Over-constrained Weierstrass iteration and the nearest consistent system

We propose a generalization of the Weierstrass iteration for over-constrained systems of equations and we prove that the proposed method is the Gauss-Newton iteration to find the nearest system which has at least $k$ common roots and which is obtained via a perturbation of prescribed structure. In the univariate case we show the connection of our method to the optimization problem formulated by Karmarkar and Lakshman for the nearest GCD. In the multivariate case we generalize the expressions of Karmarkar and Lakshman, and give explicitly several iteration functions to compute the optimum. The arithmetic complexity of the iterations is detailed

Over-constrained Weierstrass iteration and the nearest consistent system

We propose a generalization of the Weierstrass iteration for over-constrained systems of equations and we prove that the proposed method allows us to find the nearest system which has at least $k$ common roots and which is obtained via a perturbation of prescribed structure. In the univariate case we show the connection of ourmethod to the optimization problem formulated by Karmarkar and Lakshmanfor the nearest GCD. In the multivariate case we generalize the expressions of Karmarkar and Lakshman, and give a simple iterative method to compute the optimum. The arithmetic complexity of the iteration is detailed

Graph Estimation From Multi-attribute Data

Many real world network problems often concern multivariate nodal attributes such as image, textual, and multi-view feature vectors on nodes, rather than simple univariate nodal attributes. The existing graph estimation methods built on Gaussian graphical models and covariance selection algorithms can not handle such data, neither can the theories developed around such methods be directly applied. In this paper, we propose a new principled framework for estimating graphs from multi-attribute data. Instead of estimating the partial correlation as in current literature, our method estimates the partial canonical correlations that naturally accommodate complex nodal features. Computationally, we provide an efficient algorithm which utilizes the multi-attribute structure. Theoretically, we provide sufficient conditions which guarantee consistent graph recovery. Extensive simulation studies demonstrate performance of our method under various conditions. Furthermore, we provide illustrative applications to uncovering gene regulatory networks from gene and protein profiles, and uncovering brain connectivity graph from functional magnetic resonance imaging data.Comment: Extended simulation study. Added an application to a new data se

A Variational Level Set Approach for Surface Area Minimization of Triply Periodic Surfaces

In this paper, we study triply periodic surfaces with minimal surface area under a constraint in the volume fraction of the regions (phases) that the surface separates. Using a variational level set method formulation, we present a theoretical characterization of and a numerical algorithm for computing these surfaces. We use our theoretical and computational formulation to study the optimality of the Schwartz P, Schwartz D, and Schoen G surfaces when the volume fractions of the two phases are equal and explore the properties of optimal structures when the volume fractions of the two phases not equal. Due to the computational cost of the fully, three-dimensional shape optimization problem, we implement our numerical simulations using a parallel level set method software package.Comment: 28 pages, 16 figures, 3 table

Reservoir Flooding Optimization by Control Polynomial Approximations

In this dissertation, we provide novel parametrization procedures for water-flooding production optimization problems, using polynomial approximation techniques. The methods project the original infinite dimensional controls space into a polynomial subspace. Our contribution includes new parameterization formulations using natural polynomials, orthogonal Chebyshev polynomials and Cubic spline interpolation. We show that the proposed methods are well suited for black-box approach with stochastic global-search method as they tend to produce smooth control trajectories, while reducing the solution space size. We demonstrate their efficiency on synthetic two-dimensional problems and on a realistic 3-dimensional problem. By contributing with a new adjoint method formulation for polynomial approximation, we implemented the methods also with gradient-based algorithms. In addition to fine-scale simulation, we also performed reduced order modeling, where we demonstrated a synergistic effect when combining polynomial approximation with model order reduction, that leads to faster optimization with higher gains in terms of Net Present Value. Finally, we performed gradient-based optimization under uncertainty. We proposed a new multi-objective function with three components, one that maximizes the expected value of all realizations, and two that maximize the averages of distribution tails from both sides. The new objective provides decision makers with the flexibility to choose the amount of risk they are willing to take, while deciding on production strategy or performing reserves estimation (P10;P50;P90)

An Exploration of the Emergence of Pattern and Form from Constraints on Growth

Growing structures are subjects of the space in which they develop. When space is limited or growth is constrained complex patterns and formations can arise. One example of this is seen in the bark patterns of trees. The rigid outer bark layer constrains the growth of the inner layers, resulting in the formation of intricate fracture patterns. An understanding of bark pattern formation has been hampered by insufficient information regarding the biomechanical properties of bark and the corresponding difficulties in faithfully modeling bark fractures using continuum mechanics. Grasstrees, however, have a discrete bark-like structure, making them particularly well suited for computational studies. In this thesis I present a model of grasstree development capturing both primary and secondary growth. A biomechanical model based on a mass-spring network represents the surface of the trunk, permitting the emergence of fractures. This model reproduces key features of grasstree bark patterns which have the same statistical character as trees found in nature. The results support the general hypothesis that the observed bark patterns found in grasstrees may be explained in terms of mechanical fractures driven by secondary growth and that bark pattern formation is primarily a biomechanical phenomenon. Furthermore, I extend the grasstree model to analyze the patterning of discrete elements on the surface of pandanus fruit. Pandanus fruit also exhibit patterns apparently related to fracturing and constraints of space. In this case, the results show that the pattern is likely a result of a higher level mechanisms as opposed to purely biomechanical

Functional Dynamics I : Articulation Process

The articulation process of dynamical networks is studied with a functional map, a minimal model for the dynamic change of relationships through iteration. The model is a dynamical system of a function $f$, not of variables, having a self-reference term $f \circ f$, introduced by recalling that operation in a biological system is often applied to itself, as is typically seen in rules in the natural language or genes. Starting from an inarticulate network, two types of fixed points are formed as an invariant structure with iterations. The function is folded with time, until it has finite or infinite piecewise-flat segments of fixed points, regarded as articulation. For an initial logistic map, attracted functions are classified into step, folded step, fractal, and random phases, according to the degree of folding. Oscillatory dynamics are also found, where function values are mapped to several fixed points periodically. The significance of our results to prototype categorization in language is discussed.Comment: 48 pages, 15 figeres (5 gif files