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Stagewise Polytope Faces Pursuit for Recovery of Sparse Representations
"... In sparse representations or compressed sensing, we are typically interested in finding a sparse vector s which satisfies an underdetermined system of linear equations x = As. We can do this by searching for the minimum ℓ1 norm, or Basis Pursuit (BP) solution, which in standard form [2] is the left ..."
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In sparse representations or compressed sensing, we are typically interested in finding a sparse vector s which satisfies an underdetermined system of linear equations x = As. We can do this by searching for the minimum ℓ1 norm, or Basis Pursuit (BP) solution, which in standard form [2] is the lefthand side member of the following equality: min ˜s
Greedy and Randomized Versions of the Multiplicative Schwarz Method
, 2011
"... We consider sequential, i.e., GaussSeidel type, subspace correction methods for the iterative solution of symmetric positive definite variational problems, where the order of subspace correction steps is not deterministically fixed as in standard multiplicative Schwarz methods. Here, we greedily ch ..."
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We consider sequential, i.e., GaussSeidel type, subspace correction methods for the iterative solution of symmetric positive definite variational problems, where the order of subspace correction steps is not deterministically fixed as in standard multiplicative Schwarz methods. Here, we greedily choose the subspace with the largest (or at least a relatively large) residual norm for the next update step, which is also known as the GaussSouthwell method. We prove exponential convergence in the energy norm, with a reduction factor per iteration step directly related to the spectral properties, e.g., the condition number, of the underlying space splitting. To avoid the additional computational cost associated with the greedy pick, we alternatively consider choosing the next subspace randomly, and show similar estimates for the expected error reduction. We give some numerical examples, in particular applications to a Toeplitz system and to multilevel discretizations of an elliptic boundary value problem, which illustrate the theoretical estimates.