Majorization algorithms for inspecting circles, ellipses, squares, rectangles, and rhombi

In several disciplines, as diverse as shape analysis, locationtheory, quality control, archaeology, and psychometrics, it can beof interest to fit a circle through a set of points. We use theresult that it suffices to locate a center for which the varianceof the distances from the center to a set of given points isminimal. In this paper, we propose a new algorithm based oniterative majorization to locate the center. This algorithm isguaranteed to yield a series nonincreasing variances until astationary point is obtained. In all practical cases, thestationary point turns out to be a local minimum. Numericalexperiments show that the majorizing algorithm is stable and fast.In addition, we extend the method to fit other shapes, such as asquare, an ellipse, a rectangle, and a rhombus by making use ofthe class of $l_p$ distances and dimension weighting. In addition,we allow for rotations for shapes that might be rotated in theplane. We illustrate how this extended algorithm can be used as atool for shape recognition.iterative majorization;location;optimization;shape analysis

Problem-orientable numerical algorithm for modelling multi-dimensional radiative MHD flows in astrophysics -- the hierarchical solution scenario

We present a hierarchical approach for enhancing the robustness of numerical solvers for modelling radiative MHD flows in multi-dimensions. This approach is based on clustering the entries of the global Jacobian in a hierarchical manner that enables employing a variety of solution procedures ranging from a purely explicit time-stepping up to fully implicit schemes. A gradual coupling of the radiative MHD equation with the radiative transfer equation in higher dimensions is possible. Using this approach, it is possible to follow the evolution of strongly time-dependent flows with low/high accuracies and with efficiency comparable to explicit methods, as well as searching quasi-stationary solutions for highly viscous flows. In particular, it is shown that the hierarchical approach is capable of modelling the formation of jets in active galactic nuclei and reproduce the corresponding spectral energy distribution with a reasonable accuracy.Comment: 28 pages, 9 figure

Large N Dynamics of Dimensionally Reduced 4D SU(N) Super Yang-Mills Theory

We perform Monte Carlo simulations of a supersymmetric matrix model, which is obtained by dimensional reduction of 4D SU(N) super Yang-Mills theory. The model can be considered as a four-dimensional counterpart of the IIB matrix model. We extract the space-time structure represented by the eigenvalues of bosonic matrices. In particular we compare the large N behavior of the space-time extent with the result obtained from a low energy effective theory. We measure various Wilson loop correlators which represent string amplitudes and we observe a nontrivial universal scaling in N. We also observe that the Eguchi-Kawai equivalence to ordinary gauge theory does hold at least within a finite range of scale. Comparison with the results for the bosonic case clarifies the role of supersymmetry in the large N dynamics. It does affect the multi-point correlators qualitatively, but the Eguchi-Kawai equivalence is observed even in the bosonic case.Comment: 35 pages, 17 figure

Data Structures and Algorithms for Efficient Solution of Simultaneous Linear Equations from 3-D Ice Sheet Models

Two current software packages for solving large systems of sparse simultaneous l~neare equations are evaluated in terms of their applicability to solving systems of equations generated by the University of Maine Ice Sheet Model. SuperLU, the first package, has been developed by researchers at the University of California at Berkeley and the Lawrence Berkeley National Laboratory. UMFPACK, the second package, has been developed by T. A. Davis of the University of Florida who has ties with the U. C. Berkeley researchers as well as European researchers. Both packages are direct solvers that use LU factorization with forward and backward substitution. The University of Maine Ice Sheet Model uses the finite element method to solve partial differential equations that describe ice thickness, velocity,and temperature throughout glaciers as functions of position and t~me. The finite element method generates systems of linear equations having tens of thousands of variables and one hundred or so non-zero coefficients per equation. Matrices representing these systems of equations may be strictly banded or banded with right and lower borders. In order to efficiently Interface the software packages with the ice sheet model, a modified compressed column data structure and supporting routines were designed and written. The data structure interfaces directly with both software packages and allows the ice sheet model to access matrix coefficients by row and column number in roughly 100 nanoseconds while only storing non-zero entries of the matrix. No a priori knowledge of the matrix\u27s sparsity pattern is required. Both software packages were tested with matrices produced by the model and performance characteristics were measured arid compared with banded Gaussian elimination. When combined with high performance basic linear algebra subprograms (BLAS), the packages are as much as 5 to 7 times faster than banded Gaussian elimination. The BLAS produced by K. Goto of the University of Texas was used. Memory usage by the packages varted from slightly more than banded Gaussian elimination with UMFPACK, to as much as a 40% savings with SuperLU. In addition, the packages provide componentwise backward error measures and estimates of the matrix\u27s condition number. SuperLU is available for parallel computers as well as single processor computers. UMPACK is only for single processor computers. Both packages are also capable of efficiently solving the bordered matrix problem

Majorization algorithms for inspecting circles, ellipses, squares, rectangles, and rhombi

In several disciplines, as diverse as shape analysis, location theory, quality control, archaeology, and psychometrics, it can be of interest to fit a circle through a set of points. We use the result that it suffices to locate a center for which the variance of the distances from the center to a set of given points is minimal. In this paper, we propose a new algorithm based on iterative majorization to locate the center. This algorithm is guaranteed to yield a series nonincreasing variances until a stationary point is obtained. In all practical cases, the stationary point turns out to be a local minimum. Numerical experiments show that the majorizing algorithm is stable and fast. In addition, we extend the method to fit other shapes, such as a square, an ellipse, a rectangle, and a rhombus by making use of the class of $l_p$ distances and dimension weighting. In addition, we allow for rotations for shapes that might be rotated in the plane. We illustrate how this extended algorithm can be used as a tool for shape recognition

Spectral Ewald Acceleration of Stokesian Dynamics for polydisperse suspensions

In this work we develop the Spectral Ewald Accelerated Stokesian Dynamics (SEASD), a novel computational method for dynamic simulations of polydisperse colloidal suspensions with full hydrodynamic interactions. SEASD is based on the framework of Stokesian Dynamics (SD) with extension to compressible solvents, and uses the Spectral Ewald (SE) method [Lindbo & Tornberg, J. Comput. Phys. 229 (2010) 8994] for the wave-space mobility computation. To meet the performance requirement of dynamic simulations, we use Graphic Processing Units (GPU) to evaluate the suspension mobility, and achieve an order of magnitude speedup compared to a CPU implementation. For further speedup, we develop a novel far-field block-diagonal preconditioner to reduce the far-field evaluations in the iterative solver, and SEASD-nf, a polydisperse extension of the mean-field Brownian approximation of Banchio & Brady [J. Chem. Phys. 118 (2003) 10323]. We extensively discuss implementation and parameter selection strategies in SEASD, and demonstrate the spectral accuracy in the mobility evaluation and the overall $\mathcal{O}(N\log N)$ computation scaling. We present three computational examples to further validate SEASD and SEASD-nf in monodisperse and bidisperse suspensions: the short-time transport properties, the equilibrium osmotic pressure and viscoelastic moduli, and the steady shear Brownian rheology. Our validation results show that the agreement between SEASD and SEASD-nf is satisfactory over a wide range of parameters, and also provide significant insight into the dynamics of polydisperse colloidal suspensions.Comment: 39 pages, 21 figure