Scalable iterative methods for sampling from massive Gaussian random vectors

Sampling from Gaussian Markov random fields (GMRFs), that is multivariate Gaussian ran- dom vectors that are parameterised by the inverse of their covariance matrix, is a fundamental problem in computational statistics. In this paper, we show how we can exploit arbitrarily accu- rate approximations to a GMRF to speed up Krylov subspace sampling methods. We also show that these methods can be used when computing the normalising constant of a large multivariate Gaussian distribution, which is needed for both any likelihood-based inference method. The method we derive is also applicable to other structured Gaussian random vectors and, in particu- lar, we show that when the precision matrix is a perturbation of a (block) circulant matrix, it is still possible to derive O(n log n) sampling schemes.Comment: 17 Pages, 4 Figure

Nested Fork-Join Queuing Networks and Their Application to Mobility Airfield Operations Analysis

A single-chain nested fork-join queuing network (FJQN) model of mobility airfield ground processing is proposed. In order to analyze the queuing network model, advances on two fronts are made. First, a general technique for decomposing nested FJQNs with probabilistic forks is proposed, which consists of incorporating feedback loops into the embedded Markov chain of the synchronization station, then using Marie\u27s Method to decompose the network. Numerical studies show this strategy to be effective, with less than two percent relative error in the approximate performance measures in most realistic cases. The second contribution is the identification of a quick, efficient method for solving for the stationary probabilities of the λn/Ck/r/N queue. Unpreconditioned Conjugate Gradient Squared is shown to be the method of choice in the context of decomposition using Marie\u27s Method, thus broadening the class of networks where the method is of practical use. The mobility airfield model is analyzed using the strategies described above, and accurate approximations of airfield performance measures are obtained in a fraction of the time needed for a simulation study. The proposed airfield modeling approach is especially effective for quick-look studies and sensitivity analysis

Bayesian inference for structured additive regression models for large-scale problems with applications to medical imaging

In der angewandten Statistik können Regressionsmodelle mit hochdimensionalen Koeffizienten auftreten, die sich nicht mit gewöhnlichen Computersystemen schätzen lassen. Dies betrifft unter anderem die Analyse digitaler Bilder unter Berücksichtigung räumlich-zeitlicher Abhängigkeiten, wie sie innerhalb der medizinisch-biologischen Forschung häufig vorkommen. In der vorliegenden Arbeit wird ein Verfahren formuliert, das in der Lage ist, Regressionsmodelle mit hochdimensionalen Koeffizienten und nicht-normalverteilten Zielgrößen unter moderaten Anforderungen an die benötigte Hardware zu schätzen. Hierzu wird zunächst im Rahmen strukturiert additiver Regressionsmodelle aufgezeigt, worin die Limitationen aktueller Inferenzansätze bei der Anwendung auf hochdimensionale Problemstellungen liegen, sowie Möglichkeiten diskutiert, diese zu umgehen. Darauf basierend wird ein Algorithmus formuliert, dessen Stärken und Schwächen anhand von Simulationsstudien analysiert werden. Darüber hinaus findet das Verfahren Anwendung in drei verschiedenen Bereichen der medizinisch-biologischen Bildgebung und zeigt dadurch, dass es ein vielversprechender Kandidat für die Beantwortung hochdimensionaler Fragestellungen ist.In applied statistics regression models with high-dimensional coefficients can occur which cannot be estimated using ordinary computers. Amongst others, this applies to the analysis of digital images taking spatio-temporal dependencies into account as they commonly occur within bio-medical research. In this thesis a procedure is formulated which allows to fit regression models with high-dimensional coefficients and non-normal response values requiring only moderate computational equipment. To this end, limitations of different inference strategies for structured additive regression models are demonstrated when applied to high-dimensional problems and possible solutions are discussed. Based thereon an algorithm is formulated whose strengths and weaknesses are subsequently analyzed using simulation studies. Furthermore, the procedure is applied to three different fields of bio-medical imaging from which can be concluded that the algorithm is a promising candidate for answering high-dimensional problems

[Activity of Institute for Computer Applications in Science and Engineering]

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science

Semiannual report

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science during the period 1 Oct. 1994 - 31 Mar. 1995

Implementing Probabilistic Numerical Solvers for Differential Equations

The numerical solution of differential equations underpins a large share of simulation methods that are used in the natural sciences and engineering, both in research and in industrial applications. The usability of a differential equation model depends, often crucially, on the choice of the simulation algorithm. Probabilistic numerical algorithms promise to combine efficient simulations with well-calibrated uncertainty quantification. Being able to handle various sources of uncertainty without a severely increased computational burden simplifies the combination of differential equation models with, for example, observational data and thereby improves the fusion of mechanistic and statistical information. However, until now, the general usability of probabilistic numerical solvers had not reached a level comparable to non-probabilistic approaches. A lack of numerical stability and scalability, combined with a strong focus on ordinary differential equations and initial value problems, put probabilistic numerical algorithms out of the scope of an implementation in the physical and the life sciences, which would require the efficient simulation of dynamics that may exhibit spatiotemporal patterns or could be constrained by boundary information. This thesis explains a series of contributions to the solution of this problem by discussing the implementation of a class of probabilistic numerical differential equation solvers that shares many features with collocation methods and with Gaussian filtering and smoothing: 1. A set of instructions for the numerically stable implementation of probabilistic numerical differential equation solvers that scales to high-dimensional problems. 2. The generalisation of solvers for ordinary-differential-equation-based initial value problems to boundary value problems and partial differential equations. Many of the techniques have already been implemented successfully in various software libraries for probabilistic numerical differential equation solvers. Altogether, the contributions improve the usability of existing and future probabilistic numerical algorithms. The simulation of challenging differential equation models and an application of the probabilistic numerical paradigm to real-world problems is no longer out of reach

Online QoS/Revenue Management for Third Generation Mobile Communication Networks

This thesis shows how online management of both quality of service (QoS) and provider revenue can be performed in third generation (3G) mobile networks by adaptive control of system parameters to changing traffic conditions. As a main result, this approach is based on a novel call admission control and bandwidth degradation scheme for real-time traffic. The admission controller considers real-time calls with two priority levels: calls of high priority have a guaranteed bit-rate, whereas calls of low priority can be temporarily degraded to a lower bit-rate in order to reduce forced termination of calls due to a handover failure. A second contribution constitutes the development of a Markov model for the admission controller that incorporates important features of 3G mobile networks, such as code division multiple access (CDMA) intra- and inter-cell interference and soft handover. Online evaluation of the Markov model enables a periodical adjustment of the threshold for maximal call degradation according to the currently measured traffic in the radio access network and a predefined goal for optimization. Using distinct optimization goals, this allows optimization of both QoS and provider revenue. Performance studies illustrate the effectiveness of the proposed approach and show that QoS and provider revenue can be increased significantly with a moderate degradation of low-priority calls. Compared with existing admission control policies, the overall utilization of cell capacity is significantly improved using the proposed degradation scheme, which can be considered as an 'on demand' reservation of cell capacity.To enable online QoS/revenue management of both real-time and non real-time services, accurate analytical traffic models for non real-time services are required. This thesis identifies the batch Markovian arrival process (BMAP) as the analytically tractable model of choice for the joint characterization of packet arrivals and packet lengths. As a key idea, the BMAP is customized such that different packet lengths are represented by batch sizes of arrivals. Thus, the BMAP enables the 'two-dimensional', i.e., joint, characterization of packet arrivals and packet lengths, and is able to capture correlations between the packet arrival process and the packet length process. A novel expectation maximization (EM) algorithm is developed, and it is shown how to utilize the randomization technique and a stable calculation of Poisson jump probabilities effectively for computing time-dependent conditional expectations of a continuous-time Markov chain required by the expectation step of the EM algorithm. This methodological work enables the EM algorithm to be both efficient and numerical robust and constitutes an important step towards effective, analytically/numerically tractable traffic models. Case studies of measured IP traffic with different degrees of traffic burstiness evidently demonstrate the advantages of the BMAP modeling approach over other widely used analytically tractable models and show that the joint characterization of packet arrivals and packet lengths is decisively for realistic traffic modeling at packet level

Experiments with two-stage iterative solvers and preconditioned Krylov subspace methods on nearly completely decomposable Markov chains

Ankara : Department of Computer Engineering and Information Science and the Institute of Engineering and Science of Bilkent University, 1997.Thesis (Master's) -- Bilkent University, 1997.Includes bibliographical references leaves 121-124Gueaieb, WailM.S