White Noise Dispersion Research Articles

Noise immunity algorithm for embedding a digital watermark in medical images

When storing and transferring printed medical materials, such as tomograms or radiographs, there is a need to protect additional information from unauthorized access. This information includes personal data of the patient and a summary of the medical history, and it can be added to the medical image (container) in the form of a watermark. Existing algorithms for embedding digital watermarks (DWM) in graphic objects distort the initial characteristics of the container image, which in the case of medical images can lead to a misdiagnosis. This study aimed to develop a distortion- and noise-resistant algorithm for embedding the DWM in the spatial domain of a medical image intended for storage on paper (as a printout). The article initially considered the possibilities of using the method of modifying the least significant bits of image pixel brightness or the LSB algorithm. Mathematical modeling in Matlab showed that the maximum brightness of the DWM guaranteeing its invisibility cannot exceed 2% of the maximum brightness of the container image for monochrome images, and 6% for color images. The maximum value of the white noise dispersion, at which it is possible to single out a DWM with a correlation coefficient of at least 0.9 was 0.0001. We prorose a new noise immunе algorithm (NIA) for embedding the DWM in the subpixels of the main image, which, after extracting the DWM, are not used to build the container image. In the absence of noise, there are no distortions of the original medical image whatsoever. The essence of the NIA is as follows. The size of the original image in the form of a two-dimensional array is quadrupled by adding a subpixel in each row and column with a brightness equal to the average arithmetic brightness of neighboring pixels. A DWM with the same size as the original image is added to the resulting subpixels. Matlab modeling showed that the DWM would remain invisible at a relative brightness of approximately 5% for monochrome container images and 15% for color images. For the NIA algorithm, the maximum value of the white noise dispersion to obtain a correlation coefficient of 0.9 is 0.005, which means that the noise immunity of the proposed method is significantly higher than that of the LSB-based algorithm.

On the wellposedness of periodic nonlinear Schrödinger equations with white noise dispersion

On the wellposedness of periodic nonlinear Schrödinger equations with white noise dispersion

Strong Rates of Convergence of a Splitting Scheme for Schrödinger Equations with Nonlocal Interaction Cubic Nonlinearity and White Noise Dispersion

Strong Rates of Convergence of a Splitting Scheme for Schrödinger Equations with Nonlocal Interaction Cubic Nonlinearity and White Noise Dispersion

The nonlinear Schrödinger equation with white noise dispersion on quantum graphs

The nonlinear Schrödinger equation with white noise dispersion on quantum graphs

Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion

We study a Lie splitting scheme for a nonlinear Schrodinger-type equation with random dispersion. The main result is an approximation of the local error. Then we can deduce sharp order estimates, for instance in the case of a white noise dispersion.

Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion

<p style='text-indent:20px;'>In this article, the asymptotic behavior of the solution to the following one dimensional Schrödinger equations with white noise dispersion <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ idu + u_{xx}\circ dW+ |u|^{p-1}udt = 0 $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>is studied. Here the equation is written in the { Stratonovich} formulation, and <inline-formula><tex-math id="M1">\begin{document}$ W(t) $\end{document}</tex-math></inline-formula> is a standard real valued Brownian motion. After establishing the global well-posedness, theoretical proof and numerical investigations are provided showing that, for a deterministic small enough initial data in <inline-formula><tex-math id="M2">\begin{document}$ L^1_x\cap H^1_x $\end{document}</tex-math></inline-formula>, the expectation of the <inline-formula><tex-math id="M3">\begin{document}$ L^\infty_x $\end{document}</tex-math></inline-formula> norm of the solutions decay to zero at <inline-formula><tex-math id="M4">\begin{document}$ O(t^{-\frac14}) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M5">\begin{document}$ t $\end{document}</tex-math></inline-formula> goes to <inline-formula><tex-math id="M6">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula>, as soon as <inline-formula><tex-math id="M7">\begin{document}$ p&gt;7 $\end{document}</tex-math></inline-formula>.

Multirevolution Integrators for Differential Equations with Fast Stochastic Oscillations

We introduce a new methodology based on the multirevolution idea for constructing integrators for stochastic differential equations in the situation where the fast oscillations themselves are driven by a Stratonovich noise. Applications include in particular highly-oscillatory Kubo oscillators and spatial discretizations of the nonlinear Schr\"odinger equation with fast white noise dispersion. We construct a method of weak order two with computational cost and accuracy both independent of the stiffness of the oscillations. A geometric modification that conserves exactly quadratic invariants is also presented.

Numerical Analysis of the Midpoint Scheme for the Generalized Benjamin-Bona-Mahony Equation with White Noise Dispersion

Numerical Analysis of the Midpoint Scheme for the Generalized Benjamin-Bona-Mahony Equation with White Noise Dispersion

Exponential integrators for nonlinear Schr\xf6dinger equations with white noise dispersion

This article deals with the numerical integration in time of the nonlinear Schrödinger equation with power law nonlinearity and random dispersion. We introduce a new explicit exponential integrator for this purpose that integrates the noisy part of the equation exactly. We prove that this scheme is of mean-square order 1 and we draw consequences of this fact. We compare our exponential integrator with several other numerical methods from the literature. We finally propose a second exponential integrator, which is implicit and symmetric and, in contrast to the first one, preserves the L^2-norm of the solution.

Stochastic symplectic and multi-symplectic methods for nonlinear Schrödinger equation with white noise dispersion

We indicate that the nonlinear Schrödinger equation with white noise dispersion possesses stochastic symplectic and multi-symplectic structures. Based on these structures, we propose the stochastic symplectic and multi-symplectic methods, which preserve the continuous and discrete charge conservation laws, respectively. Moreover, we show that the proposed methods are convergent with temporal order one in probability. Numerical experiments are presented to verify our theoretical results.

Generalized regularized long wave equation with white noise dispersion

In this article, we address the generalized BBM equation with white noise dispersion which reads $$\begin{aligned} du-du_{xx}+u_x \circ dW+ u^pu_xdt=0, \end{aligned}$$ in the Stratonovich formulation, where W(t) is a standard real valued Brownian motion. We first investigate the well-posedness of the initial value problem for this equation. We then prove theoretically and numerically that for a deterministic initial data, the expectation of the \(L^\infty _x\) norm of the solutions decays to zero at \(O(t^{-\frac{1}{6}})\) as t approaches to \(+\infty \), by assuming that \(p>8\) and that the initial data is small in \(L^1_x\cap H^4_x\). This decay rate matches the one for solutions of the linear equation with white noise dispersion.

A Random Schrödinger Equation with Time-Oscillating Nonlinearity and Linear Dissipation/Gain

This paper is devoted to a random dispersion Schrodinger equation including time-oscillating nonlinearity and dissipation/gain: \(i\mathrm{d}u + \frac{1}{\varepsilon } m\left( \frac{t}{\varepsilon ^{2}}\right) \partial _{xx}u\mathrm{d}t + \phi _{1}\left( \frac{t}{\varepsilon ^{2}}\right) |u|^{2\sigma }u \mathrm{d}t + i\phi _{2}\left( \frac{t}{\varepsilon ^{2}}\right) u\mathrm{d}t = 0\). The main aim is to prove the equation converges to a white noise dispersion Schrodinger equation with the average of time-oscillating nonlinearity and dissipation/gain. Firstly, the existence of the local solution for the limit equation is established for \(0 0\) if \(d = 1\) or 2). Then, the convergence is proved in one dimension for \(\sigma >0\).

Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion

This article is devoted to the numerical study of a nonlinear Schrodinger equation in which the coefficient in front of the group velocity dispersion is multiplied by a real valued Gaussian white noise. We first perform the numerical analysis of a semi-discrete Crank-Nicolson scheme in the case when the continuous equation possesses a unique global solution. We prove that the strong order of convergence in probability is equal to one in this case. In a second step, we numerically investigate, in space dimension one, the behavior of the solutions of the equation for different power nonlinearities, corresponding to subcritical, critical or supercritical nonlinearities in the deterministic case. Numerical evidence of a change in the critical power due to the presence of the noise is pointed out.

1D quintic nonlinear Schrödinger equation with white noise dispersion

In this article, we improve the Strichartz estimates obtained in A. de Bouard, A. Debussche (2010) [12] for the Schrödinger equation with white noise dispersion in one dimension. This allows us to prove global well posedness when a quintic critical nonlinearity is added to the equation. We finally show that the white noise dispersion is the limit of smooth random dispersion.

The nonlinear Schrödinger equation with white noise dispersion

Under certain scaling the nonlinear Schrödinger equation with random dispersion converges to the nonlinear Schrödinger equation with white noise dispersion. The aim of this work is to prove that this latter equation is globally well posed in L 2 or H 1 . The main ingredient is the generalization of the classical Strichartz estimates. Additionally, we justify rigorously the formal limit described above.