Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A^(-α/2)b, where A ∈ ℝ^(n×n) is a large, sparse symmetric positive definite matrix and b ∈ ℝ^n is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LL^T is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L^(-T)z, with x = A^(-1/2)z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form ϕn = A^(-α/2)b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t^(-α/2) and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A^(-α/2)b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.

Urban dynamic origin-destination matrices estimation

The aim of this paper is to explore a new approach to obtain better traffic demand (Origin-Destination, OD matrices) for dense urban networks. From reviewing existing methods, from static to dynamic OD matrix evaluation, possible deficiencies in the approach could be identified: traffic assignment details for complex urban network and lacks in dynamic approach. To improve the global process of traffic demand estimation, this paper is focussing on a new methodology to determine dynamic OD matrices for urban areas characterized by complex route choice situation and high level of traffic controls. An iterative bi-level approach will be used, the Lower level (traffic assignment) problem will determine, dynamically, the utilisation of the network by vehicles using heuristic data from mesoscopic traffic simulator and the Upper level (matrix adjustment) problem will proceed to an OD estimation using optimization Kalman filtering technique. In this way, a full dynamic and continuous estimation of the final OD matrix could be obtained. First results of the proposed approach and remarks are presented.

Mineralized human primary osteoblast matrices as a model system to analyse interactions of prostate cancer cells with the bone microenvironment

Prostate cancer metastasis is reliant on the reciprocal interactions between cancer cells and the bone niche/micro-environment. The production of suitable matrices to study metastasis, carcinogenesis and in particular prostate cancer/bone micro-environment interaction has been limited to specific protein matrices or matrix secreted by immortalised cell lines that may have undergone transformation processes altering signaling pathways and modifying gene or receptor expression. We hypothesize that matrices produced by primary human osteoblasts are a suitable means to develop an in vitro model system for bone metastasis research mimicking in vivo conditions. We have used a decellularized matrix secreted from primary human osteoblasts as a model for prostate cancer function in the bone micro-environment. We show that this collagen I rich matrix is of fibrillar appearance, highly mineralized, and contains proteins, such as osteocalcin, osteonectin and osteopontin, and growth factors characteristic of bone extracellular matrix (ECM). LNCaP and PC3 cells grown on this matrix, adhere strongly, proliferate, and express markers consistent with a loss of epithelial phenotype. Moreover, growth of these cells on the matrix is accompanied by the induction of genes associated with attachment, migration, increased invasive potential, Ca2+ signaling and osteolysis. In summary, we show that growth of prostate cancer cells on matrices produced by primary human osteoblasts mimics key features of prostate cancer bone metastases and thus is a suitable model system to study the tumor/bone micro-environment interaction in this disease.

Osteogenic induction of human bone marrow-derived mesenchymal progenitor cells in novel synthetic polymer-hydrogel matrices

The aim of this project was to investigate the in vitro osteogenic potential of human mesenchymal progenitor cells in novel matrix architectures built by means of a three-dimensional bioresorbable synthetic framework in combination with a hydrogel. Human mesenchymal progenitor cells (hMPCs) were isolated from a human bone marrow aspirate by gradient centrifugation. Before in vitro engineering of scaffold-hMPC constructs, the adipogenic and osteogenic differentiation potential was demonstrated by staining of neutral lipids and induction of bone-specific proteins, respectively. After expansion in monolayer cultures, the cells were enzymatically detached and then seeded in combination with a hydrogel into polycaprolactone (PCL) and polycaprolactone-hydroxyapatite (PCL-HA) frameworks. This scaffold design concept is characterized by novel matrix architecture, good mechanical properties, and slow degradation kinetics of the framework and a biomimetic milieu for cell delivery and proliferation. To induce osteogenic differentiation, the specimens were cultured in an osteogenic cell culture medium and were maintained in vitro for 6 weeks. Cellular distribution and viability within three-dimensional hMPC bone grafts were documented by scanning electron microscopy, cell metabolism assays, and confocal laser microscopy. Secretion of the osteogenic marker molecules type I procollagen and osteocalcin was analyzed by semiquantitative immunocytochemistry assays. Alkaline phosphatase activity was visualized by p-nitrophenyl phosphate substrate reaction. During osteogenic stimulation, hMPCs proliferated toward and onto the PCL and PCL-HA scaffold surfaces and metabolic activity increased, reaching a plateau by day 15. The temporal pattern of bone-related marker molecules produced by in vitro tissue-engineered scaffold-cell constructs revealed that hMPCs differentiated better within the biomimetic matrix architecture along the osteogenic lineage.

In vitro characterization of natural and synthetic dermal matrices cultured with human dermal fibroblasts

The ideal dermal matrix should be able to provide the right biological and physical environment to ensure homogenous cell and extracellular matrix (ECM) distribution, as well as the right size and morphology of the neo-tissue required. Four natural and synthetic 3D matrices were evaluated in vitro as dermal matrices, namely (1) equine collagen foam, TissuFleece®, (2) acellular dermal replacement, Alloderm®, (3) knitted poly(lactic-co-glycolic acid) (10:90)–poly(-caprolactone) (PLGA–PCL) mesh, (4) chitosan scaffold. Human dermal fibroblasts were cultured on the specimens over 3 weeks. Cell morphology, distribution and viability were assessed by electron microscopy, histology and confocal laser microscopy. Metabolic activity and DNA synthesis were analysed via MTS metabolic assay and [3H]-thymidine uptake, while ECM protein expression was determined by immunohistochemistry. TissuFleece®, Alloderm® and PLGA–PCL mesh supported cell attachment, proliferation and neo-tissue formation. However, TissuFleece® contracted to 10% of the original size while Alloderm® supported cell proliferation predominantly on the surface of the material. PLGA–PCL mesh promoted more homogenous cell distribution and tissue formation. Chitosan scaffolds did not support cell attachment and proliferation. These results demonstrated that physical characteristics including porosity and mechanical stability to withstand cell contraction forces are important in determining the success of a dermal matrix material.

Engineered silk fibroin protein 3D matrices for in vitro tumor model

3D in vitro model systems that are able to mimic the in vivo microenvironment are now highly sought after in cancer research. Antheraea mylitta silk fibroin protein matrices were investigated as potential biomaterial for in vitro tumor modeling. We compared the characteristics of MDA-MB-231 cells on A. mylitta, Bombyx mori silk matrices, Matrigel, and tissue culture plates. The attachment and morphology of the MDA-MB-231 cell line on A. mylitta silk matrices was found to be better than on B. mori matrices and comparable to Matrigel and tissue culture plates. The cells grown in all 3D cultures showed more MMP-9 activity, indicating a more invasive potential. In comparison to B. mori fibroin, A. mylitta fibroin not only provided better cell adhesion, but also improved cell viability and proliferation. Yield coefficient of glucose consumed to lactate produced by cells on 3D A. mylitta fibroin was found to be similar to that of cancer cells in vivo. LNCaP prostate cancer cells were also cultured on 3D A. mylitta fibroin and they grew as clumps in long term culture. The results indicate that A. mylitta fibroin scaffold can provide an easily manipulated microenvironment system to investigate individual factors such as growth factors and signaling peptides, as well as evaluation of anticancer drugs.

Structural and textural variations in carbonaceous chondrite matrices

Recent studies of C2 carbonaceous chondrite matrices using high resolu­tion transmission electron microscopy (HRTEM)have shown that structural details of the matrix minerals can be imaged [1-4]. The Murchison and Mighei matrices contain minerals having ordered and disordered mixed-layer structures [1,3,4] in addition to chrysotile- and lizardite-type structures [2].

Towards the retrieval of accurate OD matrices from Bluetooth data : lessons learned from 2 years of data

The Bluetooth technology is being increasingly used to track vehicles throughout their trips, within urban networks and across freeway stretches. One important opportunity offered by this type of data is the measurement of Origin-Destination patterns, emerging from the aggregation and clustering of individual trips. In order to obtain accurate estimations, however, a number of issues need to be addressed, through data filtering and correction techniques. These issues mainly stem from the use of the Bluetooth technology amongst drivers, and the physical properties of the Bluetooth sensors themselves. First, not all cars are equipped with discoverable Bluetooth devices and the Bluetooth-enabled vehicles may belong to some small socio-economic groups of users. Second, the Bluetooth datasets include data from various transport modes; such as pedestrian, bicycles, cars, taxi driver, buses and trains. Third, the Bluetooth sensors may fail to detect all of the nearby Bluetooth-enabled vehicles. As a consequence, the exact journey for some vehicles may become a latent pattern that will need to be extracted from the data. Finally, sensors that are in close proximity to each other may have overlapping detection areas, thus making the task of retrieving the correct travelled path even more challenging. The aim of this paper is twofold. We first give a comprehensive overview of the aforementioned issues. Further, we propose a methodology that can be followed, in order to cleanse, correct and aggregate Bluetooth data. We postulate that the methods introduced by this paper are the first crucial steps that need to be followed in order to compute accurate Origin-Destination matrices in urban road networks.

Retrieving dynamic origin-destination matrices from Bluetooth data

The Bluetooth technology is being increasingly used, among the Automated Vehicle Identification Systems, to retrieve important information about urban networks. Because the movement of Bluetooth-equipped vehicles can be monitored, throughout the network of Bluetooth sensors, this technology represents an effective means to acquire accurate time dependant Origin Destination information. In order to obtain reliable estimations, however, a number of issues need to be addressed, through data filtering and correction techniques. Some of the main challenges inherent to Bluetooth data are, first, that Bluetooth sensors may fail to detect all of the nearby Bluetooth-enabled vehicles. As a consequence, the exact journey for some vehicles may become a latent pattern that will need to be estimated. Second, sensors that are in close proximity to each other may have overlapping detection areas, thus making the task of retrieving the correct travelled path even more challenging. The aim of this paper is twofold: to give an overview of the issues inherent to the Bluetooth technology, through the analysis of the data available from the Bluetooth sensors in Brisbane; and to propose a method for retrieving the itineraries of the individual Bluetooth vehicles. We argue that estimating these latent itineraries, accurately, is a crucial step toward the retrieval of accurate dynamic Origin Destination Matrices.