Polynomial systems : graphical structure, geometry, and applications

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 199-208).Solving systems of polynomial equations is a foundational problem in computational mathematics, that has several applications in the sciences and engineering. A closely related problem, also prevalent in applications, is that of optimizing polynomial functions subject to polynomial constraints. In this thesis we propose novel methods for both of these tasks. By taking advantage of the graphical and geometrical structure of the problem, our methods can achieve higher efficiency, and we can also prove better guarantees. Various problems in areas such as robotics, power systems, computer vision, cryptography, and chemical reaction networks, can be modeled by systems of polynomial equations, and in many cases the resulting systems have a simple sparsity structure. In the first part of this thesis we represent this sparsity structure with a graph, and study the algorithmic and complexity consequences of this graphical abstraction. Our main contribution is the introduction of a novel data structure, chordal networks, that always preserves the underlying graphical structure of the system. Remarkably, many interesting families of polynomial systems admit compact chordal network representations (of size linear in the number of variables), even though the number of components is exponentially large. Our methods outperform existing techniques by orders of magnitude in applications from algebraic statistics and vector addition systems. We then turn our attention to the study of graphical structure in the computation of matrix permanents, a classical problem from computer science. We provide a novel algorithm that requires Õ(n 2[superscript w]) arithmetic operations, where [superscript w] is the treewidth of its bipartite adjacency graph. We also investigate the complexity of some related problems, including mixed discriminants, hyperdeterminants, and mixed volumes. Although seemingly unrelated to polynomial systems, our results have natural implications on the complexity of solving sparse systems. The second part of this thesis focuses on the problem of minimizing a polynomial function subject to polynomial equality constraints. This problem captures many important applications, including Max-Cut, tensor low rank approximation, the triangulation problem, and rotation synchronization. Although these problems are nonconvex, tractable semidefinite programming (SDP) relaxations have been proposed. We introduce a methodology to derive more efficient (smaller) relaxations, by leveraging the geometrical structure of the underlying variety. The main idea behind our method is to describe the variety with a generic set of samples, instead of relying on an algebraic description. Our methods are particularly appealing for varieties that are easy to sample from, such as SO(n), Grassmannians, or rank k tensors. For arbitrary varieties we can take advantage of the tools from numerical algebraic geometry. Optimization problems from applications usually involve parameters (e.g., the data), and there is often a natural value of the parameters for which SDP relaxations solve the (polynomial) problem exactly. The final contribution of this thesis is to establish sufficient conditions (and quantitative bounds) under which SDP relaxations will continue to be exact as the parameter moves in a neighborhood of the original one. Our results can be used to show that several statistical estimation problems are solved exactly by SDP relaxations in the low noise regime. In particular, we prove this for the triangulation problem, rotation synchronization, rank one tensor approximation, and weighted orthogonal Procrustes.by Diego Cifuentes.Ph. D

Problemas de identificação paramétrica

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas.Em muitas áreas das ciências aplicadas nos deparamos com o problema de estimar um conjunto de parâmetros a partir de dados obtidos experimentalmente, onde se assume que os dados provêm de modelos exponenciais. Esses dados são armazenados em matrizes, as quais servem como ponto de partida para uma série de algoritmos que extraem os parâmetros de interesse, via solução de um problema de autovalor

Active vibration control of civil engineering structures

This thesis is in the area of active vibration control of Civil Engineering structures subject to earthquake loading. Existing structural control methods and technologies including passive, active, semi-active and hybrid control are first introduced. An extensive analysis of a frame-pendulum model is developed and analysed to investigate under what conditions effective energy dissipation is achieved in Tuned Mass Damper systems and the limitation of these devices under stiffness degradation when the structure enters the inelastic region. Linear Quadratic Gaussian and H-infinity active control schemes are designed, simulated and assessed for buildings, modelled as lumped parameter systems, including base and actuator dynamics. Various aspects of the designs are extensively evaluated using multiple criteria and loading conditions and validated in large-scale benchmark problems under practical limitations and implementation constraints. A novel design method is proposed for minimising peak responses of regulated signals via a deadbeat parametrisation of all stabilising controllers in discrete-time. The method incorporates constraints on the magnitude and rate of the control signal and is solved via efficient Linear Programming methods. It is argued that this type of optimisation is more relevant for structural control, as failure occurs when maximum member displacements are exceeded. The problem of stiffness matrix estimation from experimental data is formulated as an optimisation problem and solved under various conditions (positive definiteness, tridiagonal structure) via an alternating convex projection scheme. Both static and dynamic loading is considered. The method is finally incorporated in an adaptive control scheme involving the redesign in real-time of an LQR (Linear Quadratic Regulator) active vibration controller. It is shown that the method is successful in recovering the stability and performance properties of the nominal design under conditions of significant uncertainty in the stiffness parameters

Computer Vision without Vision : Methods and Applications of Radio and Audio Based SLAM

The central problem of this thesis is estimating receiver-sender node positions from measured receiver-sender distances or equivalent measurements. This problem arises in many applications such as microphone array calibration, radio antenna array calibration, mapping and positioning using ultra-wideband and mapping and positioning using round-trip-time measurements between mobile phones and Wi-Fi-units. Previous research has explored some of these problems, creating minimal solvers for instance, but these solutions lack real world implementation. Due to the nature of using different media, finding reliable receiver-sender distances is tough, with many of the measurements being erroneous or to a worse extent missing. Therefore in this thesis, we explore using minimal solvers to create robust solutions, that encompass small erroneous measurements and work around missing and grossly erroneous measurements.This thesis focuses mainly on Time-of-Arrival measurements using radio technologies such as Two-way-Ranging in Ultra-Wideband and a new IEEE standard 802.11mc found on many WiFi modules. The methods investigated, also related to Computer Vision problems such as Stucture-from-Motion. As part of this thesis, a range of new commercial radio technologies are characterised in terms of ranging in real world enviroments. In doing so, we have shown how these technologies can be used as a more accurate alternative to the Global Positioning System in indoor enviroments. Further to these solutions, more methods are proposed for large scale problems when multiple users will collect the data, commonly known as Big Data. For these cases, more data is not always better, so a method is proposed to try find the relevant data to calibrate large systems

Matrix Polynomials and their Lower Rank Approximations

This thesis is a wide ranging work on computing a “lower-rank” approximation of a matrix polynomial using second-order non-linear optimization techniques. Two notions of rank are investigated. The first is the rank as the number of linearly independent rows or columns, which is the classical definition. The other notion considered is the lowest rank of a matrix polynomial when evaluated at a complex number, or the McCoy rank. Together, these two notions of rank allow one to compute a nearby matrix polynomial where the structure of both the left and right kernels is prescribed, along with the structure of both the infinite and finite eigenvalues. The computational theory of the calculus of matrix polynomial valued functions is developed and used in optimization algorithms based on second-order approximations. Special functions studied with a detailed error analysis are the determinant and adjoint of matrix polynomials. The unstructured and structured variants of matrix polynomials are studied in a very general setting in the context of an equality constrained optimization problem. The most general instances of these optimization problems are NP hard to approximate solutions to in a global setting. In most instances we are able to prove that solutions to our optimization problems exist (possibly at infinity) and discuss techniques in conjunction with an implementation to compute local minimizers to the problem. Most of the analysis of these problems is local and done through the Karush-Kuhn-Tucker optimality conditions for constrained optimization problems. We show that most formulations of the problems studied satisfy regularity conditions and admit Lagrange multipliers. Furthermore, we show that under some formulations that the second-order sufficient condition holds for instances of interest of the optimization problems in question. When Lagrange multipliers do not exist, we discuss why, and if it is reasonable to do so, how to regularize the problem. In several instances closed form expressions for the derivatives of matrix polynomial valued functions are derived to assist in analysis of the optimality conditions around a solution. From this analysis it is shown that variants of Newton’s method will have a local rate of convergence that is quadratic with a suitable initial guess for many problems. The implementations are demonstrated on some examples from the literature and several examples are cross-validated with different optimization formulations of the same mathematical problem. We conclude with a special application of the theory developed in this thesis is computing a nearby pair of differential polynomials with a non-trivial greatest common divisor, a non-commutative symbolic-numeric computation problem. We formulate this problem as finding a nearby structured matrix polynomial that is rank deficient in the classical sense