Graph Inference and Graph Matching

Graphs are widely used in many fields of research, ranging from natural sciences to computer and mathematical sciences. Graph inference is an area of intense research. In this dissertation, we propose several methodologies in graph inference. We focus on statistical inference using graph invariants, vertex nomination, and a divide-and-conquer graph matching technique. We present a comparative power analysis of various graph invariants for testing the hypothesis that the graph has a subgraph with higher edge probability. Given a graph drawn from a kidney-egg random graph model, the null hypothesis is that all edge probabilities are equal. The alternative hypothesis is that there exists a subset of vertices which are more likely to be adjacenct to each other than the rest of the graph. Using Monte Carlo simulations, we estimate the power of several graph invariants acting as test statistics. We discovered that for many choices of parameters in the random graph model, the scan statistic and clustering coefficient often dominate other graph invariants. However, our results indicates that none of the graph invariants considered is uniformly most powerful. Given a graph drawn from a stochastic block model where one block is of particular interest, vertex nomination is the task of creating a list of vertices such that there are an abundance of vertices from the block of interest at the top of the list. Vertex nomination is useful in situations where only a limited number of vertices can be examined and have their block membership checked. We propose several vertex nomination schemes, derive theoretical results for performance, and compare the schemes on simulated and real data. Given two graphs, graph matching is to create a mapping from one set of vertices to the other, such that the edge structure of the graphs is preserved as best as possible. We develop a new method for scaling graph matching algorithms, and prove performance guarantees. Any graph matching algorithm can be scaled using our divide-and-conquer technique. The performance of this technique is demonstrated on large simulated graphs and human brain graphs

Sedimentation-consolidation of a double porosity material

This paper studies the sedimentation-consolidation of a double porosity material, such as lumpy clay. Large displacements and finite strains are accounted for in a multidimensional setting. Fundamental equations are derived using a phenomenological approach and non-equilibrium thermodynamics, as set out by Coussy [Coussy, Poromechanics, Wiley, Chichester, 2004]. These equations particularise to three non-linear partial differential equations in one dimensional context. Numerical implementation in a finite element code is currently being undertaken

A reaction-diffusion system of a competitor-competitor-mutualist model

We investigate the homogeneous Dirichlet problem and Neumann problem to a reaction-diffusion system of a competitor-competitor-mutualist model. The existence, uniqueness, and boundedness of the solutions are established by means of the comparison principle and the monotonicity method. For the Dirichlet problem, we study the existence of trivial and nontrivial nonnegative equilibrium solutions and their stabilities. For the Neumann problem, we analyze the contant equilibrium solutions and their stabilities. The main method used in studying of the stabilities is the spectral analysis to the linearized operators. The O.D.E. problem to the same model was proposed and studied by B. Rai, H. I. Freedman, and J. F. Addicott (Math. Biosci. 65 (1983), 13-50).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/26702/1/0000250.pd

Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of parabolic type

[EN] The main objective of this work is to provide a stability and error analysis of high-order commutator-free quasi-Magnus (CFQM) exponential integrators. These time integration methods for nonautonomous linear evolution equations are formed by products of exponentials involving linear combinations of the defining operator evaluated at certain times. In comparison with other classes of time integration methods, such as Magnus integrators, an inherent advantage of CFQM exponential integrators is that structural properties of the operator are well preserved by the arising linear combinations. Employing the analytical framework of sectorial operators in Banach spaces, evolution equations of parabolic type and dissipative quantum systems are included in the scope of applications. In this context, however, numerical experiments show that CFQM exponential integrators of nonstiff order five or higher proposed in the literature suffer from poor stability properties. The given analysis delivers insight that CFQM exponential integrators are well defined and stable only if the coefficients occurring in the linear combinations satisfy a positivity condition and that an alternative approach for the design of stable high-order schemes relies on the consideration of complex coefficients. Together with suitable local error expansions, this implies that a high-order CFQM exponential integrator retains its nonstiff order of convergence under appropriate regularity and compatibility requirements on the exact solution. Numerical examples confirm the theoretical result and illustrate the favourable behaviour of novel schemes involving complex coefficients in stability and accuracy.Ministerio de Economia y Competitividad (Spain) through projects MTM2013-46553-C3 and MTM2016-77660-P (AEI/FEDER, UE) to S.B. and F.C.Blanes Zamora, S.; Casas, F.; Mechthild Thalhammer (2018). Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of parabolic type. IMA Journal of Numerical Analysis. 38(2):743-778. https://doi.org/10.1093/imanum/drx012S74377838

Schwarz Waveform Relaxation Methods for Systems of Semi-Linear Reaction-Diffusion Equations

Domain decomposition methods in science and engineering XIX, LNCSE, Springer Verlag, 2010.Schwarz waveform relaxation methods have been studied for a wide range of scalar linear partial differential equations (PDEs) of parabolic and hyperbolic type. They are based on a space-time decomposition of the computational domain and the subdomain iteration uses an overlapping decomposition in space. There are only few convergence studies for non-linear PDEs. We analyze in this paper the convergence of Schwarz waveform relaxation applied to systems of semi-linear reaction-diffusion equations. We show that the algorithm converges linearly under certain conditions over long time intervals. We illustrate our results, and further possible convergence behavior, with numerical experiments

On the linearization of some singular, nonlinear elliptic problems and applications

This paper deals with the spectrum of a linear, weighted eigenvalue problem associated with a singular, second order, elliptic operator in a bounded domain, with Dirichlet boundary data. In particular, we analyze the existence and uniqueness of principal eigenvalues. As an application, we extend the usual concepts of linearization and Frechet derivability, and the method of sub and supersolutions to some semilinear, singular elliptic problems

Learning to Translate with Products of Novices: A Suite of Open-Ended Challenge Problems for Teaching MT

Machine translation (MT) draws from several different disciplines, making it a complex subject to teach. There are excellent pedagogical texts, but problems in MT and current algorithms for solving them are best learned by doing. As a centerpiece of our MT course, we devised a series of open-ended challenges for students in which the goal was to improve performance on carefully constrained instances of four key MT tasks: alignment, decoding, evaluation, and reranking. Students brought a diverse set of techniques to the problems, including some novel solutions which performed remarkably well. A surprising and exciting outcome was that student solutions or their combinations fared competitively on some tasks, demonstrating that even newcomers to the field can help improve the state-ofthe-art on hard NLP problems while simultaneously learning a great deal. The problems, baseline code, and results are freely available.