12,632 research outputs found
Acceleration of Randomized Kaczmarz Method via the Johnson-Lindenstrauss Lemma
The Kaczmarz method is an algorithm for finding the solution to an
overdetermined consistent system of linear equations Ax=b by iteratively
projecting onto the solution spaces. The randomized version put forth by
Strohmer and Vershynin yields provably exponential convergence in expectation,
which for highly overdetermined systems even outperforms the conjugate gradient
method. In this article we present a modified version of the randomized
Kaczmarz method which at each iteration selects the optimal projection from a
randomly chosen set, which in most cases significantly improves the convergence
rate. We utilize a Johnson-Lindenstrauss dimension reduction technique to keep
the runtime on the same order as the original randomized version, adding only
extra preprocessing time. We present a series of empirical studies which
demonstrate the remarkable acceleration in convergence to the solution using
this modified approach
Randomized Local Model Order Reduction
In this paper we propose local approximation spaces for localized model order
reduction procedures such as domain decomposition and multiscale methods. Those
spaces are constructed from local solutions of the partial differential
equation (PDE) with random boundary conditions, yield an approximation that
converges provably at a nearly optimal rate, and can be generated at close to
optimal computational complexity. In many localized model order reduction
approaches like the generalized finite element method, static condensation
procedures, and the multiscale finite element method local approximation spaces
can be constructed by approximating the range of a suitably defined transfer
operator that acts on the space of local solutions of the PDE. Optimal local
approximation spaces that yield in general an exponentially convergent
approximation are given by the left singular vectors of this transfer operator
[I. Babu\v{s}ka and R. Lipton 2011, K. Smetana and A. T. Patera 2016]. However,
the direct calculation of these singular vectors is computationally very
expensive. In this paper, we propose an adaptive randomized algorithm based on
methods from randomized linear algebra [N. Halko et al. 2011], which constructs
a local reduced space approximating the range of the transfer operator and thus
the optimal local approximation spaces. The adaptive algorithm relies on a
probabilistic a posteriori error estimator for which we prove that it is both
efficient and reliable with high probability. Several numerical experiments
confirm the theoretical findings.Comment: 31 pages, 14 figures, 1 table, 1 algorith
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