Analytical Methods for Structured Matrix Computations

The design of fast algorithms is not only about achieving faster speeds but also about retaining the ability to control the error and numerical stability. This is crucial to the reliability of computed numerical solutions. This dissertation studies topics related to structured matrix computations with an emphasis on their numerical analysis aspects and algorithms. The methods discussed here are all based on rich analytical results that are mathematically justified. In chapter 2, we present a series of comprehensive error analyses to an analytical matrix compression method and it serves as a theoretical explanation of the proxy point method. These results are also important instructions on optimizing the performance. In chapter 3, we propose a non-Hermitian eigensolver by combining HSS matrix techniques with a contour-integral based method. Moreover, probabilistic analysis enables further acceleration of the method in addition to manipulating the HSS representation algebraically. An application of the HSS matrix is discussed in chapter 4 where we design a structured preconditioner for linear systems generated by AIIM. We improve the numerical stability for the matrix-free HSS construction process and make some additional modifications tailored to this particular problem

General Analysis Tool Box for Controlled Perturbation

The implementation of reliable and efficient geometric algorithms is a challenging task. The reason is the following conflict: On the one hand, computing with rounded arithmetic may question the reliability of programs while, on the other hand, computing with exact arithmetic may be too expensive and hence inefficient. One solution is the implementation of controlled perturbation algorithms which combine the speed of floating-point arithmetic with a protection mechanism that guarantees reliability, nonetheless. This paper is concerned with the performance analysis of controlled perturbation algorithms in theory. We answer this question with the presentation of a general analysis tool box. This tool box is separated into independent components which are presented individually with their interfaces. This way, the tool box supports alternative approaches for the derivation of the most crucial bounds. We present three approaches for this task. Furthermore, we have thoroughly reworked the concept of controlled perturbation in order to include rational function based predicates into the theory; polynomial based predicates are included anyway. Even more we introduce object-preserving perturbations. Moreover, the tool box is designed such that it reflects the actual behavior of the controlled perturbation algorithm at hand without any simplifying assumptions.Comment: 90 pages, 30 figure

Blind adaptive constrained reduced-rank parameter estimation based on constant modulus design for CDMA interference suppression

This paper proposes a multistage decomposition for blind adaptive parameter estimation in the Krylov subspace with the code-constrained constant modulus (CCM) design criterion. Based on constrained optimization of the constant modulus cost function and utilizing the Lanczos algorithm and Arnoldi-like iterations, a multistage decomposition is developed for blind parameter estimation. A family of computationally efficient blind adaptive reduced-rank stochastic gradient (SG) and recursive least squares (RLS) type algorithms along with an automatic rank selection procedure are also devised and evaluated against existing methods. An analysis of the convergence properties of the method is carried out and convergence conditions for the reduced-rank adaptive algorithms are established. Simulation results consider the application of the proposed techniques to the suppression of multiaccess and intersymbol interference in DS-CDMA systems

Tamper-Resistant Arithmetic for Public-Key Cryptography

Cryptographic hardware has found many uses in many ubiquitous and pervasive security devices with a small form factor, e.g. SIM cards, smart cards, electronic security tokens, and soon even RFIDs. With applications in banking, telecommunication, healthcare, e-commerce and entertainment, these devices use cryptography to provide security services like authentication, identification and confidentiality to the user. However, the widespread adoption of these devices into the mass market, and the lack of a physical security perimeter have increased the risk of theft, reverse engineering, and cloning. Despite the use of strong cryptographic algorithms, these devices often succumb to powerful side-channel attacks. These attacks provide a motivated third party with access to the inner workings of the device and therefore the opportunity to circumvent the protection of the cryptographic envelope. Apart from passive side-channel analysis, which has been the subject of intense research for over a decade, active tampering attacks like fault analysis have recently gained increased attention from the academic and industrial research community. In this dissertation we address the question of how to protect cryptographic devices against this kind of attacks. More specifically, we focus our attention on public key algorithms like elliptic curve cryptography and their underlying arithmetic structure. In our research we address challenges such as the cost of implementation, the level of protection, and the error model in an adversarial situation. The approaches that we investigated all apply concepts from coding theory, in particular the theory of cyclic codes. This seems intuitive, since both public key cryptography and cyclic codes share finite field arithmetic as a common foundation. The major contributions of our research are (a) a generalization of cyclic codes that allow embedding of finite fields into redundant rings under a ring homomorphism, (b) a new family of non-linear arithmetic residue codes with very high error detection probability, (c) a set of new low-cost arithmetic primitives for optimal extension field arithmetic based on robust codes, and (d) design techniques for tamper resilient finite state machines

Optimization and Applications of Modern Wireless Networks and Symmetry

Due to the future demands of wireless communications, this book focuses on channel coding, multi-access, network protocol, and the related techniques for IoT/5G. Channel coding is widely used to enhance reliability and spectral efficiency. In particular, low-density parity check (LDPC) codes and polar codes are optimized for next wireless standard. Moreover, advanced network protocol is developed to improve wireless throughput. This invokes a great deal of attention on modern communications

Minimizing Communication for Eigenproblems and the Singular Value Decomposition

Algorithms have two costs: arithmetic and communication. The latter represents the cost of moving data, either between levels of a memory hierarchy, or between processors over a network. Communication often dominates arithmetic and represents a rapidly increasing proportion of the total cost, so we seek algorithms that minimize communication. In \cite{BDHS10} lower bounds were presented on the amount of communication required for essentially all $O(n^3)$-like algorithms for linear algebra, including eigenvalue problems and the SVD. Conventional algorithms, including those currently implemented in (Sca)LAPACK, perform asymptotically more communication than these lower bounds require. In this paper we present parallel and sequential eigenvalue algorithms (for pencils, nonsymmetric matrices, and symmetric matrices) and SVD algorithms that do attain these lower bounds, and analyze their convergence and communication costs.Comment: 43 pages, 11 figure