## Journals

- Advances in Mathematics of Communications
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- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
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- Mathematical Foundations of Computing
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DCDS

The problem of effectively combining data with a
mathematical model constitutes a major challenge
in applied mathematics. It is particular
challenging for high-dimensional dynamical systems
where data is received sequentially in time and
the objective is to estimate the system state
in an on-line fashion; this situation arises, for
example, in weather forecasting.
The sequential particle filter
is then impractical and

*ad hoc*filters, which employ some form of Gaussian approximation, are widely used. Prototypical of these*ad hoc*filters is the 3DVAR method. The goal of this paper is to analyze the 3DVAR method, using the Lorenz '63 model to exemplify the key ideas. The situation where the data is partial and noisy is studied, and both discrete time and continuous time data streams are considered. The theory demonstrates how the widely used technique of variance inflation acts to stabilize the filter, and hence leads to asymptotic accuracy.
IPI

We consider the inverse problem of estimating a function $u$
from noisy, possibly nonlinear, observations.
We adopt a Bayesian approach to the problem. This
approach has a long history for inversion, dating back to 1970,
and has, over the last decade, gained importance as a practical tool.
However most of the existing theory has been
developed for Gaussian prior measures. Recently Lassas,
Saksman and Siltanen (Inv. Prob. Imag. 2009)
showed how to construct Besov prior measures,
based on wavelet expansions with random coefficients, and used these
prior measures to study linear inverse problems. In this
paper we build on this development of Besov priors to include
the case of nonlinear measurements. In doing so a key technical tool,
established here, is a Fernique-like theorem for
Besov measures. This theorem enables us to
identify appropriate conditions on the forward solution
operator which, when matched to properties of the prior
Besov measure, imply the well-definedness and well-posedness
of the posterior measure. We then consider the application of these
results to the inverse problem of finding the diffusion coefficient
of an elliptic partial differential equation, given noisy
measurements of its solution.

IPI

We provide a rigorous Bayesian formulation of the EIT problem in an infinite dimensional setting, leading to well-posedness in the Hellinger metric with respect to the data. We focus particularly on the reconstruction of binary fields where the interface between different media is the primary unknown. We consider three different prior models -- log-Gaussian, star-shaped and level set. Numerical simulations based on the implementation of MCMC are performed, illustrating the advantages and disadvantages of each type of prior in the reconstruction, in the
case where the true conductivity is a binary field, and exhibiting the properties of the resulting posterior distribution.

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