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Sampling bounds for 2-D vector field tomography
The tomographic mapping of a 2-D vector field from line-integral data in the discrete domain requires the uniform sampling of the continuous Radon domain parameter space. In this paper we use sampling theory and derive limits for the sampling steps of the Radon parameters, so that no information is lost. It is shown that if Ξx is the sampling interval of the reconstruction region and xmax is the maximum value of domain parameter x, the steps one should use to sample Radon parameters Ο and ΞΈ should be: ΞΟβ€ Ξx/β2 and ΞΞΈβ€Ξx/((β2+2)|xmax|). Experiments show that when the proposed sampling bounds are violated, the reconstruction accuracy of the vector field deteriorates. We further demonstrate that the employment of a scanning geometry that satisfies the proposed sampling requirements also increases the resilience to noise
Multilevel Double Loop Monte Carlo and Stochastic Collocation Methods with Importance Sampling for Bayesian Optimal Experimental Design
An optimal experimental set-up maximizes the value of data for statistical
inferences and predictions. The efficiency of strategies for finding optimal
experimental set-ups is particularly important for experiments that are
time-consuming or expensive to perform. For instance, in the situation when the
experiments are modeled by Partial Differential Equations (PDEs), multilevel
methods have been proven to dramatically reduce the computational complexity of
their single-level counterparts when estimating expected values. For a setting
where PDEs can model experiments, we propose two multilevel methods for
estimating a popular design criterion known as the expected information gain in
simulation-based Bayesian optimal experimental design. The expected information
gain criterion is of a nested expectation form, and only a handful of
multilevel methods have been proposed for problems of such form. We propose a
Multilevel Double Loop Monte Carlo (MLDLMC), which is a multilevel strategy
with Double Loop Monte Carlo (DLMC), and a Multilevel Double Loop Stochastic
Collocation (MLDLSC), which performs a high-dimensional integration by
deterministic quadrature on sparse grids. For both methods, the Laplace
approximation is used for importance sampling that significantly reduces the
computational work of estimating inner expectations. The optimal values of the
method parameters are determined by minimizing the average computational work,
subject to satisfying the desired error tolerance. The computational
efficiencies of the methods are demonstrated by estimating the expected
information gain for Bayesian inference of the fiber orientation in composite
laminate materials from an electrical impedance tomography experiment. MLDLSC
performs better than MLDLMC when the regularity of the quantity of interest,
with respect to the additive noise and the unknown parameters, can be
exploited
Low rank matrix recovery from rank one measurements
We study the recovery of Hermitian low rank matrices from undersampled measurements via nuclear norm minimization. We
consider the particular scenario where the measurements are Frobenius inner
products with random rank-one matrices of the form for some
measurement vectors , i.e., the measurements are given by . The case where the matrix to be recovered
is of rank one reduces to the problem of phaseless estimation (from
measurements, via the PhaseLift approach,
which has been introduced recently. We derive bounds for the number of
measurements that guarantee successful uniform recovery of Hermitian rank
matrices, either for the vectors , , being chosen independently
at random according to a standard Gaussian distribution, or being sampled
independently from an (approximate) complex projective -design with .
In the Gaussian case, we require measurements, while in the case
of -designs we need . Our results are uniform in the
sense that one random choice of the measurement vectors guarantees
recovery of all rank -matrices simultaneously with high probability.
Moreover, we prove robustness of recovery under perturbation of the
measurements by noise. The result for approximate -designs generalizes and
improves a recent bound on phase retrieval due to Gross, Kueng and Krahmer. In
addition, it has applications in quantum state tomography. Our proofs employ
the so-called bowling scheme which is based on recent ideas by Mendelson and
Koltchinskii.Comment: 24 page
TV-min and Greedy Pursuit for Constrained Joint Sparsity and Application to Inverse Scattering
This paper proposes a general framework for compressed sensing of constrained
joint sparsity (CJS) which includes total variation minimization (TV-min) as an
example. TV- and 2-norm error bounds, independent of the ambient dimension, are
derived for the CJS version of Basis Pursuit and Orthogonal Matching Pursuit.
As an application the results extend Cand`es, Romberg and Tao's proof of exact
recovery of piecewise constant objects with noiseless incomplete Fourier data
to the case of noisy data.Comment: Mathematics and Mechanics of Complex Systems (2013
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