115 research outputs found
New Coherence and RIP Analysis for Weak Orthogonal Matching Pursuit
In this paper we define a new coherence index, named the global 2-coherence,
of a given dictionary and study its relationship with the traditional mutual
coherence and the restricted isometry constant. By exploring this relationship,
we obtain more general results on sparse signal reconstruction using greedy
algorithms in the compressive sensing (CS) framework. In particular, we obtain
an improved bound over the best known results on the restricted isometry
constant for successful recovery of sparse signals using orthogonal matching
pursuit (OMP).Comment: arXiv admin note: substantial text overlap with arXiv:1307.194
High Order Methods for a Class of Volterra Integral Equations with Weakly Singular Kernels
The solution of the Volterra integral equation, where , and are smooth functions, can be represented as ,, where , are, smooth and satisfy a system of Volterra integral equations. In this paper, numerical schemes for the solution of (*) are suggested which calculate via , in a neighborhood of the origin and use (*) on the rest of the interval . In this way, methods of arbitrarily high order can be derived. As an example, schemes based on the product integration analogue of Simpson's rule are treated in detail. The schemes are shown to be convergent of order . Asymptotic error estimates are derived in order to examine the numerical stability of the methods
A Note on Error Bounds for Pseudo Skeleton Approximations of Matrices
Due to their importance in both data analysis and numerical algorithms, low
rank approximations have recently been widely studied. They enable the handling
of very large matrices. Tight error bounds for the computationally efficient
Gaussian elimination based methods (skeleton approximations) are available. In
practice, these bounds are useful for matrices with singular values which
decrease quickly. Using the Chebyshev norm, this paper provides improved bounds
for the errors of the matrix elements. These bounds are substantially better in
the practically relevant cases where the eigenvalues decrease polynomially.
Results are proven for general real rectangular matrices. Even stronger bounds
are obtained for symmetric positive definite matrices. A simple example is
given, comparing these new bounds to earlier ones.Comment: 8 pages, 1 figur
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