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Mathematical analysis of Liquid Cristal with stochastic perturbations

FWF

The effect of random external perturbation on the dynamics of the nematic liquid crystals has been the subject of numerous theoretical and experimental studies in Chemistry, Engineering and Physics. Many prominent scientists have shown, for instance, that in the presence of noise the liquid crystal will leave an unstable state quite fast and goes to a stable one. However, these previous works just looked at the alignment of the molecules and have neglected the effect of the hydrodynamic flow. However, it is pointed out by de Gennes and J.~Prost in the book “The Physics of Liquid Crystals”, published by Clarendon Press, Oxford in 1993, that the fluid flow disturbs the alignment and conversely a change in the alignment will induce a flow in the nematic liquid crystal. Hence for a full understanding of the effect of fluctuating external fields on the behavior of the liquid crystals one needs to take into account the dynamics of director field and the velocity of the fluid. Motivated by these facts, in this project we intend to initiate and perform mathematical analysis on stochastic equations arisen from the hydrodynamics of incompressible nematic liquid crystals under the influence of random external perturbations. Our first aim is to study basic questions such as the existence and uniqueness of solution of the above system and related problems. The second aim will be the investigation of the long-time behavior of the above system. Here we are mainly interested in the existence of stationary solution or invariant measure. We will also study the uniqueness of the invariant measure. The last but not least aim is to analyze and implement some numerical approximations of our stochastic models.

The numerical Approximation of the pressure in the stochastic 2d Navier Stokes Equation

FWF

Stochastic Partial Differential Equations have become one of the most popular tools in understanding and investigating mathematically the hydrodynamic turbulence. To model turbulent fluids, mathematicians often use the stochastic Navier-Stokes equation obtained from adding a noise term in the dynamical equations of the fluids. Since the pioneering work of Bensoussan and Temam on stochastic Navier-Stokes equations with Wiener noise and related problems has been the object of intense analytical investigations which have generated several important results. However, the numerical study of these equations is really at its infancy and most of the results so far are about numerical approximation of the velocity field only. In this project we address the approximation of the pressure term in the stochastic Navier-Stokes equations. For this purpose we mainly use the projection method. The Projection method, initiated by Chorin and Temam, consists in splitting the original equation in two sub-equations which have lower complexity and are easier to solve numerically. At a first step we will first study numerically the pressure term for the linear stochastic Stokes equations with divergence free noise. For the second step of our investigation we want to extend our result obtained in the first step to extend to the fully stochastic Navier-Stokes equations by adding the nonlinearity. With the help of our results obtained in our first goal, our objective is to verify the convergence rate for the pressure term. However, due to the incompressibility constraint, the non-Lipschitz nonlinearity and the stochastic forcing we expect that we will encounter some problems. Finally, if time permits, one can investigate other methods, and verify if there is a possibility of improving the order of rate of convergence. Also, the question of non–divergence free noise will be addressed.

Numerical Analysis of nonlinear filtering with Levy noise

FWF

In engineering one of the most important areas of application of nonlinear filtering is positioning, navigation and tracking problems. In positioning, one is often interested in estimating its own position, while one is moving around, provided by data from some sensors. To get an idea, imagine a ship on its tour through the ocean having some problems with the GPS signals. First, the signals are corrupted by noise, and secondly, there are some gaps where the ship does not receive any signal. The ship on the way to its harbor is traveling in a certain direction. However, due to the waves the ship does not follow exactly its direction and deviates slightly. Now the problem is to estimate the ship's position using the historical data of the journey given by the GPS signals. In navigation, beside the position also velocity, acceleration, attitude and heading are included in the filtering problem. Here, the task is to calculate automatically the route, or, to say it better, its direction and its speed in order to navigate the ship to its destination. In target tracking, the position of another object is to be estimated based on measurement of relative range and angles to one's own position. The target can be again a ship, or a hostile drone in the air. Now, using cheap low quality sensors, the signal may be noisy. Here, the aim in nonlinear filtering is estimating the position of the target from the incomplete and noisy measurements. The aim in stochastic filtering is to reconstruct information about an unobserved (random) process X, called the signal process, given the current available observations of a certain noisy transformation of that process. However, in many applications the noise is often rough, or lives at a time scale which is comparable to the time scale of X and Y quite short. Although, modeling an earthquake one is faced by a noise which has jumps. Or, modeling the change of climate, the time scale of the noise is much faster than the time scale of the dynamic describing the climate. Thereby, the noise appears to have jumps. In finance, high frequency data are modeled successfully by Levy processes. In other words, there are many examples, where the Gaussian noise is replaced by a Levy noise in order to improve the model or to cover features which are not covered by the Gaussian noise. There exist several results where the problem is analyzed, if the observation or signal process is perturbed by Levy noise. Nevertheless, in practice, theoretical results are often not sufficient. Applying the theory means calculating the density on a computer, and this means, performing numerical calculations or simulations. Here, it is necessary to provide strategies, proof of consistency and stability, and error estimates. The aim of the project is to provide some numerical schemes for nonlinear filtering in case the stochastic perturbation comes from a Levy noise and to prove their convergence.

Nonlinear Filtering with Levy noise
Razafimandimby Paul Andre; Fernando Bandhisattambige
FWF
2012-2015

The non linear filtering theory is a cross domain between Stochastic Analysis and Statistics. The aim in nonlinear filtering is to reconstruct information about an unobserved random process X, called the signal process, via the current available observations of a certain noisy process Y, called the observable process. The signal process is well reconstructed if the optimal filter ” p “or the unnormalized filter “V” is calculated. By the unnormalized filter, we mean the conditional expectation of X, known the history of Y. The unnormalized filter is equivalent to the optimal filter via the Kllianpur-Striebel formula. Furthermore, the unnormalized filter V satisfies a stochastic partial differential equation called the Wong-Zakai equation. The main task in filtering theory is to study the well posedness and the properties of the unnormalized filter V as well as to calculate it numerically. In the classical theory, the signal and the observable processes satisfy stochastic differential equations perturbed by Gaussian noises. The aim of this project is to extend the filtering theory, known for the Wiener process to the framework of the Lévy process. A topic which is treated partially for special cases such as for the degenerate case and for the Poisson random noise, but still far to reach the whole general theory. Here, the key tools to get the unnormalized filter will be completely different. In fact, the leading operator in the Wong-Zakai equation will be a pseudo differential operator instead of the second order operator known for the Gaussian case and the equation will be perturbed by a general Lévy noise.

Simulation of a Lévy Random Walk
Simonov Illia
FWF
2011-2014

coming soon …

The Numeric of SPDEs with unbounded Nonlinearities
Giri Ankik
FWF
2009-2013

coming soon …

Ergodische Eigenschaften Systeme mit Levy Rauschen
Hausenblas Erika;
P 20705
2008-2011

coming soon …

Numerik Stochastischer Partieller Differentialgleichungen
Hausenblas, Erika;
P 17273
2004-2008

coming soon ….

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projects/start.txt · Last modified: 2016/08/12 08:12 by admin

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