2,231 research outputs found
Inverse Problems in a Bayesian Setting
In a Bayesian setting, inverse problems and uncertainty quantification (UQ)
--- the propagation of uncertainty through a computational (forward) model ---
are strongly connected. In the form of conditional expectation the Bayesian
update becomes computationally attractive. We give a detailed account of this
approach via conditional approximation, various approximations, and the
construction of filters. Together with a functional or spectral approach for
the forward UQ there is no need for time-consuming and slowly convergent Monte
Carlo sampling. The developed sampling-free non-linear Bayesian update in form
of a filter is derived from the variational problem associated with conditional
expectation. This formulation in general calls for further discretisation to
make the computation possible, and we choose a polynomial approximation. After
giving details on the actual computation in the framework of functional or
spectral approximations, we demonstrate the workings of the algorithm on a
number of examples of increasing complexity. At last, we compare the linear and
nonlinear Bayesian update in form of a filter on some examples.Comment: arXiv admin note: substantial text overlap with arXiv:1312.504
Inverse problems and uncertainty quantification
In a Bayesian setting, inverse problems and uncertainty quantification (UQ) -
the propagation of uncertainty through a computational (forward) model - are
strongly connected. In the form of conditional expectation the Bayesian update
becomes computationally attractive. This is especially the case as together
with a functional or spectral approach for the forward UQ there is no need for
time-consuming and slowly convergent Monte Carlo sampling. The developed
sampling-free non-linear Bayesian update is derived from the variational
problem associated with conditional expectation. This formulation in general
calls for further discretisation to make the computation possible, and we
choose a polynomial approximation. After giving details on the actual
computation in the framework of functional or spectral approximations, we
demonstrate the workings of the algorithm on a number of examples of increasing
complexity. At last, we compare the linear and quadratic Bayesian update on the
small but taxing example of the chaotic Lorenz 84 model, where we experiment
with the influence of different observation or measurement operators on the
update.Comment: 25 pages, 17 figures. arXiv admin note: text overlap with
arXiv:1201.404
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