Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations

Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach

Coded Caching for a Large Number Of Users

Information theoretic analysis of a coded caching system is considered, in which a server with a database of N equal-size files, each F bits long, serves K users. Each user is assumed to have a local cache that can store M files, i.e., capacity of MF bits. Proactive caching to user terminals is considered, in which the caches are filled by the server in advance during the placement phase, without knowing the user requests. Each user requests a single file, and all the requests are satisfied simultaneously through a shared error-free link during the delivery phase. First, centralized coded caching is studied assuming both the number and the identity of the active users in the delivery phase are known by the server during the placement phase. A novel group-based centralized coded caching (GBC) scheme is proposed for a cache capacity of M = N/K. It is shown that this scheme achieves a smaller delivery rate than all the known schemes in the literature. The improvement is then extended to a wider range of cache capacities through memory-sharing between the proposed scheme and other known schemes in the literature. Next, the proposed centralized coded caching idea is exploited in the decentralized setting, in which the identities of the users that participate in the delivery phase are assumed to be unknown during the placement phase. It is shown that the proposed decentralized caching scheme also achieves a delivery rate smaller than the state-of-the-art. Numerical simulations are also presented to corroborate our theoretical results

Still spring was spring

The works came out of an exploration of looking, time and place. A strange tension always occurred to me every time I returned home. For a month, I resumed my early morning schedule in high school on a daily basis. The route between school and home constructs most of my memories in the city. The practice of repeating the old routine is my way of trying to understand my relationship with this place, to probe into the separation and intimacy that constantly contradict within me. What has kept you away and brought you back, why, I ask myself. Relying on returning to the past, I recollect my attachment and connections to what seemingly was forgotten to me

An application of Hoffman graphs for spectral characterizations of graphs

In this paper, we present the first application of Hoffman graphs for spectral characterizations of graphs. In particular, we show that the $2$-clique extension of the $(t+1)\times(t+1)$-grid is determined by its spectrum when $t$ is large enough. This result will help to show that the Grassmann graph $J_2(2D,D)$ is determined by its intersection numbers as a distance regular graph, if $D$ is large enough

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

From Common to Special: When Multi-Attribute Learning Meets Personalized Opinions

Visual attributes, which refer to human-labeled semantic annotations, have gained increasing popularity in a wide range of real world applications. Generally, the existing attribute learning methods fall into two categories: one focuses on learning user-specific labels separately for different attributes, while the other one focuses on learning crowd-sourced global labels jointly for multiple attributes. However, both categories ignore the joint effect of the two mentioned factors: the personal diversity with respect to the global consensus; and the intrinsic correlation among multiple attributes. To overcome this challenge, we propose a novel model to learn user-specific predictors across multiple attributes. In our proposed model, the diversity of personalized opinions and the intrinsic relationship among multiple attributes are unified in a common-to-special manner. To this end, we adopt a three-component decomposition. Specifically, our model integrates a common cognition factor, an attribute-specific bias factor and a user-specific bias factor. Meanwhile Lasso and group Lasso penalties are adopted to leverage efficient feature selection. Furthermore, theoretical analysis is conducted to show that our proposed method could reach reasonable performance. Eventually, the empirical study carried out in this paper demonstrates the effectiveness of our proposed method

The relationships between PM2.5 and meteorological factors in China: Seasonal and regional variations

The interactions between PM2.5 and meteorological factors play a crucial role in air pollution analysis. However, previous studies that have researched the relationships between PM2.5 concentration and meteorological conditions have been mainly confined to a certain city or district, and the correlation over the whole of China remains unclear. Whether or not spatial and seasonal variations exit deserves further research. In this study, the relationships between PM2.5 concentration and meteorological factors were investigated in 74 major cities in China for a continuous period of 22 months from February 2013 to November 2014, at season, year, city, and regional scales, and the spatial and seasonal variations were analyzed. The meteorological factors were relative humidity (RH), temperature (TEM), wind speed (WS), and surface pressure (PS). We found that spatial and seasonal variations of their relationships with PM2.5 do exist. Spatially, RH is positively correlated with PM2.5 concentration in North China and Urumqi, but the relationship turns to negative in other areas of China. WS is negatively correlated with PM2.5 everywhere expect for Hainan Island. PS has a strong positive relationship with PM2.5 concentration in Northeast China and Mid-south China, and in other areas the correlation is weak. Seasonally, the positive correlation between PM2.5 concentration and RH is stronger in winter and spring. TEM has a negative relationship with PM2.5 in autumn and the opposite in winter. PS is more positively correlated with PM2.5 in autumn than in other seasons. Our study investigated the relationships between PM2.5 and meteorological factors in terms of spatial and seasonal variations, and the conclusions about the relationships between PM2.5 and meteorological factors are more comprehensive and precise than before.Comment: 3 tables, 13 figure