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Two particular cases of the overdetermined gravimetry-gradiometry problem are discussed: (i) the case of a latitude-dependant statistical weight for gradiometric data, corresponding to a data distribution coming from satellite polar orbits, (ii) the case of a volume distribution, instead of a surface distribution, for satellite gradiometric data. In both cases a discussion of numerical methods for solving the problem with realistic data is started; for case (i), an analytic solution is found under...

Some non-linear s. p. d. e's that are second order in time.

Dalang, Robert C., Mueller, Carl (2003)

Electronic Journal of Probability [electronic only]

Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs

Abdellah Chkifa, Albert Cohen, Ronald DeVore, Christoph Schwab (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number of involved parameter is large. This paper considers a model class of second order, linear, parametric, elliptic PDEs on a bounded domain D with diffusion coefficients depending on the parameters in an affine manner. For such models, it was shown in [9, 10] that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations...

Spatial smoothness of the stationary solutions of the 3D Navier-Stokes equations.

Odasso, Cyril (2006)

Electronic Journal of Probability [electronic only]

SPDEs with coloured noise: Analytic and stochastic approaches

Marco Ferrante, Marta Sanz-Solé (2006)

ESAIM: Probability and Statistics

We study strictly parabolic stochastic partial differential equations on $ℝ^{d}$ , d ≥ 1, driven by a Gaussian noise white in time and coloured in space. Assuming that the coefficients of the differential operator are random, we give sufficient conditions on the correlation of the noise ensuring Hölder continuity for the trajectories of the solution of the equation. For self-adjoint operators with deterministic coefficients, the mild and weak formulation of the equation are related, deriving...

Spectral statistics for random Schrödinger operators in the localized regime

François Germinet, Frédéric Klopp (2014)

Journal of the European Mathematical Society

We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy $E$ in the localized phase. Assume the density of states function is not too flat near $E$ . Restrict it to some large cube $Λ$ . Consider now $I_{Λ}$ , a small energy interval centered at $E$ that asymptotically contains infintely many eigenvalues when the volume of the cube $Λ$ grows to infinity. We prove that, with probability one in the large volume...

Stability of semilinear equations with boundary and pointwise noise

Bohdan Maslowski (1995)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Statistical Solutions of PDEs by Nonstandard Densities.

M. Capinski, J. Cutland (1991)

Monatshefte für Mathematik

Statistically stationary solutions to the 3-D Navier-Stokes equation do not show singularities.

Flandoli, Franco, Romito, Marco (2001)

Electronic Journal of Probability [electronic only]

Stochastic averaging lemmas for kinetic equations

Pierre-Louis Lions, Benoît Perthame, Panagiotis E. Souganidis (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

We develop a class of averaging lemmas for stochastic kinetic equations. The velocity is multiplied by a white noise which produces a remarkable change in time scale.Compared to the deterministic case and as far as we work in $L^{2}$ , the nature of regularity on averages is not changed in this stochastic kinetic equation and stays in the range of fractional Sobolev spaces at the price of an additional expectation. However all the exponents are changed; either time decay rates are slower (when the right...

Stochastic differential equations in Hilbert spaces

Anna Chojnowska-Michalik (1979)

Banach Center Publications

Stochastic evolution equations driven by Liouville fractional Brownian motion

Zdzisław Brzeźniak, Jan van Neerven, Donna Salopek (2012)

Czechoslovak Mathematical Journal

Let $H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of $ℒ (H, E)$ -valued functions with respect to $H$ -cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter $0 < β < 1$ . For $0 < β < \frac{1}{2}$ we show that a function $Φ : (0, T) \to ℒ (H, E)$ is stochastically integrable with respect to an $H$ -cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an $H$ -cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations...

Stochastic Inverse Problem with Noisy Simulator. Application to aeronautical model

Nabil Rachdi, Jean-Claude Fort, Thierry Klein (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

Inverse problem is a current practice in engineering where the goal is to identify parameters from observed data through numerical models. These numerical models, also called Simulators, are built to represent the phenomenon making possible the inference. However, such representation can include some part of variability or commonly called uncertainty (see [4]), arising from some variables of the model. The phenomenon we study is the fuel mass needed to link two given countries with a commercial...

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