## Journals

- Advances in Mathematics of Communications
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IPI

In this paper, we consider the reconstruction of time-varying concentration distributions under
nonstationary flow conditions.
Previous studies have shown that the state estimation approach that is based on stochastic process
evolution models, facilitates reconstructions of rapidly time-varying targets.
However, only cases with stationary velocity fields, or cases in which the velocity field can be
completely specified by a velocity profile, have been studied.
While simultaneous estimation of the time-varying concentration and low-dimensional representations
of the flow field itself has been shown to be possible to some extent, this would be computationally
too heavy for on-line process estimation and control.
On the other hand, using an incorrect flow model in the evolution model may induce intolerable
estimation errors.
In this paper, we consider an approach in which the state evolution model is written to correspond
to a stationary flow, while the actual flow is nonstationary.
The associated modelling errors are handled by constructing the state noise process to accommodate
to this discrepancy.
We carry out a numerical feasibility study with different Reynolds numbers and show that the
approach yields significant reduction of estimation errors and simultaneously facilitates using
computationally efficient reduced order models.

IPI

In fluorescence diffuse optical tomography (fDOT), the reconstruction of the fluorophore
concentration inside the target body is usually carried out using a normalized Born approximation model where
the measured fluorescent emission data is scaled by measured excitation data. One of the benefits of the model is that it can tolerate inaccuracy in the absorption and scattering distributions that are used in the construction of the forward model to some extent. In this paper, we employ the recently proposed Bayesian approximation error approach to fDOT for compensating for the
modeling errors caused by the inaccurately known optical properties of the target in combination with the normalized Born approximation model. The approach is evaluated using a simulated test case with
different amount of error in the optical properties. The results show that the Bayesian approximation error approach improves
the tolerance of fDOT imaging against modeling errors caused by inaccurately known absorption and scattering of the target.

IPI

We consider a source identification problem related to determination of
contaminant source parameters in lake environments using remote sensing
measurements.
We carry out a numerical example case
study in which we employ the statistical inversion approach for
the determination of the source parameters. In the simulation study
a pipeline breaks on the bottom of a lake and only low-resolution
remote sensing measurements are available.
We also describe how model uncertainties and especially errors that are
related to model reduction are taken into account in the overall
statistical model of the system.
The results indicate that the estimates may be
heavily misleading if the statistics of the model errors are not
taken into account.

IPI

Quantitative photoacoustic tomography is a hybrid imaging method,
combining near-infrared optical and ultrasonic imaging. One of the
interests of the method is the reconstruction of the optical
absorption coefficient within the target. The measurement depends
also on the uninteresting but often unknown optical scattering
coefficient. In this work, we apply the approximation error method
for handling uncertainty related to the unknown scattering and
reconstruct the absorption only. This way the number of unknown
parameters can be reduced in the inverse problem in comparison to
the case of estimating all the unknown parameters. The
approximation error approach is evaluated with data simulated using
the diffusion approximation and Monte Carlo method. Estimates are
inspected in four two-dimensional cases with biologically relevant
parameter values. Estimates obtained with the approximation error
approach are compared to estimates where both the absorption and
scattering coefficient are reconstructed, as well to estimates where
the absorption is reconstructed, but the scattering is assumed
(incorrect) fixed value. The approximation error approach is found
to give better estimates for absorption in comparison to estimates
with the conventional measurement error model using fixed
scattering. When the true scattering contains stronger variations,
improvement of the approximation error method over fixed scattering
assumption is more significant.

IPI

Electrical impedance tomography (EIT) is a non-invasive imaging
modality in which the internal conductivity distribution
is reconstructed
based on boundary voltage measurements.
In this work, we consider the
application of EIT to non-destructive testing (NDT) of materials and,
especially, crack detection.
The main goal is to estimate the location, depth
and orientation of a crack in three dimensions.
We formulate the crack detection task as a shape estimation problem for
boundaries imposed with Neumann zero boundary conditions.
We propose an adaptive meshing algorithm that iteratively
seeks the

*maximum a posteriori*estimate for the shape of the crack. The approach is tested both numerically and experimentally. In all test cases, the EIT measurements are collected using a set of electrodes attached on only a single planar surface of the target -- this is often the only realizable configuration in NDT of large building structures, such as concrete walls. The results show that with the proposed computational method, it is possible to recover the position and size of the crack, even in cases where the background conductivity is inhomogeneous.## Year of publication

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