81 research outputs found
On Modified l_1-Minimization Problems in Compressed Sensing
Sparse signal modeling has received much attention recently because of its application in medical imaging, group testing and radar technology, among others. Compressed sensing, a recently coined term, has showed us, both in theory and practice, that various
signals of interest which are sparse or approximately sparse can be efficiently recovered by using far fewer samples than suggested by Shannon sampling theorem.
Sparsity is the only prior information about an unknown signal assumed in traditional compressed sensing techniques. But in many applications, other kinds of prior information are also available, such as partial knowledge of the support, tree structure
of signal and clustering of large coefficients around a small set of coefficients.
In this thesis, we consider compressed sensing problems with prior information on the support of the signal, together with sparsity. We modify regular l_1 -minimization problems considered in compressed sensing, using this extra information. We call these
modified l_1 -minimization problems.
We show that partial knowledge of the support helps us to weaken sufficient conditions for the recovery of sparse signals using modified ` 1 minimization problems. In case
of deterministic compressed sensing, we show that a sharp condition for sparse recovery
can be improved using modified ` 1 minimization problems. We also derive algebraic necessary and sufficient condition for modified basis pursuit problem and use an open source algorithm known as l_1 -homotopy algorithm to perform some numerical experiments and compare the performance of modified Basis Pursuit Denoising with the regular Basis Pursuit Denoising
Sparse-Based Estimation Performance for Partially Known Overcomplete Large-Systems
We assume the direct sum o for the signal subspace. As a result of
post- measurement, a number of operational contexts presuppose the a priori
knowledge of the LB -dimensional "interfering" subspace and the goal is to
estimate the LA am- plitudes corresponding to subspace . Taking into account
the knowledge of the orthogonal "interfering" subspace \perp, the Bayesian
estimation lower bound is de-
rivedfortheLA-sparsevectorinthedoublyasymptoticscenario,i.e. N,LA,LB -> \infty
with a finite asymptotic ratio. By jointly exploiting the Compressed Sensing
(CS) and the Random Matrix Theory (RMT) frameworks, closed-form expressions for
the lower bound on the estimation of the non-zero entries of a sparse vector of
interest are derived and studied. The derived closed-form expressions enjoy
several interesting features: (i) a simple interpretable expression, (ii) a
very low computational cost especially in the doubly asymptotic scenario, (iii)
an accurate prediction of the mean-square-error (MSE) of popular sparse-based
estimators and (iv) the lower bound remains true for any amplitudes vector
priors. Finally, several idealized scenarios are compared to the derived bound
for a common output signal-to-noise-ratio (SNR) which shows the in- terest of
the joint estimation/rejection methodology derived herein.Comment: 10 pages, 5 figures, Journal of Signal Processin
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Bifidelity Methods for Polynomial Chaos Expansions
This thesis provides an in-depth evaluation of two multi fidelity uncertainty quantification techniques, highlighting the key characteristics, benefits, and shortcomings therein. Physics based simulations subject to uncertain inputs are used to demonstrate the efficacy of each technique in reducing the computational cost of generating a polynomial chaos (PC) approximation of a simulated quantity of interest(QoI). Considered is a weighted L1 minimization technique, wherein a priori estimates on the decay of PC coefficients are used to generate sparse PC approximations of the QoI. Also considered is a stochastic basis reduction method, which identifies a subspace that spans the PC basis by principle component analysis of the covariance of the QoI. Numerical tests were conducted upon 2 airfoil simulations subject to 6 uncertain inputs (one at high Mach number, one at low) and a lithium ion battery simulation subject to 17 uncertain inputs to evaluate each method. The examples studied illustrate the main characteristics of each method and provide insight to their applicability to UQ in numerical simulations. Appreciable reductions in computational resources were observed in all cases when compared to direct simulation of a high fidelity model
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