Unconventional computations of generalized inverses

Import 22/07/2015V tomto textu se budeme zabývat inverzními maticemi a jak takové matice spočítat. Poté, pomocí podobného algoritmu, zkusíme spočítat zobecněné inverze matic, které jsou regulární. Budeme hledat způsob, kterým převedeme singulární matici na regulární, abychom mohli spočítat její inverzi a poté jak najít blok takové inverze, který bude právě onou zobecněnou inverzí. Další věcí kterou zde zmíníme je aplikace zobecněné inverze a její využití při řešení Stouksova problému.This text deals with inverse matrices, where we discuss some of the ways how to calculate an inverse to a matrix. Then, using a similar algorithm, we try to calculate generalized inverses of matrices that are singular. We will find a way, how to make such matrix singular, so that it's inverse can be calculated and then how a block of this inverse is the generalized inverse. Another thing we are going to discuss is the application of generalized inverses in solving the Stokes problem.470 - Katedra aplikované matematikyvýborn

Krylov subspace methods and their generalizations for solving singular linear operator equations with applications to continuous time Markov chains

Viele Resultate über MR- und OR-Verfahren zur Lösung linearer Gleichungssysteme bleiben (in leicht modifizierter Form) gültig, wenn der betrachtete Operator nicht invertierbar ist. Neben dem für reguläre Probleme charakteristischen Abbruchverhalten, kann bei einem singulären Gleichungssystem auch ein so genannter singulärer Zusammenbruch auftreten. Für beide Fälle werden verschiedene Charakterisierungen angegeben. Die Unterrauminverse, eine spezielle verallgemeinerte Inverse, beschreibt die Näherungen eines MR-Unterraumkorrektur-Verfahrens. Für Krylov-Unterräume spielt die Drazin-Inverse eine Schlüsselrolle. Bei Krylov-Unterraum-Verfahren kann a-priori entschieden werden, ob ein regulärer oder ein singulärer Abbruch auftritt. Wir können zeigen, dass ein Krylov-Verfahren genau dann für beliebige Startwerte eine Lösung des linearen Gleichungssystems liefert, wenn der Index der Matrix nicht größer als eins und das Gleichungssystem konsistent ist. Die Berechnung stationärer Zustandsverteilungen zeitstetiger Markov-Ketten mit endlichem Zustandsraum stellt eine praktische Aufgabe dar, welche die Lösung eines singulären linearen Gleichungssystems erfordert. Die Eigenschaften der Übergangs-Halbgruppe folgen aus einfachen Annahmen auf rein analytischem und matrixalgebrischen Wege. Insbesondere ist die erzeugende Matrix eine singuläre M-Matrix mit Index 1. Ist die Markov-Kette irreduzibel, so ist die stationäre Zustandsverteilung eindeutig bestimmt

Alternating least squares as moving subspace correction

In this note we take a new look at the local convergence of alternating optimization methods for low-rank matrices and tensors. Our abstract interpretation as sequential optimization on moving subspaces yields insightful reformulations of some known convergence conditions that focus on the interplay between the contractivity of classical multiplicative Schwarz methods with overlapping subspaces and the curvature of low-rank matrix and tensor manifolds. While the verification of the abstract conditions in concrete scenarios remains open in most cases, we are able to provide an alternative and conceptually simple derivation of the asymptotic convergence rate of the two-sided block power method of numerical algebra for computing the dominant singular subspaces of a rectangular matrix. This method is equivalent to an alternating least squares method applied to a distance function. The theoretical results are illustrated and validated by numerical experiments.Comment: 20 pages, 4 figure

Projected Krylov methods for solving non-symmetric two-by-two block linear systems arising from fictitious domain formulations

The paper deals with the solution of large non-symmetric two-by-two block linear systems with a singular leading submatrix. Our algorithm consists of two levels. The outer level combines the Schur complement reduction with the orthogonal projectors that leads to the linear equation on subspaces. To solve this equation, we use a Krylov-type method representing the inner level of the algorithm. We propose a general technique how to get from the standard Krylov methods their projected variants generating iterations on subspaces. Then we derive the projected GMRES. The efficiency of our approach is illustrated by examples arising from the combination of the fictitious domain and FETI method

Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation" algorithm capable of harnessing this exponential advantage, that can apply polynomial transformations to the singular values of a block of a unitary, generalizing the optimal Hamiltonian simulation results of Low and Chuang. The proposed quantum circuits have a very simple structure, often give rise to optimal algorithms and have appealing constant factors, while usually only use a constant number of ancilla qubits. We show that singular value transformation leads to novel algorithms. We give an efficient solution to a certain "non-commutative" measurement problem and propose a new method for singular value estimation. We also show how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum. Finally, as a quantum machine learning application we show how to efficiently implement principal component regression. "Singular value transformation" is conceptually simple and efficient, and leads to a unified framework of quantum algorithms incorporating a variety of quantum speed-ups. We illustrate this by showing how it generalizes a number of prominent quantum algorithms, including: optimal Hamiltonian simulation, implementing the Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude amplification, robust oblivious amplitude amplification, fast QMA amplification, fast quantum OR lemma, certain quantum walk results and several quantum machine learning algorithms. In order to exploit the strengths of the presented method it is useful to know its limitations too, therefore we also prove a lower bound on the efficiency of singular value transformation, which often gives optimal bounds.Comment: 67 pages, 1 figur

An introduction to the theory of generalized matrix invertibility

Literature survey on pseudo invertibility of matrice

Measuring the confinement of probabilistic systems

AbstractIn this paper we lay the semantic basis for a quantitative security analysis of probabilistic systems by introducing notions of approximate confinement based on various process equivalences. We re-cast the operational semantics classically expressed via probabilistic transition systems (PTS) in terms of linear operators and we present a technique for defining approximate semantics as probabilistic abstract interpretations of the PTS semantics. An operator norm is then used to quantify this approximation. This provides a quantitative measure ɛ of the indistinguishability of two processes and therefore of their confinement. In this security setting a statistical interpretation is then given of the quantity ɛ which relates it to the number of tests needed to breach the security of the system

Properties of the matrix A- XY

As a management problem the identification of stakeholders is not easily solved. It comprises a modelling and a normative issue, which need to be solved in connection with each other. In stakeholder literature knowledge can be found, e.g. on various stakeholder categorizations, that could be useful for the modelling issue. However, the normative issue remains unresolved. Furthermore, the modelling of the so-called stakeholder category “the affected” is even more difficult. Nevertheless, this group holds justified interests in aspects of organizational activity and are, for that reason, legitimate stakeholders. In this article it is explored to what extent Critical Systems Heuristics can help resolving the managerial problem of identifying stakeholders, particularly the affected. Critical Systems Heuristics can be viewed a modelling methodology. The normative aspect of modelling is crucial in this methodology. Using the distinction between “the involved” and “the affected” a variety of boundary judgments is discussed. Special attention is given to the so-called “witness” as a representative of the affected