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L1 Norm (l1 + norm)

Distribution by Scientific Domains
Mathematics and Statistics83%

Selected Abstracts

Compressed sensing in hyperpolarized 3He Lung MRI

MAGNETIC RESONANCE IN MEDICINE, Issue 4 2010
Salma Ajraoui
Abstract In this work, the application of compressed sensing techniques to the acquisition and reconstruction of hyperpolarized 3He lung MR images was investigated. The sparsity of 3He lung images in the wavelet domain was investigated through simulations based on fully sampled Cartesian two-dimensional and three-dimensional 3He lung ventilation images, and the k -spaces of 2D and 3D images were undersampled randomly and reconstructed by minimizing the L1 norm. The simulation results show that temporal resolution can be readily improved by a factor of 2 for two-dimensional and 4 to 5 for three-dimensional ventilation imaging with 3He with the levels of signal to noise ratio (SNR) (,19) typically obtained. The feasibility of producing accurate functional apparent diffusion coefficient (ADC) maps from undersampled data acquired with fewer radiofrequency pulses was also demonstrated, with the preservation of quantitative information (mean ADCcs , mean ADCfull , 0.16 cm2 sec,1). Prospective acquisition of 2-fold undersampled two-dimensional 3He images with a compressed sensing k -space pattern was then demonstrated in a healthy volunteer, and the results were compared to the equivalent fully sampled images (SNRcs = 34, SNRfull = 19). Magn Reson Med 63:1059,1069, 2010. © 2010 Wiley-Liss, Inc. [source]

Testing monotone high-dimensional distributions,,

RANDOM STRUCTURES AND ALGORITHMS, Issue 1 2009
Ronitt Rubinfeld
Abstract A monotone distribution P over a (partially) ordered domain has P(y) , P(x) if y , x in the order. We study several natural problems of testing properties of monotone distributions over the n -dimensional Boolean cube, given access to random draws from the distribution being tested. We give a poly(n)-time algorithm for testing whether a monotone distribution is equivalent to or , -far (in the L1 norm) from the uniform distribution. A key ingredient of the algorithm is a generalization of a known isoperimetric inequality for the Boolean cube. We also introduce a method for proving lower bounds on testing monotone distributions over the n -dimensional Boolean cube, based on a new decomposition technique for monotone distributions. We use this method to show that our uniformity testing algorithm is optimal up to polylog(n) factors, and also to give exponential lower bounds on the complexity of several other problems (testing whether a monotone distribution is identical to or , -far from a fixed known monotone product distribution and approximating the entropy of an unknown monotone distribution). © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009 [source]

Distributions for which div v = F has a continuous solution

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2008
Thierry De Pauw
The equation div v = F has a continuous weak solution in an open set U , ,m if and only if the distribution F satisfies the following condition: the F(,i) converge to 0 for every sequence {,i} of test functions such that the support of each ,i is contained in a fixed compact subset of U, and in the L1 norm, {,i} converges to 0 and {,,i} is bounded. © 2007 Wiley Periodicals, Inc. [source]

Rigidity and gamma convergence for solid-solid phase transitions with SO(2) invariance

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2006
Sergio Conti
The singularly perturbed two-well problem in the theory of solid-solid phase transitions takes the form where u : , , ,n , ,n is the deformation, and W vanishes for all matrices in K = SO(n)A , SO(n)B. We focus on the case n = 2 and derive, by means of Gamma convergence, a sharp-interface limit for I,. The proof is based on a rigidity estimate for low-energy functions. Our rigidity argument also gives an optimal two-well Liouville estimate: if ,u has a small BV norm (compared to the diameter of the domain), then, in the L1 sense, either the distance of ,u from SO(2)A or the one from SO(2)B is controlled by the distance of ,u from K. This implies that the oscillation of ,u in weak L1 is controlled by the L1 norm of the distance of ,u to K. © 2006 Wiley Periodicals, Inc. [source]

Hyperbolic limit of the Jin-Xin relaxation model

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 5 2006
Stefano Bianchini
We consider the special Jin-Xin relaxation model We assume that the initial data () are sufficiently smooth and close to () in L, and have small total variation. Then we prove that there exists a solution () with uniformly small total variation for all t , 0, and this solution depends Lipschitz-continuously in the L1 norm with respect to time and the initial data. Letting , the solution converges to a unique limit, providing a relaxation limit solution to the quasi-linear, nonconservative system These limit solutions generate a Lipschitz semigroup on a domain containing the functions with small total variation and close to . This is precisely the Riemann semigroup determined by the unique Riemann solver compatible with (0.1). © 2005 Wiley Periodicals, Inc. [source]

On L1 decay problem for the dissipative wave equation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2003
Kosuke Ono
Abstract We study the decay estimates of solutions to the Cauchy problem for the dissipative wave equation in one, two, and three dimensions. The representation formulas of the solutions provide the sharp decay rates on L1 norms and also Lp norms. Copyright © 2003 John Wiley & Sons, Ltd. [source]