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.

Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions

We consider time-space fractional reaction diffusion equations in two dimensions. This equation is obtained from the standard reaction diffusion equation by replacing the first order time derivative with the Caputo fractional derivative, and the second order space derivatives with the fractional Laplacian. Using the matrix transfer technique proposed by Ilic, Liu, Turner and Anh [Fract. Calc. Appl. Anal., 9:333--349, 2006] and the numerical solution strategy used by Yang, Turner, Liu, and Ilic [SIAM J. Scientific Computing, 33:1159--1180, 2011], the solution of the time-space fractional reaction diffusion equations in two dimensions can be written in terms of a matrix function vector product $f(A)b$ at each time step, where $A$ is an approximate matrix representation of the standard Laplacian. We use the finite volume method over unstructured triangular meshes to generate the matrix $A$, which is therefore non-symmetric. However, the standard Lanczos method for approximating $f(A)b$ requires that $A$ is symmetric. We propose a simple and novel transformation in which the standard Lanczos method is still applicable to find $f(A)b$, despite the loss of symmetry. Numerical results are presented to verify the accuracy and efficiency of our newly proposed numerical solution strategy.

Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach.

A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.

A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of fractional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction–diffusion equations. The computational heart of this approach is the efficient computation of a matrix-function-vector product f(A)bf(A)b, where A A is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is show-cased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.

Gamma ray-induced bystander effect in tumour glioblastoma cells: a specific study on cell survival, cytokine release and cytokine receptors.

Recent experimental evidence has challenged the paradigm according to which radiation traversal through the nucleus of a cell is a prerequisite for producing genetic changes or biological responses. Thus, unexposed cells in the vicinity of directly irradiated cells or recipient cells of medium from irradiated cultures can also be affected. The aim of the present study was to evaluate, by means of the medium transfer technique, whether interleukin-8 and its receptor (CXCR1) may play a role in the bystander effect after gamma irradiation of T98G cells in vitro. In fact the cell specificity in inducing the bystander effect and in receiving the secreted signals that has been described suggests that not only the ability to release the cytokines but also the receptor profiles are likely to modulate the cell responses and the final outcome. The dose and time dependence of the cytokine release into the medium, quantified using an enzyme linked immunosorbent assay, showed that radiation causes alteration in the release of interleukin-8 from exposed cells in a dose-independent but time-dependent manner. The relative receptor expression was also affected in exposed and bystander cells.

Polarization spectral synthesis for Type Ia supernova explosion models

We present a Monte Carlo radiative transfer technique for calculating synthetic spectropolarimetry for multidimensional supernova explosion models. The approach utilizes 'virtual-packets' that are generated during the propagation of the Monte Carlo quanta and used to compute synthetic observables for specific observer orientations. Compared to extracting synthetic observables by direct binning of emergent Monte Carlo quanta, this virtual-packet approach leads to a substantial reduction in the Monte Carlo noise. This is not only vital for calculating synthetic spectropolarimetry (since the degree of polarization is typically very small) but also useful for calculations of light curves and spectra. We first validate our approach via application of an idealized test code to simple geometries. We then describe its implementation in the Monte Carlo radiative transfer code ARTIS and present test calculations for simple models for Type Ia supernovae. Specifically, we use the well-known one-dimensional W7 model to verify that our scheme can accurately recover zero polarization from a spherical model, and to demonstrate the reduction in Monte Carlo noise compared to a simple packet-binning approach. To investigate the impact of aspherical ejecta on the polarization spectra, we then use ARTIS to calculate synthetic observables for prolate and oblate ellipsoidal models with Type Ia supernova compositions.

Avaliação clínica de pacientes submetidos à colocação de implantes zigomáticos pela técnica de Stella & Warner

This study aimed to evaluate patients who underwent placement of zygomatic implants technique by Stella & Warner, considering the survival of conventional and zygomatic implants, sinus health and level of patient satisfaction in relation to oral rehabilitation. We evaluated 28 patients where 14 had received conventional and zygomatic implants, being rehabilitated with implant-fixed dentures (group 1) and 14 were rehabilitated only with conventional implants and implant-fixed dentures (group 2). The study had four phases, represented by radiographic evaluation of implants (stage I), clinical evaluation (stage II), assessing the health of the maxillary sinus (stage III) and a questionnaire to measure satisfaction of rehabilitation with fixed prosthesis implant Total -backed (stage IV). Group 2 underwent only stage IV, while group 1 participated in all stages. Descriptive analysis and statistics were performed, using the t test for independent samples in the evaluation of phase IV. The results demonstrated that the technique of Stella & Warner proved effective, allowing a high survival rate of conventional implants and zygomatic (100%), considering a minimum follow-up of 15 months and maximum 53 months after prosthetic rehabilitation. There were no pathological changes in tissues periimplants conventional and zygomatic implants analyzed. Radiographic findings showed satisfactory levels bone implants in the oral rehabilitation with conventional zygomatic implants and a good positioning of the apex of the zygomatic implants over the zygomatic bone. The presence of the zygomatic implant did not cause sinus and the t test showed a satisfaction index lower in group 1 compared with group 2. The zygomatic implant placement technique by Stella & Warner proved to be a predictable technique with high survival rate in patients with atrophic jaws, necessitating long-term follow-up to confirm the initial findings of the study

Marking with pigments for identification of flies in experimental populations of Megaselia scalaris Loew

A técnica de marcação de insetos de Tadei & Mourão (1976) é, até o momento, o único método experimental que possibilita determinar a idade real de cada indivíduo na população e, conseqüentemente, determinar a estrutura etária da mesma. Para isto propomos um aprimoramento dessa técnica, utilizada aqui para determinar a estrutura etária de populações da linhagem geográfica SR do díptero forídeo Megaselia scalaris Loew, mantidas pela técnica da transferência seriada em câmaras com temperatura constante de 25 ± 1,0ºC e 20 ± 1,0ºC. O estabelecimento da estrutura etária permitiu calcular a longevidade real das moscas e detectar o efeito ambiental temperatura, sendo fator determinante neste trabalho a marcação dos insetos, pois se não o fosse, teríamos somente estimativas e, dependendo do erro cometido na estimação, o efeito do fator de interesse (temperatura) poderia não ser detectado.

Estimation of population profiles of two strains of the fly Megaselia scalaris (Diptera: Phoridae) by bootstrap simulation

A partir de perfis populacionais experimentais de linhagens do díptero forídeo Megaselia scalaris, foi determinado o número mínimo de perfis amostrais que devem ser repetidos, via processo de simulação bootstrap, para se ter uma estimativa confiável do perfil médio populacional e apresentar estimativas do erro-padrão como medida da precisão das simulações realizadas. Os dados originais são provenientes de populações experimentais fundadas com as linhagens SR e R4, com três réplicas cada, e que foram mantidas por 33 semanas pela técnica da transferência seriada em câmara de temperatura constante (25 ± 1,0ºC). A variável usada foi tamanho populacional e o modelo adotado para cada perfíl foi o de um processo estocástico estacionário. Por meio das simulações, os perfis de três populações experimentais foram amplificados, determinando-se, dessa forma, o tamanho mínimo de amostra. Fixado o tamanho de amostra, simulações bootstrap foram realizadas para construção de intervalos de confiança e comparação dos perfis médios populacionais das duas linhagens. Os resultados mostram que com o tamanho de amostra igual a 50 inicia-se o processo de estabilização dos valores médios.

Radiografia do implante coclear: técnica adaptada para aparelho portátil

Considerando a indisponibilidade de equipamentos avançados de aquisição de imagens nos centros cirúrgicos da maioria dos centros hospitalares e a importância fundamental que têm para o cirurgião uma visualização imediata do implante coclear logo após sua inserção, uma boa opção é a utilização da radiografia convencional. OBJETIVO: Descrever um método radiográfico rápido prático e de baixo custo, que permita avaliar não só a posição, mas também a integridade dos eletrodos, na instalação do implante coclear. MATERIAL E MÉTODO: Foram analisadas radiografias de 262 pacientes submetidos à cirurgia de implante coclear entre Março/2005 e Outubro/2008, com radiografia transoperatória, logo após a inserção dos eletrodos. As radiografias foram analisadas pelo cirurgião no transoperatório e, posteriormente, pelo médico radiologista. RESULTADOS: Foram analisadas 524 radiografias das quais, 95,61% apresentavam técnica adequada, com posicionamento do paciente dentro da técnica descrita neste estudo e boa visualização dos eletrodos, sendo consideradas satisfatórias e 4,39% apresentavam técnica inadequada e/ou visualização insatisfatória dos eletrodos, sendo consideradas insatisfatórias. CONCLUSÃO: Apesar dos aparelhos de Raios X portáteis possuírem limitações, utilizando técnicas e acessórios adequados, é possível conseguir radiografias com resultados satisfatórios para visualização dos implantes cocleares.