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Felix Otto - One of the best experts on this subject based on the ideXlab platform.

  • quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics
    Inventiones Mathematicae, 2015
    Co-Authors: Antoine Gloria, Stefan Neukamm, Felix Otto
    Abstract:

    We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice \(\mathbb Z^d\) with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the Infinite-Dimensional Space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions \(d>2\)) and prove new optimal estimates for regularized versions of the corrector (in dimensions \(d\ge 2\)). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients.

  • quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics
    Inventiones Mathematicae, 2015
    Co-Authors: Antoine Gloria, Stefan Neukamm, Felix Otto
    Abstract:

    We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice \(\mathbb Z^d\) with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the Infinite-Dimensional Space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions \(d>2\)) and prove new optimal estimates for regularized versions of the corrector (in dimensions \(d\ge 2\)). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients.

  • quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics long version
    2013
    Co-Authors: Antoine Gloria, Stefan Neukamm, Felix Otto
    Abstract:

    We study the effective large-scale behavior of discrete elliptic equations on the lattice $\mathbb Z^d$ with random coefficients. The theory of stochastic homogenization relates the random but stationary field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the Infinite-Dimensional Space of admissible coefficient fields. In this contribution we develop quantitative methods for the corrector problem assuming that the ensemble of coefficient fields satisfies a spectral gap estimate w.~r.~t. a Glauber dynamics. As a main result we prove an optimal estimate for the decay in time of the parabolic equation associated to the corrector problem (i.~e. for the ''random environment as seen from a random walker''). As a corollary we obtain existence and moment bounds for stationary correctors (in dimension $d>2$) and optimal estimates for regularized versions of the corrector (in dimensions $d\geq 2$). We also give a self-contained proof for a new estimate on the gradient of the parabolic, variable-coefficient Green's function, which is a crucial analytic ingredient in our method. As an application, we study the approximation of the homogenized coefficients via a representative volume element. The approximation introduces two types of errors. Based on our quantitative methods, we develop an error analysis that gives optimal bounds in terms of scaling in the size of the representative volume element --- even for large ellipticity ratios.

Antoine Gloria - One of the best experts on this subject based on the ideXlab platform.

  • quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics
    Inventiones Mathematicae, 2015
    Co-Authors: Antoine Gloria, Stefan Neukamm, Felix Otto
    Abstract:

    We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice \(\mathbb Z^d\) with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the Infinite-Dimensional Space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions \(d>2\)) and prove new optimal estimates for regularized versions of the corrector (in dimensions \(d\ge 2\)). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients.

  • quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics
    Inventiones Mathematicae, 2015
    Co-Authors: Antoine Gloria, Stefan Neukamm, Felix Otto
    Abstract:

    We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice \(\mathbb Z^d\) with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the Infinite-Dimensional Space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions \(d>2\)) and prove new optimal estimates for regularized versions of the corrector (in dimensions \(d\ge 2\)). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients.

  • quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics long version
    2013
    Co-Authors: Antoine Gloria, Stefan Neukamm, Felix Otto
    Abstract:

    We study the effective large-scale behavior of discrete elliptic equations on the lattice $\mathbb Z^d$ with random coefficients. The theory of stochastic homogenization relates the random but stationary field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the Infinite-Dimensional Space of admissible coefficient fields. In this contribution we develop quantitative methods for the corrector problem assuming that the ensemble of coefficient fields satisfies a spectral gap estimate w.~r.~t. a Glauber dynamics. As a main result we prove an optimal estimate for the decay in time of the parabolic equation associated to the corrector problem (i.~e. for the ''random environment as seen from a random walker''). As a corollary we obtain existence and moment bounds for stationary correctors (in dimension $d>2$) and optimal estimates for regularized versions of the corrector (in dimensions $d\geq 2$). We also give a self-contained proof for a new estimate on the gradient of the parabolic, variable-coefficient Green's function, which is a crucial analytic ingredient in our method. As an application, we study the approximation of the homogenized coefficients via a representative volume element. The approximation introduces two types of errors. Based on our quantitative methods, we develop an error analysis that gives optimal bounds in terms of scaling in the size of the representative volume element --- even for large ellipticity ratios.

Stefan Neukamm - One of the best experts on this subject based on the ideXlab platform.

  • quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics
    Inventiones Mathematicae, 2015
    Co-Authors: Antoine Gloria, Stefan Neukamm, Felix Otto
    Abstract:

    We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice \(\mathbb Z^d\) with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the Infinite-Dimensional Space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions \(d>2\)) and prove new optimal estimates for regularized versions of the corrector (in dimensions \(d\ge 2\)). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients.

  • quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics
    Inventiones Mathematicae, 2015
    Co-Authors: Antoine Gloria, Stefan Neukamm, Felix Otto
    Abstract:

    We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice \(\mathbb Z^d\) with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the Infinite-Dimensional Space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions \(d>2\)) and prove new optimal estimates for regularized versions of the corrector (in dimensions \(d\ge 2\)). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients.

  • quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics long version
    2013
    Co-Authors: Antoine Gloria, Stefan Neukamm, Felix Otto
    Abstract:

    We study the effective large-scale behavior of discrete elliptic equations on the lattice $\mathbb Z^d$ with random coefficients. The theory of stochastic homogenization relates the random but stationary field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the Infinite-Dimensional Space of admissible coefficient fields. In this contribution we develop quantitative methods for the corrector problem assuming that the ensemble of coefficient fields satisfies a spectral gap estimate w.~r.~t. a Glauber dynamics. As a main result we prove an optimal estimate for the decay in time of the parabolic equation associated to the corrector problem (i.~e. for the ''random environment as seen from a random walker''). As a corollary we obtain existence and moment bounds for stationary correctors (in dimension $d>2$) and optimal estimates for regularized versions of the corrector (in dimensions $d\geq 2$). We also give a self-contained proof for a new estimate on the gradient of the parabolic, variable-coefficient Green's function, which is a crucial analytic ingredient in our method. As an application, we study the approximation of the homogenized coefficients via a representative volume element. The approximation introduces two types of errors. Based on our quantitative methods, we develop an error analysis that gives optimal bounds in terms of scaling in the size of the representative volume element --- even for large ellipticity ratios.

Röckner Michael - One of the best experts on this subject based on the ideXlab platform.

  • On a Class of Infinite-Dimensional Singular Stochastic Control Problems
    2019
    Co-Authors: Federico Salvatore, Ferrari Giorgio, Riedel Frank, Röckner Michael
    Abstract:

    We study a class of Infinite-Dimensional singular stochastic control problems with applications in economic theory and finance. The control process linearly affects an abstract evolution equation on a suitable partially-ordered Infinite-Dimensional Space X, it takes values in the positive cone of X, and it has right-continuous and nondecreasing paths. We first provide a rigorous formulation of the problem by properly defining the controlled dynamics and integrals with respect to the control process. We then exploit the concave structure of our problem and derive necessary and sufficient first-order conditions for optimality. The latter are finally exploited in a specification of the model where we find an explicit expression of the optimal control. The techniques used are those of semigroup theory, vector-valued integration, convex analysis, and general theory of stochastic processes.Comment: 21 page

  • On a Class of Infinite-Dimensional Singular Stochastic Control Problems
    Center for Mathematical Economics, 2019
    Co-Authors: Federico Salvatore, Ferrari Giorgio, Riedel Frank, Röckner Michael
    Abstract:

    Federico S, Ferrari G, Riedel F, Röckner M. On a Class of Infinite-Dimensional Singular Stochastic Control Problems. Center for Mathematical Economics Working Papers. Vol 614. Bielefeld: Center for Mathematical Economics; 2019.We study a class of Infinite-Dimensional singular stochastic control problems with applications in economic theory and finance. The control process linearly affects an abstract evolution equation on a suitable partially-ordered Infinite-Dimensional Space X, it takes values in the positive cone of X, and it has right-continuous and nondecreasing paths. We first provide a rigorous formulation of the problem by properly defining the controlled dynamics and integrals with respect to the control process. We then exploit the concave structure of our problem and derive necessary and sufficient first-order conditions for optimality. The latter are finally exploited in a specification of the model where we find an explicit expression of the optimal control. The techniques used are those of semigroup theory, vector-valued integration, convex analysis, and general theory of stochastic processes

Gianfranco Pierobon - One of the best experts on this subject based on the ideXlab platform.

  • performance of quantum data transmission systems in the presence of thermal noise
    IEEE Transactions on Communications, 2010
    Co-Authors: Gianfranco Cariolaro, Gianfranco Pierobon
    Abstract:

    In the literature the performance of quantum data transmission systems is usually evaluated in the absence of thermal noise. A more realistic approach taking into account the thermal noise is intrinsically more difficult because it requires dealing with Glauber coherent states in an Infinite-Dimensional Space. In particular, the exact evaluation of the optimal measurement operators is a very difficult task, and numerical approximation is unavoidable. The paper faces the problem by approximating the P-representation of the noisy quantum states with a large but finite numbers of terms and applying to them the square root measurement (SRM) approach. Comparisons with cases where the exact solution are known show that the SRM approach gives quite accurate results. As application, the performance of quadrature amplitude modulation (QAM) and phase shift keying (PSK) systems is considered. In spite of the fact that the SRM approach is not optimal and overestimates the error probability, also in these cases the quantum detection maintains its superiority with respect to the classical homodyne detection.

  • performance of quantum data transmission systems in the presence of thermal noise
    arXiv: Quantum Physics, 2009
    Co-Authors: Gianfranco Cariolaro, Gianfranco Pierobon
    Abstract:

    In the literature the performance of quantum data transmission systems is usually evaluated in the absence of thermal noise. A more realistic approach taking into account the thermal noise is intrinsically more difficult because it requires dealing with Glauber coherent states in an infinite--dimensional Space. In particular, the exact evaluation of the optimal measurement operators is a very difficult task, and numerical approximation is unavoidable. The paper faces the problem by approximating the P-representation of the noisy quantum states with a large but finite number of terms and applying to them the square root measurement (SRM) approach. Comparisons with the exact solution obtained with convex semidefinite programming show that the SRM approach gives quite accurate results. As application, the performance of quadrature amplitude modulation (QAM) and phase shift keying (PSK) systems is considered. In spite of the fact that the SRM approach is not optimal and overestimates the error probability, also in these cases the quantum detection maintains its superiority with respect to the classical homodyne detection.