28,474 research outputs found
Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification
This paper analyses the following question: let , be
the Galerkin matrices corresponding to finite-element discretisations of the
exterior Dirichlet problem for the heterogeneous Helmholtz equations
. How small must and be (in terms of -dependence) for
GMRES applied to either or
to converge in a -independent number of
iterations for arbitrarily large ? (In other words, for to be
a good left- or right-preconditioner for ?). We prove results
answering this question, give theoretical evidence for their sharpness, and
give numerical experiments supporting the estimates.
Our motivation for tackling this question comes from calculating quantities
of interest for the Helmholtz equation with random coefficients and .
Such a calculation may require the solution of many deterministic Helmholtz
problems, each with different and , and the answer to the question above
dictates to what extent a previously-calculated inverse of one of the Galerkin
matrices can be used as a preconditioner for other Galerkin matrices
Systematic Study of Accuracy of Wall-Modeled Large Eddy Simulation using Uncertainty Quantification Techniques
The predictive accuracy of wall-modeled large eddy simulation is studied by
systematic simulation campaigns of turbulent channel flow. The effect of wall
model, grid resolution and anisotropy, numerical convective scheme and
subgrid-scale modeling is investigated. All of these factors affect the
resulting accuracy, and their action is to a large extent intertwined. The wall
model is of the wall-stress type, and its sensitivity to location of velocity
sampling, as well as law of the wall's parameters is assessed. For efficient
exploration of the model parameter space (anisotropic grid resolution and wall
model parameter values), generalized polynomial chaos expansions are used to
construct metamodels for the responses which are taken to be measures of the
predictive error in quantities of interest (QoIs). The QoIs include the mean
wall shear stress and profiles of the mean velocity, the turbulent kinetic
energy, and the Reynolds shear stress. DNS data is used as reference. Within
the tested framework, a particular second-order accurate CFD code (OpenFOAM),
the results provide ample support for grid and method parameters
recommendations which are proposed in the present paper, and which provide good
results for the QoIs. Notably, good results are obtained with a grid with
isotropic (cubic) hexahedral cells, with cells per , where
is the channel half-height (or thickness of the turbulent boundary
layer). The importance of providing enough numerical dissipation to obtain
accurate QoIs is demonstrated. The main channel flow case investigated is , but extension to a wide range of -numbers is
considered. Use of other numerical methods and software would likely modify
these recommendations, at least slightly, but the proposed framework is fully
applicable to investigate this as well
Discovering an active subspace in a single-diode solar cell model
Predictions from science and engineering models depend on the values of the
model's input parameters. As the number of parameters increases, algorithmic
parameter studies like optimization or uncertainty quantification require many
more model evaluations. One way to combat this curse of dimensionality is to
seek an alternative parameterization with fewer variables that produces
comparable predictions. The active subspace is a low-dimensional linear
subspace defined by important directions in the model's input space; input
perturbations along these directions change the model's prediction more, on
average, than perturbations orthogonal to the important directions. We describe
a method for checking if a model admits an exploitable active subspace, and we
apply this method to a single-diode solar cell model with five input
parameters. We find that the maximum power of the solar cell has a dominant
one-dimensional active subspace, which enables us to perform thorough parameter
studies in one dimension instead of five
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