2,770 research outputs found
The ROMES method for statistical modeling of reduced-order-model error
This work presents a technique for statistically modeling errors introduced
by reduced-order models. The method employs Gaussian-process regression to
construct a mapping from a small number of computationally inexpensive `error
indicators' to a distribution over the true error. The variance of this
distribution can be interpreted as the (epistemic) uncertainty introduced by
the reduced-order model. To model normed errors, the method employs existing
rigorous error bounds and residual norms as indicators; numerical experiments
show that the method leads to a near-optimal expected effectivity in contrast
to typical error bounds. To model errors in general outputs, the method uses
dual-weighted residuals---which are amenable to uncertainty control---as
indicators. Experiments illustrate that correcting the reduced-order-model
output with this surrogate can improve prediction accuracy by an order of
magnitude; this contrasts with existing `multifidelity correction' approaches,
which often fail for reduced-order models and suffer from the curse of
dimensionality. The proposed error surrogates also lead to a notion of
`probabilistic rigor', i.e., the surrogate bounds the error with specified
probability
Compressive sensing adaptation for polynomial chaos expansions
Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of
the underlying Gaussian germ. Several rotations have been proposed in the
literature resulting in adaptations with different convergence properties. In
this paper we present a new adaptation mechanism that builds on compressive
sensing algorithms, resulting in a reduced polynomial chaos approximation with
optimal sparsity. The developed adaptation algorithm consists of a two-step
optimization procedure that computes the optimal coefficients and the input
projection matrix of a low dimensional chaos expansion with respect to an
optimally rotated basis. We demonstrate the attractive features of our
algorithm through several numerical examples including the application on
Large-Eddy Simulation (LES) calculations of turbulent combustion in a HIFiRE
scramjet engine.Comment: Submitted to Journal of Computational Physic
Minimum mean square distance estimation of a subspace
We consider the problem of subspace estimation in a Bayesian setting. Since
we are operating in the Grassmann manifold, the usual approach which consists
of minimizing the mean square error (MSE) between the true subspace and its
estimate may not be adequate as the MSE is not the natural metric in
the Grassmann manifold. As an alternative, we propose to carry out subspace
estimation by minimizing the mean square distance (MSD) between and its
estimate, where the considered distance is a natural metric in the Grassmann
manifold, viz. the distance between the projection matrices. We show that the
resulting estimator is no longer the posterior mean of but entails
computing the principal eigenvectors of the posterior mean of .
Derivation of the MMSD estimator is carried out in a few illustrative examples
including a linear Gaussian model for the data and a Bingham or von Mises
Fisher prior distribution for . In all scenarios, posterior distributions
are derived and the MMSD estimator is obtained either analytically or
implemented via a Markov chain Monte Carlo simulation method. The method is
shown to provide accurate estimates even when the number of samples is lower
than the dimension of . An application to hyperspectral imagery is finally
investigated
Recent Progress in Image Deblurring
This paper comprehensively reviews the recent development of image
deblurring, including non-blind/blind, spatially invariant/variant deblurring
techniques. Indeed, these techniques share the same objective of inferring a
latent sharp image from one or several corresponding blurry images, while the
blind deblurring techniques are also required to derive an accurate blur
kernel. Considering the critical role of image restoration in modern imaging
systems to provide high-quality images under complex environments such as
motion, undesirable lighting conditions, and imperfect system components, image
deblurring has attracted growing attention in recent years. From the viewpoint
of how to handle the ill-posedness which is a crucial issue in deblurring
tasks, existing methods can be grouped into five categories: Bayesian inference
framework, variational methods, sparse representation-based methods,
homography-based modeling, and region-based methods. In spite of achieving a
certain level of development, image deblurring, especially the blind case, is
limited in its success by complex application conditions which make the blur
kernel hard to obtain and be spatially variant. We provide a holistic
understanding and deep insight into image deblurring in this review. An
analysis of the empirical evidence for representative methods, practical
issues, as well as a discussion of promising future directions are also
presented.Comment: 53 pages, 17 figure
Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches
Imaging spectrometers measure electromagnetic energy scattered in their
instantaneous field view in hundreds or thousands of spectral channels with
higher spectral resolution than multispectral cameras. Imaging spectrometers
are therefore often referred to as hyperspectral cameras (HSCs). Higher
spectral resolution enables material identification via spectroscopic analysis,
which facilitates countless applications that require identifying materials in
scenarios unsuitable for classical spectroscopic analysis. Due to low spatial
resolution of HSCs, microscopic material mixing, and multiple scattering,
spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus,
accurate estimation requires unmixing. Pixels are assumed to be mixtures of a
few materials, called endmembers. Unmixing involves estimating all or some of:
the number of endmembers, their spectral signatures, and their abundances at
each pixel. Unmixing is a challenging, ill-posed inverse problem because of
model inaccuracies, observation noise, environmental conditions, endmember
variability, and data set size. Researchers have devised and investigated many
models searching for robust, stable, tractable, and accurate unmixing
algorithms. This paper presents an overview of unmixing methods from the time
of Keshava and Mustard's unmixing tutorial [1] to the present. Mixing models
are first discussed. Signal-subspace, geometrical, statistical, sparsity-based,
and spatial-contextual unmixing algorithms are described. Mathematical problems
and potential solutions are described. Algorithm characteristics are
illustrated experimentally.Comment: This work has been accepted for publication in IEEE Journal of
Selected Topics in Applied Earth Observations and Remote Sensin
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