Iterative Solvers for Large, Dense Matrices

Stochastic Interpolation (SI) uses a continuous, centrally symmetric probability distribution function to interpolate a given set of data points, and splits the interpolation operator into a discrete deconvolution followed by a discrete convolution of the data. The method is particularly effective for large data sets, as it does not suffer from the problem of oversampling, where too many data points cause the interpolating function to oscillate wildly. Rather, the interpolation improves every time more data points are added. The method relies on the inversion of relatively large, dense matrices to solve Annx = b for x. Based on the probability distribution function chosen, the matrix Ann may have specific properties that make the problem of solving for x tractable. The iterative Shulz Jones Mayer (SJM) method relies on an initial guess, which is iterated to approximate A�1 nn . We present initial guesses that are guaranteed to converge quadratically for several classes of matrices, including diagonally and tri-diagonally dominant matrices and the structured matrices we encounter in the implementation of SI. We improve the method, creating the Polynomial Shulz Jones Mayer method, and take advantage of the more efficient matrix operations possible for Toeplitz matrices. We calculate error bounds and use those to improve the method’s accuracy, resulting in a method requiring O(nlogn) operations that returns x with double precision. The use of SI and PSJM is illustrated in interpolating functions and images in grey scale and color

Balanced k-Colorings

While discrepancy theory is normally only studied in the context of 2-colorings, we explore the problem of k-coloring, for k 2, a set of vertices to minimize imbalance among a family of subsets of vertices. The imbalance is the maximum, over all subsets in the family, of the largest difference between the size of any two color classes in that subset. The discrepancy is the minimum possible imbalance. We show that the discrepancy is always at most 4d \Gamma 3, where d (the "dimension") is the maximum number of subsets containing a common vertex. For 2-colorings, the bound on the discrepancy is at most maxf2d \Gamma 3; 2g. Finally, we prove that several restricted versions of computing the discrepancy are NP-complete. Key words: Discrepancy, Balance Theorem, NP completeness Corresponding author Email addresses: [email protected] (Therese C. Biedl), [email protected] (Eowyn Cenek), [email protected] (Timothy M. Chan), [email protected] (Erik D. Demaine), [email protected] (Martin L. Demaine), [email protected] (Rudolf Fleischer), [email protected] (Ming-Wei Wang). 1 The research was mainly done while the author was a PhD student at the University of Waterloo, Department of Computer Science 2 The research was mainly done while the author was a visiting associate professor at the University of Waterloo, Department of Computer Science Preprint submitted to Elsevier Science 28 September 2001

On simultaneous planar graph embeddings

AbstractWe consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another in which the mapping is not given. We present positive and negative results for the two versions of the problem. Among the positive results with given mapping, we show that we can embed two paths on an n×n grid, and two caterpillar graphs on a 3n×3n grid. Among the negative results with given mapping, we show that it is not always possible to simultaneously embed three paths or two general planar graphs. If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n)×O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O(n2)×O(n2) grid