2,884 research outputs found
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
Parameter Identification in a Probabilistic Setting
Parameter identification problems are formulated in a probabilistic language,
where the randomness reflects the uncertainty about the knowledge of the true
values. This setting allows conceptually easily to incorporate new information,
e.g. through a measurement, by connecting it to Bayes's theorem. The unknown
quantity is modelled as a (may be high-dimensional) random variable. Such a
description has two constituents, the measurable function and the measure. One
group of methods is identified as updating the measure, the other group changes
the measurable function. We connect both groups with the relatively recent
methods of functional approximation of stochastic problems, and introduce
especially in combination with the second group of methods a new procedure
which does not need any sampling, hence works completely deterministically. It
also seems to be the fastest and more reliable when compared with other
methods. We show by example that it also works for highly nonlinear non-smooth
problems with non-Gaussian measures.Comment: 29 pages, 16 figure
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
Towards Efficient Maximum Likelihood Estimation of LPV-SS Models
How to efficiently identify multiple-input multiple-output (MIMO) linear
parameter-varying (LPV) discrete-time state-space (SS) models with affine
dependence on the scheduling variable still remains an open question, as
identification methods proposed in the literature suffer heavily from the curse
of dimensionality and/or depend on over-restrictive approximations of the
measured signal behaviors. However, obtaining an SS model of the targeted
system is crucial for many LPV control synthesis methods, as these synthesis
tools are almost exclusively formulated for the aforementioned representation
of the system dynamics. Therefore, in this paper, we tackle the problem by
combining state-of-the-art LPV input-output (IO) identification methods with an
LPV-IO to LPV-SS realization scheme and a maximum likelihood refinement step.
The resulting modular LPV-SS identification approach achieves statical
efficiency with a relatively low computational load. The method contains the
following three steps: 1) estimation of the Markov coefficient sequence of the
underlying system using correlation analysis or Bayesian impulse response
estimation, then 2) LPV-SS realization of the estimated coefficients by using a
basis reduced Ho-Kalman method, and 3) refinement of the LPV-SS model estimate
from a maximum-likelihood point of view by a gradient-based or an
expectation-maximization optimization methodology. The effectiveness of the
full identification scheme is demonstrated by a Monte Carlo study where our
proposed method is compared to existing schemes for identifying a MIMO LPV
system
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