1,099 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
Polarization Phenomena by Deuteron Fragmentation into Pions
The fragmentation of deuterons into pions emitted forward in the kinematic
region forbidden for free nucleon-nucleon collisions is analyzed. The inclusive
relativistic invariant spectrum of pions and the tensor analyzing power T_{20}
are investigated within the framework of an impulse approximation using
different kinds of the deuteron wave function. The influence of P-wave
inclusion in the deuteron wave function is studied, too. The invariant spectrum
is shown to be more sensitive to the amplitude of the process
than the tensor analyzing power T_{20}. It is shown that the inclusion of the
non-nucleon degrees of freedom in a deuteron results a satisfactory description
of experimental data about the inclusive pion spectrum and improves the
description of data about T_{20}. According to the experimental data, T_{20}
has the positive sign and very small values, less than 0.2, what contradicts to
the theoretical calculations ignoring these degrees of freedom.Comment: 18 pages, 8 eps figures, 1 picture - svjour.cls required; enlarged
new version with corrections and additional figure. The Abstract and the
section "Summary and outlook" have been also corrected. Final version to
appear in Eur.Phys.J. A. A talk given at the International Workshop
"Symmetries and Spin" (July 17-22, Prague, Czech Republic
Parameter Estimation via Conditional Expectation --- A Bayesian Inversion
When a mathematical or computational model is used to analyse some system, it
is usual that some parameters resp.\ functions or fields in the model are not
known, and hence uncertain. These parametric quantities are then identified by
actual observations of the response of the real system. In a probabilistic
setting, Bayes's theory is the proper mathematical background for this
identification process. The possibility of being able to compute a conditional
expectation turns out to be crucial for this purpose. We show how this
theoretical background can be used in an actual numerical procedure, and
shortly discuss various numerical approximations
Polynomial Chaos Expansion of random coefficients and the solution of stochastic partial differential equations in the Tensor Train format
We apply the Tensor Train (TT) decomposition to construct the tensor product
Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic
elliptic diffusion PDE with the stochastic Galerkin discretization, and to
compute some quantities of interest (mean, variance, exceedance probabilities).
We assume that the random diffusion coefficient is given as a smooth
transformation of a Gaussian random field. In this case, the PCE is delivered
by a complicated formula, which lacks an analytic TT representation. To
construct its TT approximation numerically, we develop the new block TT cross
algorithm, a method that computes the whole TT decomposition from a few
evaluations of the PCE formula. The new method is conceptually similar to the
adaptive cross approximation in the TT format, but is more efficient when
several tensors must be stored in the same TT representation, which is the case
for the PCE. Besides, we demonstrate how to assemble the stochastic Galerkin
matrix and to compute the solution of the elliptic equation and its
post-processing, staying in the TT format.
We compare our technique with the traditional sparse polynomial chaos and the
Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial
degree is bounded for each random variable independently. This provides higher
accuracy than the sparse polynomial set or the Monte Carlo method, but the
cardinality of the tensor product set grows exponentially with the number of
random variables. However, when the PCE coefficients are implicitly
approximated in the TT format, the computations with the full tensor product
polynomial set become possible. In the numerical experiments, we confirm that
the new methodology is competitive in a wide range of parameters, especially
where high accuracy and high polynomial degrees are required.Comment: This is a major revision of the manuscript arXiv:1406.2816 with
significantly extended numerical experiments. Some unused material is remove
To be or not to be intrusive? The solution of parametric and stochastic equations - the "plain vanilla" Galerkin case
In parametric equations - stochastic equations are a special case - one may
want to approximate the solution such that it is easy to evaluate its
dependence of the parameters. Interpolation in the parameters is an obvious
possibility, in this context often labeled as a collocation method. In the
frequent situation where one has a "solver" for the equation for a given
parameter value - this may be a software component or a program - it is evident
that this can independently solve for the parameter values to be interpolated.
Such uncoupled methods which allow the use of the original solver are classed
as "non-intrusive". By extension, all other methods which produce some kind of
coupled system are often - in our view prematurely - classed as "intrusive". We
show for simple Galerkin formulations of the parametric problem - which
generally produce coupled systems - how one may compute the approximation in a
non-intusive way
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