24,173 research outputs found
A Symmetry Preserving Algorithm for Matrix Scaling
International audienceWe present an iterative algorithm which asymptotically scales the -norm of each row and each column of a matrix to one. This scaling algorithm preserves symmetry of the original matrix and shows fast linear convergence with an asymptotic rate of . We discuss extensions of the algorithm to the one-norm, and by inference to other norms. For the 1-norm case, we show again that convergence is linear, with the rate dependent on the spectrum of the scaled matrix. We demonstrate experimentally that the scaling algorithm improves the conditioning of the matrix and that it helps direct solvers by reducing the need for pivoting. In particular, for symmetric matrices the theoretical and experimental results highlight the potential of the proposed algorithm over existing alternatives.Nous dĂ©crivons un algorithme itĂ©ratif qui, asymptotiquement, met une matrice Ă l'Ă©chelle de telle sorte que chaque ligne et chaque colonne est de taille 1 dans la norme infini. Cet algorithme prĂ©serve la symĂ©trie. De plus, il converge assez rapidement avec un taux asymptotique de 1/2. Nous discutons la gĂ©nĂ©ralisation de l'algorithme Ă la norme 1 et, par infĂ©rence, Ă d'autres normes. Pour le cas de la norme 1, nous Ă©tablissons que l'algorithme converge avec un taux linĂ©aire. Nous dĂ©montrons expĂ©rimentalement que notre algorithme amĂ©liore le conditionnement de la matrice et qu'il aide les mĂ©thodes directes de rĂ©solution en rĂ©duisant le pivotage. ParticuliĂšrement pour des matrices symĂ©triques, nos rĂ©sultats thĂ©oriques et expĂ©rimentaux mettent en valeur l'intĂ©rĂȘt de notre algorithme par rapport aux algorithmes existants
Scaling Sparse Matrices for Optimization Algorithms
To iteratively solve large scale optimization problems in various contexts like planning, operations, design etc., we need to generate descent directions that are based on linear system solutions. Irrespective of the optimization algorithm or the solution method employed for the linear systems, ill conditioning introduced by problem characteristics or the algorithm or both need to be addressed. In [GL01] we used an intuitive heuristic approach in scaling linear systems that improved performance of a large scale interior point algorithm significantly. We saw a factor of 10*3* improvements in condition number estimates. In this paper, given our experience with optimization problems from a variety of application backgrounds like economics, finance, engineering, planning etc., we examine the theoretical basis for scaling while solving the linear systems. Our goal is to develop reasonably "good" scaling schemes with sound theoretical basis. We introduce concepts and define "good" scaling schemes in section (1), as well as explain related work in this area. Scaling has been studied extensively and though there is a broad agreement on its importance, the same cannot be said about what constitutes good scaling. A theoretical framework to scale an m x n real matrix is established in section (2). We use the first order conditions associated with the Euclidean metric to develop iterative schemes in section (2.3) that approximate solution in O(mn) time for real matrice. We discuss symmetry preserving scale factors for an n x n symmetric matrix in (3). The importance of symmetry preservation is discussed in section (3.1). An algorithm to directly compute symmetry preserving scale factors in O(n2) time based on Euclidean metric is presented in section (3.4) We also suggest scaling schemes based on rectilinear norm in section (2.4). Though all p-norms are theoretically equivalent, the importance of outliers increases as p increases. For barrier methods, due to large diagnal corrections, we believe that the taxicab metric (p = 1) may be more appropriate. We develop a linear programming model for it and look at a "reduced" dual that can be formulated as a minimum cost flow problem on networks. We are investigating algorithms to solve it in O(mn) time that we require for an efficient scaling procedure. We hope that in future special structure of the "reduced" dual could be exploited to solve it quickly. The dual information can then be used to compute the required scale factors. We discuss Manhattan metric for symmetric matrices in section (3.5) and as in the case of real matrices, we are unable to propose an efficient computational scheme for this metric. We look at a linearized ideal penalty function that only uses deviations out of the desired range in section (2.5). If we could use such a metric to generate an efficient solution, then we would like to see impact of changing the range on the numerical behavior.
Symmetry-guided nonrigid registration: the case for distortion correction in multidimensional photoemission spectroscopy
Image symmetrization is an effective strategy to correct symmetry distortion
in experimental data for which symmetry is essential in the subsequent
analysis. In the process, a coordinate transform, the symmetrization transform,
is required to undo the distortion. The transform may be determined by image
registration (i.e. alignment) with symmetry constraints imposed in the
registration target and in the iterative parameter tuning, which we call
symmetry-guided registration. An example use case of image symmetrization is
found in electronic band structure mapping by multidimensional photoemission
spectroscopy, which employs a 3D time-of-flight detector to measure electrons
sorted into the momentum (, ) and energy () coordinates. In
reality, imperfect instrument design, sample geometry and experimental settings
cause distortion of the photoelectron trajectories and, therefore, the symmetry
in the measured band structure, which hinders the full understanding and use of
the volumetric datasets. We demonstrate that symmetry-guided registration can
correct the symmetry distortion in the momentum-resolved photoemission
patterns. Using proposed symmetry metrics, we show quantitatively that the
iterative approach to symmetrization outperforms its non-iterative counterpart
in the restored symmetry of the outcome while preserving the average shape of
the photoemission pattern. Our approach is generalizable to distortion
corrections in different types of symmetries and should also find applications
in other experimental methods that produce images with similar features
Simulation of anyons with tensor network algorithms
Interacting systems of anyons pose a unique challenge to condensed matter
simulations due to their non-trivial exchange statistics. These systems are of
great interest as they have the potential for robust universal quantum
computation, but numerical tools for studying them are as yet limited. We show
how existing tensor network algorithms may be adapted for use with systems of
anyons, and demonstrate this process for the 1-D Multi-scale Entanglement
Renormalisation Ansatz (MERA). We apply the MERA to infinite chains of
interacting Fibonacci anyons, computing their scaling dimensions and local
scaling operators. The scaling dimensions obtained are seen to be in agreement
with conformal field theory. The techniques developed are applicable to any
tensor network algorithm, and the ability to adapt these ansaetze for use on
anyonic systems opens the door for numerical simulation of large systems of
free and interacting anyons in one and two dimensions.Comment: Fixed typos, matches published version. 16 pages, 21 figures, 4
tables, RevTeX 4-1. For a related work, see arXiv:1006.247
Existence and Stability of Symmetric Periodic Simultaneous Binary Collision Orbits in the Planar Pairwise Symmetric Four-Body Problem
We extend our previous analytic existence of a symmetric periodic
simultaneous binary collision orbit in a regularized fully symmetric equal mass
four-body problem to the analytic existence of a symmetric periodic
simultaneous binary collision orbit in a regularized planar pairwise symmetric
equal mass four-body problem. We then use a continuation method to numerically
find symmetric periodic simultaneous binary collision orbits in a regularized
planar pairwise symmetric 1, m, 1, m four-body problem for between 0 and 1.
Numerical estimates of the the characteristic multipliers show that these
periodic orbits are linearly stability when , and are
linearly unstable when .Comment: 6 figure
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