A randomized Kaczmarz algorithm with exponential convergence

The Kaczmarz method for solving linear systems of equations is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Despite the popularity of this method, useful theoretical estimates for its rate of convergence are still scarce. We introduce a randomized version of the Kaczmarz method for consistent, overdetermined linear systems and we prove that it converges with expected exponential rate. Furthermore, this is the first solver whose rate does not depend on the number of equations in the system. The solver does not even need to know the whole system, but only a small random part of it. It thus outperforms all previously known methods on general extremely overdetermined systems. Even for moderately overdetermined systems, numerical simulations as well as theoretical analysis reveal that our algorithm can converge faster than the celebrated conjugate gradient algorithm. Furthermore, our theory and numerical simulations confirm a prediction of Feichtinger et al. in the context of reconstructing bandlimited functions from nonuniform sampling

Hybrid deterministic/stochastic algorithm for large sets of rate equations

We propose a hybrid algorithm for the time integration of large sets of rate equations coupled by a relatively small number of degrees of freedom. A subset containing fast degrees of freedom evolves deterministically, while the rest of the variables evolves stochastically. The emphasis is put on the coupling between the two subsets, in order to achieve both accuracy and efficiency. The algorithm is tested on the problem of nucleation, growth and coarsening of clusters of defects in iron, treated by the formalism of cluster dynamics. We show that it is possible to obtain results indistinguishable from fully deterministic and fully stochastic calculations, while speeding up significantly the computations with respect to these two cases.Comment: 9 pages, 7 figure

Beyond Gr\"obner Bases: Basis Selection for Minimal Solvers

Many computer vision applications require robust estimation of the underlying geometry, in terms of camera motion and 3D structure of the scene. These robust methods often rely on running minimal solvers in a RANSAC framework. In this paper we show how we can make polynomial solvers based on the action matrix method faster, by careful selection of the monomial bases. These monomial bases have traditionally been based on a Gr\"obner basis for the polynomial ideal. Here we describe how we can enumerate all such bases in an efficient way. We also show that going beyond Gr\"obner bases leads to more efficient solvers in many cases. We present a novel basis sampling scheme that we evaluate on a number of problems

A JavaScript API for the Ice Sheet System Model: towards on online interactive model for the Cryosphere Community

Abstract. Earth System Models (ESMs) are becoming increasingly complex, requiring extensive knowledge and experience to deploy and use in an efficient manner. They run on high-performance architectures that are significantly different from the everyday environments that scientists use to pre and post-process results (i.e. MATLAB, Python). This results in models that are hard to use for non specialists, and that are increasingly specific in their application. It also makes them relatively inaccessible to the wider science community, not to mention to the general public. Here, we present a new software/model paradigm that attempts to bridge the gap between the science community and the complexity of ESMs, by developing a new JavaScript Application Program Interface (API) for the Ice Sheet System Model (ISSM). The aforementioned API allows Cryosphere Scientists to run ISSM on the client-side of a webpage, within the JavaScript environment. When combined with a Web server running ISSM (using a Python API), it enables the serving of ISSM computations in an easy and straightforward way. The deep integration and similarities between all the APIs in ISSM (MATLAB, Python, and now JavaScript) significantly shortens and simplifies the turnaround of state-of-the-art science runs and their use by the larger community. We demonstrate our approach via a new Virtual Earth System Laboratory (VESL) Web site