Error Analysis of Approximated PCRLBs for Nonlinear Dynamics. (arXiv:1005.5348v1 [stat.AP])

Error Analysis of Approximated PCRLBs for Nonlinear Dynamics. (arXiv:1005.5348v1 [stat.AP]): "

In practical nonlinear filtering, the assessment of achievable filtering
performance is important. In this paper, we focus on the problem of efficiently
approximate the posterior Cramer-Rao lower bound (CRLB) in a recursive manner.
By using Gaussian assumptions, two types of approximations for calculating the
CRLB are proposed: An exact model using the state estimate as well as a
Taylor-series-expanded model using both of the state estimate and its error
covariance, are derived. Moreover, the difference between the two approximated
CRLBs is also formulated analytically. By employing the particle filter (PF)
and the unscented Kalman filter (UKF) to compute, simulation results reveal
that the approximated CRLB using mean-covariance-based model outperforms that
using the mean-based exact model. It is also shown that the theoretical
difference between the estimated CRLBs can be improved through an improved
filtering method.

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Stochastic Exponential Integrators for a Finite Element Discretization of SPDEs. (arXiv:1005.5315v1 [math.NA])

Stochastic Exponential Integrators for a Finite Element Discretization of SPDEs. (arXiv:1005.5315v1 [math.NA]): "

We consider the numerical approximation of general semilinear parabolic
stochastic partial differential equations (SPDEs) driven by additive space-time
noise. In contrast to the standard time stepping methods which uses basic
increments of the noise and the approximation of the exponential function by a
rational fraction, we introduce a new scheme, designed for finite elements,
finite volumes or finite differences space discretization, similar to the
schemes in \cite{Jentzen3,Jentzen4} for spectral methods and \cite{GTambue} for
finite element methods. We use the projection operator, the smoothing effect of
the positive definite self-adjoint operator and linear functionals of the noise
in Fourier space to obtain higher order approximations. We consider noise that
is white in time and either in $H^1$ or $H^2$ in space and give convergence
proofs in the mean square $L^{2}$ norm for a diffusion reaction equation and in
mean square $ H^{1}$ norm in the presence of an advection term. For the
exponential integrator we rely on computing the exponential of a non-diagonal
matrix. In our numerical results we use two different efficient techniques: the
real fast \Leja points and Krylov subspace techniques. We present results for a
linear reaction diffusion equation in two dimensions as well as a nonlinear
example of two-dimensional stochastic advection diffusion reaction equation
motivated from realistic porous media flow.

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Numerical convergence of a one step approximation of an intrgro-differential equation. (arXiv:1005.5344v1 [math.NA])

Numerical convergence of a one step approximation of an intrgro-differential equation. (arXiv:1005.5344v1 [math.NA]): "

We consider a linear partial integro-differential equation that arises in the
modeling of various physical and biological processes. We study the problem in
a spatial periodic domain. We analyze numerical stability and numerical
convergence of a one step approximation of the problem with smooth and
non-smooth initial functions.

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