure sheaf for this analytic group Structure coincides with the direct limit $$\mathop {\lim }\limits_ \to$$ C ^ω( G _α) of the sheaves of germs of analytic functions on the G _α.

Carlo Cattani - One of the best experts on this subject based on the ideXlab platform.