1,133 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 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
Recommended from our members
Non-nested multi-grid solvers for mixed divergence-free Scott-Vogelius discretizations
Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure
Recommended from our members
Robust arbitrary order mixed finite element methods for the incompressible Stokes equations
Standard mixed finite element methods for the incompressible
Navier-Stokes equations that relax the divergence constraint are not robust
against large irrotational forces in the momentum balance and the velocity
error depends on the continuous pressure. This robustness issue can be
completely cured by using divergence-free mixed finite elements which deliver
pressure-independent velocity error estimates. However, the construction of
H1-conforming, divergence-free mixed finite element methods is rather
difficult. Instead, we present a novel approach for the construction of
arbitrary order mixed finite element methods which deliver
pressure-independent velocity errors. The approach does not change the trial
functions but replaces discretely divergence-free test functions in some
operators of the weak formulation by divergence-free ones. This modification
is applied to inf-sup stable conforming and nonconforming mixed finite
element methods of arbitrary order in two and three dimensions. Optimal
estimates for the incompressible Stokes equations are proved for the H1 and
L2 errors of the velocity and the L2 error of the pressure. Moreover, both
velocity errors are pressure-independent, demonstrating the improved
robustness. Several numerical examples illustrate the results
Robust arbitrary order mixed finite element methods for the incompressible Stokes equations
Standard mixed finite element methods for the incompressible Navier-Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H1-conforming, divergence-free mixed finite element methods is rather difficult. Instead, we present a novel approach for the construction of arbitrary order mixed finite element methods which deliver pressure-independent velocity errors. The approach does not change the trial functions but replaces discretely divergence-free test functions in some operators of the weak formulation by divergence-free ones. This modification is applied to inf-sup stable conforming and nonconforming mixed finite element methods of arbitrary order in two and three dimensions. Optimal estimates for the incompressible Stokes equations are proved for the H1 and L2 errors of the velocity and the L2 error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results
Charakterisierung des selektiven Extraktionsverhaltens von Titan
Die Frage der Weiterverarbeitung von industriell anfallenden wertmetallhaltigen Reststoffen gewinnt einen immer höheren Stellenwert. Die Untersuchungen dieser Arbeit befassen sich mit einem titanhaltigen Rückstand der Pigmentindustrie. Zur Aufbereitung wurde ein hydrometallurgischer Prozess zur Entfernung von Störelementen und zur gleichzeitigen Anreicherung von Titan entwickelt. Eine gezielte Abtrennung von Titan bezüglich der Elemente Eisen und Vanadium kann mittels Solventextraktion durch das Extraktionsmittel DEHPA (Di-(2-ethylhexyl)phosphorsäure) erfolgen. Durch Extraktionsversuche zeigt sich, dass Titan bevorzugt extrahiert werden kann und es wird der ideale Arbeitspunkt für die Extraktion von Titan erarbeitet. Dabei zeigt sich, dass acht DEHPA-Moleküle für eine vollständige Extraktion von Titan nötig sind. Weiterhin wurden spektroskopische Messungen mittels GALDI-MS und NMR-Spektroskopie bezüglich des Vorliegens von Titan mit DEHPA durchgeführt. In der organischen Phase kommt es hauptsächlich zum Vorliegen von [Ti]4+ und [Ti2O2]4+. Weiterhin kommt es zu einer gleichartigen Anordnung von acht DEHPA-Molekülen. Die Arbeit liefert nicht nur einen Mehrwert für das Verständnis der Komplexierung von Titan durch DEHPA, sondern stellt aufgrund der gewählten Parameter einen direkten Bezug zum entwickelten Aufbereitungsprozess dar
- …