38 research outputs found
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
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
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