22,288 research outputs found
Integral curves of noisy vector fields and statistical problems in diffusion tensor imaging: nonparametric kernel estimation and hypotheses testing
Let be a vector field in a bounded open set .
Suppose that is observed with a random noise at random points that are independent and uniformly distributed in The problem
is to estimate the integral curve of the differential equation
starting at a given
point and to develop statistical tests for the hypothesis that
the integral curve reaches a specified set We develop an
estimation procedure based on a Nadaraya--Watson type kernel regression
estimator, show the asymptotic normality of the estimated integral curve and
derive differential and integral equations for the mean and covariance function
of the limit Gaussian process. This provides a method of tracking not only the
integral curve, but also the covariance matrix of its estimate. We also study
the asymptotic distribution of the squared minimal distance from the integral
curve to a smooth enough surface . Building upon this, we
develop testing procedures for the hypothesis that the integral curve reaches
. The problems of this nature are of interest in diffusion tensor
imaging, a brain imaging technique based on measuring the diffusion tensor at
discrete locations in the cerebral white matter, where the diffusion of water
molecules is typically anisotropic. The diffusion tensor data is used to
estimate the dominant orientations of the diffusion and to track white matter
fibers from the initial location following these orientations. Our approach
brings more rigorous statistical tools to the analysis of this problem
providing, in particular, hypothesis testing procedures that might be useful in
the study of axonal connectivity of the white matter.Comment: Published in at http://dx.doi.org/10.1214/009053607000000073 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
H-matrix accelerated second moment analysis for potentials with rough correlation
We consider the efficient solution of partial differential equationsfor strongly elliptic operators with constant coefficients and stochastic Dirichlet data by the boundary integral equation method. The computation of the solution's two-point correlation is well understood if the two-point correlation of the Dirichlet data is known and sufficiently smooth.Unfortunately, the problem becomes much more involved in case of rough data. We will show that the concept of the H-matrix arithmetic provides a powerful tool to cope with this problem. By employing a parametric surface representation, we end up with an H-matrix arithmetic based on balanced cluster trees. This considerably simplifies the implementation and improves the performance of the H-matrix arithmetic. Numerical experiments are provided to validate and quantify the presented methods and algorithms
An error indicator-based adaptive reduced order model for nonlinear structural mechanics -- application to high-pressure turbine blades
The industrial application motivating this work is the fatigue computation of
aircraft engines' high-pressure turbine blades. The material model involves
nonlinear elastoviscoplastic behavior laws, for which the parameters depend on
the temperature. For this application, the temperature loading is not
accurately known and can reach values relatively close to the creep
temperature: important nonlinear effects occur and the solution strongly
depends on the used thermal loading. We consider a nonlinear reduced order
model able to compute, in the exploitation phase, the behavior of the blade for
a new temperature field loading. The sensitivity of the solution to the
temperature makes {the classical unenriched proper orthogonal decomposition
method} fail. In this work, we propose a new error indicator, quantifying the
error made by the reduced order model in computational complexity independent
of the size of the high-fidelity reference model. In our framework, when the
{error indicator} becomes larger than a given tolerance, the reduced order
model is updated using one time step solution of the high-fidelity reference
model. The approach is illustrated on a series of academic test cases and
applied on a setting of industrial complexity involving 5 million degrees of
freedom, where the whole procedure is computed in parallel with distributed
memory
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