Faster Algorithms for Multivariate Interpolation with Multiplicities and Simultaneous Polynomial Approximations

The interpolation step in the Guruswami-Sudan algorithm is a bivariate interpolation problem with multiplicities commonly solved in the literature using either structured linear algebra or basis reduction of polynomial lattices. This problem has been extended to three or more variables; for this generalization, all fast algorithms proposed so far rely on the lattice approach. In this paper, we reduce this multivariate interpolation problem to a problem of simultaneous polynomial approximations, which we solve using fast structured linear algebra. This improves the best known complexity bounds for the interpolation step of the list-decoding of Reed-Solomon codes, Parvaresh-Vardy codes, and folded Reed-Solomon codes. In particular, for Reed-Solomon list-decoding with re-encoding, our approach has complexity $\mathcal{O}\tilde{~}(\ell^{\omega-1}m^2(n-k))$, where $\ell,m,n,k$ are the list size, the multiplicity, the number of sample points and the dimension of the code, and $\omega$ is the exponent of linear algebra; this accelerates the previously fastest known algorithm by a factor of $\ell / m$.Comment: Version 2: Generalized our results about Problem 1 to distinct multiplicities. Added Section 4 which details several applications of our results to the decoding of Reed-Solomon codes (list-decoding with re-encoding technique, Wu algorithm, and soft-decoding). Reorganized the sections, added references and corrected typo

GPU Accelerated Approach to Numerical Linear Algebra and Matrix Analysis with CFD Applications

A GPU accelerated approach to numerical linear algebra and matrix analysis with CFD applications is presented. The works objectives are to (1) develop stable and efficient algorithms utilizing multiple NVIDIA GPUs with CUDA to accelerate common matrix computations, (2) optimize these algorithms through CPU/GPU memory allocation, GPU kernel development, CPU/GPU communication, data transfer and bandwidth control to (3) develop parallel CFD applications for Navier Stokes and Lattice Boltzmann analysis methods. Special consideration will be given to performing the linear algebra algorithms under certain matrix types (banded, dense, diagonal, sparse, symmetric and triangular). Benchmarks are performed for all analyses with baseline CPU times being determined to find speed-up factors and measure computational capability of the GPU accelerated algorithms. The GPU implemented algorithms used in this work along with the optimization techniques performed are measured against preexisting work and test matrices available in the NIST Matrix Market. CFD analysis looked to strengthen the assessment of this work by providing a direct engineering application to analysis that would benefit from matrix optimization techniques and accelerated algorithms. Overall, this work desired to develop optimization for selected linear algebra and matrix computations performed with modern GPU architectures and CUDA developer which were applied directly to mathematical and engineering applications through CFD analysis

Application of Computer Algebra in List Decoding

The amount of data that we use in everyday life (social media, stock analysis, satellite communication etc.) are increasing day by day. As a result, the amount of data needs to be traverse through electronic media as well as to store are rapidly growing and there exist several environmental effects that can damage these important data during travelling or while in storage devices. To recover correct information from noisy data, we do use error correcting codes. The most challenging work in this area is to have a decoding algorithm that can decode the code quite fast, in addition with the existence of the code that can tolerate highest amount of noise, so that we can have it in practice. List decoding is an active research area for last two decades. This research popularise in coding theory after the breakthrough work by Madhu Sudan where he used list decoding technique to correct errors that exceeds half the minimum distance of Reed Solomon codes. Towards the direction of code development that can reach theoretical limit of error correction, Guruswami-Rudra introduced folded Reed Solomon codes that reached at $1 - R - \epsilon.$ To decode this codes, one has to first interpolate a multivariate polynomial first and then have to factor out all possible roots. The difficulties that lies here are efficient interpolation, dealing with multiplicities smartly and efficient factoring. This thesis deals with all these cases in order to have folded Reed Solomon codes in practice

Multidimensional Wavelets and Computer Vision

This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing

The Sixth Copper Mountain Conference on Multigrid Methods, part 1

The Sixth Copper Mountain Conference on Multigrid Methods was held on 4-9 Apr. 1993, at Copper Mountain, CO. This book is a collection of many of the papers presented at the conference and as such represents the conference proceedings. NASA LaRC graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth