Approximation of Stochastic Partial Differential Equations by a Kernel-based Collocation Method

In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations (SPDEs). Using an implicit time stepping scheme, we transform stochastic parabolic equations into stochastic elliptic equations. Our main attention is concentrated on the numerical solution of the elliptic equations at each time step. The estimator of the solution of the elliptic equations is given as a linear combination of reproducing kernels derived from the differential and boundary operators of the SPDE centered at collocation points to be chosen by the user. The random expansion coefficients are computed by solving a random system of linear equations. Numerical experiments demonstrate the feasibility of the method.Comment: Updated Version in International Journal of Computer Mathematics, Closed to Ye's Doctoral Thesi

Well-posedness of the transport equation by stochastic perturbation

We consider the linear transport equation with a globally Holder continuous and bounded vector field. While this deterministic PDE may not be well-posed, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influece of noise. The key tool is a differentiable stochastic flow constructed and analysed by means of a special transformation of the drift of Ito-Tanaka type.Comment: Addition of new part

Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity

The strong existence and the pathwise uniqueness of solutions with (Formula presented.) -vorticity of the 2D stochastic Euler equations are proved. The noise is multiplicative and it involves the first derivatives. A Lagrangian approach is implemented, where a stochastic flow solving a nonlinear flow equation is constructed. The stability under regularizations is also proved

Interacting Banks of Bayesian Matched Filters

There exist a number of powerful methods for detecting small low observable targets with stationary dynamics in image sequences provided by IR and other imaging sensors (see e.g. 12). However, these methods need to be extended to handle maneuvering targets. In this paper, we demonstrate that banks of interacting Bayesian filters (BIBF) can be utilized for this purpose. We are considering target dynamics modeled by jump-linear systems. In contrast to previous studies, we do not assume that the mode jump process is a Markov chain. In particular, we allow the probabilities of jumps to be conditioned on the state variable. Then, we present a computationally efficient (real time) algorithm for detection and tracking of low observable agile targets. A comparison of BIBF and IMM approaches is carried out in a simple example