On the optimal filtering of diffusion processes

Summary Let x(t) be a diffusion process satisfying a stochastic differential equation and let the observed process y(t) be related to x(t) by dy(t) = g(x(t)) + dw(t) where w(t) is a Brownian motion. The problem considered is that of finding the conditional probability of x(t) conditioned on the observed path y(s), 0st. Results on the Radon-Nikodym derivative of measures induced by diffusions processes are applied to derive equations which determine the required conditional probabilities.     

Propagation property for anisotropic nonlinear diffusion equation with convection

We consider propagation property for anisotropic diffusion equation with convection in 2 dimension,

t_∂(um)−x_1∂(|x_1∂u|^p₁−1x_1∂u)−x_2∂(|x_2∂u|^p₂−1x_2∂u)+uα−1x_1∂u=0,     

Inference for an anisotropic diffusion model

The vector sum of a white noise in an unknown hyperspace and an Ornstein-Uhlenbeck process in an unknown line is observed through sharp linear test functions over a finite time span. The parameters associated with the white noise (including the hyperplane) are determinable with precision and index the measure-equivalence classes in the relevant sample space. An intraclass relative density provides a basis for Bayesian inference of the remaining parameters.     

Strongly anisotropic diffusion problems; asymptotic analysis

The subject matter of this paper concerns anisotropic diffusion equations: we consider heat equations whose diffusion matrices have disparate eigenvalues. We determine first and second order approximations, we study the well-posedness of them and establish convergence results. The analysis relies on averaging techniques, which have been used previously for studying transport equations whose advection fields have disparate components.     

Continuous level anisotropic diffusion for noise removal

This paper presents a new approach to anisotropic diffusion and noise removal. Several functionals are introduced to a variational model. The diffusion behavior is governed by a nonlinear partial differential equation. A dynamic threshold function plays an important role in the continuous level anisotropic diffusion and a related optimization problem is presented. The noise can be removed while the edge well preserved. Multi-level noise or multi-level edge can be handled automatically. Finally, the accuracy and efficiency of the proposed method are verified by several numerical experiments.     

Complex diffusion filtering of three dimensional turbulent flows

The linear and nonlinear complex diffusion filtering methods are proposed to extract the organized coherent part as well as the random incoherent part from forced and decaying turbulent flows. An attempt to examine the robustness of the two methods in filtering the turbulent flow field without the transformation into the frequency domain is carried out. The velocity fields of the forced and decaying cases are decomposed into coherent and incoherent parts in the spatial domain. The complex diffusion process can be obtained by combining the linear diffusion equation and the free particle Schrodinger equation. The imaginary parts in the two methods serve as a robust edge-detector with increasing confidence in time. The numerical implementations of the 3D linear and nonlinear complex diffusion partial differential equations are done using the finite difference method. The flatness, skewness and spectrum of the extracted fields are also studied for each filtering method. Results show that the two filtering methods can identify the coherent fields and preserve the features of the turbulent fields. Comparisons to the wavelet and Fourier decompositions are also considered.     

A solution to anisotropic suboptimal filtering problem by convex optimization

This paper considers a robust filtering problem for a linear discrete time invariant system with measured and estimated outputs. The system is exposed to random disturbances with imprecisely known distributions generated by an unknown stable shaping filter from the Gaussian white noise. The stochastic uncertainty of the input disturbance is measured by the mean anisotropy functional. The estimation error is quantified by the anisotropic norm which is a stochastic analogue of the H _∞ norm. A sufficient condition for an estimator to exist and ensure that the error is less than a given threshold value is derived in form of a convex inequality on the determinant of a positive definite matrix and two linear matrix inequalities. The suboptimal problem setting results to a set of the estimators ensuring the anisotropic norm of the error to be strictly bounded thereby providing some additional degree of freedom to impose some additional constraints on the estimator performance specification.     

A finite volume scheme for anisotropic diffusion problems

A new finite volume for the discretization of anisotropic diffusion problems on general unstructured meshes in any space dimension is presented. The convergence of the approximate solution and its discrete gradient is proven. The efficiency of the scheme is illustrated by numerical results. To cite this article: R. Eymard et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Analysis of a Large Eddy Simulation model based on anisotropic filtering

We consider a new Large Eddy Simulation model, derived with the introduction of suitable horizontal (anisotropic) differential filters. One main advantage of this filtering is that, for channel flows, there is no need for artificial boundary conditions. Hence, we can deal with some realistic problems, equipped with Dirichlet boundary conditions, in special bounded domains (at least those bounded only in one direction). Recent numerical results for a similar model, based on a derivation with wave-number asymptotics, are also recalled. After a detailed analysis of the properties of the differential filter, we prove that the resulting initial–boundary value problem is well-posed in suitable anisotropic Sobolev spaces, giving a strong mathematical support to the model we propose. Some remarks on higher-accuracy Approximate Deconvolution Models are also given in the last section.     

A Carleman estimate of some anisotropic space-fractional diffusion equations

This paper is concerned with Carleman estimates for some anisotropic space-fractional diffusion equations, which are important tools for investigating the corresponding control and inverse problems. By employing a special weight function and the nonlocal vector calculus, we prove a Carleman estimate and apply it to build a stability result for a backward diffusion problem.     

Stochastic partial differential equations and filtering of diffusion processes

We establish basic results on existence and uniqueness for the solution of stochastic PDE's. We express the solution of a backward linear stochastic PDE in terms of the conditional law of a partially observed Markov diffusion process. It then follows that the adjoint forward stochastic PDE governs the evolution of the “unnormalized conditional density”     

Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion

We establish global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with anisotropic viscosity acting only in the horizontal direction, which arises in ocean dynamics models. Global well-posedness for this system was proven by Danchin and Paicu; however, an additional smoothness assumption on the initial density was needed to prove uniqueness. They stated that it is not clear whether uniqueness holds without this additional assumption. The present work resolves this question and we establish uniqueness without this additional assumption. Furthermore, the proof provided here is more elementary; we use only tools available in the standard theory of Sobolev spaces, and without resorting to para-product calculus. We use a new approach by defining an auxiliary “stream-function” associated with the density, analogous to the stream-function associated with the vorticity in 2D incompressible Euler equations, then we adapt some of the ideas of Yudovich for proving uniqueness for 2D Euler equations.     

Nonlinear source in diffusion filtering methods for image processing

A diffusion filtering algorithm is proposed based on the solution to an initial-boundary value problem for the two-dimensional diffusion equation with a special nonlinear source.     

Image edge detection with the use of the Tikhonov regularization method

The study considers application of the Tikhonov regularization method for smoothing of 1D signals, noise suppression, and determination of objects’ edges in images. An analytic solution to the Euler equation is obtained in the 1D case for certain stabilizers. Application of the considered method for determination of objects’ edges in noisy images is analyzed.     

A finite element method for the numerical solution of convection-dominated anisotropic diffusion equations

Summary. The proposed method is based on an additive decomposition of the differential operator and the subsequent fitted discretization of the resulting components. For standard situations, the derived stability and error estimates in the energy norm qualitatively coincide with well-known estimates. In the case of small diffusion, a uniform error estimate with reduced order is obtained. Received August 7, 1997 / Revised version received July 15, 1998 / Published online December 6, 1999     

Markov chain approximations to filtering equations for reflecting diffusion processes

Herein, we consider direct Markov chain approximations to the Duncan–Mortensen–Zakai equations for nonlinear filtering problems on regular, bounded domains. For clarity of presentation, we restrict our attention to reflecting diffusion signals with symmetrizable generators. Our Markov chains are constructed by employing a wide band observation noise approximation, dividing the signal state space into cells, and utilizing an empirical measure process estimation. The upshot of our approximation is an efficient, effective algorithm for implementing such filtering problems. We prove that our approximations converge to the desired conditional distribution of the signal given the observation. Moreover, we use simulations to compare computational efficiency of this new method to the previously developed branching particle filter and interacting particle filter methods. This Markov chain method is demonstrated to outperform the two-particle filter methods on our simulated test problem, which is motivated by the fish farming industry.