On Sampling Spatially-Correlated Random Fields for Complex Geometries
Résumé
Extracting spatial heterogeneities from patient-specific data
is challenging. In most cases, it is unfeasible to achieve an arbitrary level
of detail and accuracy. This lack of perfect knowledge can be treated
as an uncertainty associated with the estimated parameters and thus be
modeled as a spatially-correlated random field superimposed to them. In
order to quantify the effect of this uncertainty on the simulation outputs,
it is necessary to generate several realizations of these random fields.
This task is far from trivial, particularly in the case of complex geometries.
Here, we present two different approaches to achieve this. In the
first method, we use a stochastic partial differential equation, yielding
a method which is general and fast, but whose underlying correlation
function is not readily available. In the second method, we propose a
geodesic-based modification of correlation kernels used in the truncated
Karhunen-Loève expansion with pivoted Cholesky factorization, which
renders the method efficient even for complex geometries, provided that
the correlation length is not too small. Both methods are tested on a few
examples and cardiac applications.
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