30 research outputs found
Adaptative Step Size Selection for Homotopy Methods to Solve Polynomial Equations
Given a C^1 path of systems of homogeneous polynomial equations f_t, t in
[a,b] and an approximation x_a to a zero zeta_a of the initial system f_a, we
show how to adaptively choose the step size for a Newton based homotopy method
so that we approximate the lifted path (f_t,zeta_t) in the space of (problems,
solutions) pairs.
The total number of Newton iterations is bounded in terms of the length of
the lifted path in the condition metric
The complexity and geometry of numerically solving polynomial systems
These pages contain a short overview on the state of the art of efficient
numerical analysis methods that solve systems of multivariate polynomial
equations. We focus on the work of Steve Smale who initiated this research
framework, and on the collaboration between Stephen Smale and Michael Shub,
which set the foundations of this approach to polynomial system--solving,
culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo,
Peter Buergisser and Felipe Cucker
Complexity of Sparse Polynomial Solving 2: Renormalization
Renormalized homotopy continuation on toric varieties is introduced as a tool
for solving sparse systems of polynomial equations, or sparse systems of
exponential sums. The cost of continuation depends on a renormalized condition
length, defined as a line integral of the condition number along all the lifted
renormalized paths.
The theory developed in this paper leads to a continuation algorithm tracking
all the solutions between two generic systems with the same structure. The
algorithm is randomized, in the sense that it follows a random path between the
two systems. The probability of success is one. In order to produce an expected
cost bound, several invariants depending solely of the supports of the
equations are introduced. For instance, the mixed area is a quermassintegral
that generalizes surface area in the same way that mixed volume generalizes
ordinary volume. The facet gap measures for each direction in the 0-fan, how
close is the supporting hyperplane to the nearest vertex. Once the supports are
fixed, the expected cost depends on the input coefficients solely through two
invariants: the renormalized toric condition number and the imbalance of the
absolute values of the coefficients. This leads to a non-uniform complexity
bound for polynomial solving in terms of those two invariants. Up to
logarithms, the expected cost is quadratic in the first invariant and linear in
the last one.Comment: 90 pages. Major revision from the previous versio
Non-linear model predictive energy management strategies for stand-alone DC microgrids
Due to substantial generation and demand fluctuations in stand-alone green micro-grids, energy management strategies (EMSs) are becoming essential for the power
sharing purpose and regulating the microgrids voltage. The classical EMSs track the maximum power points (MPPs) of wind and PV branches independently and rely on batteries, as slack terminals, to absorb any possible excess energy. However, in order to protect batteries from being overcharged by realizing the constant current-constant voltage (IU) charging regime as well as to consider the wind turbine operational constraints, more flexible multivariable and non-linear strategies, equipped with a power curtailment feature, are necessary to control microgrids.
This dissertation work comprises developing an EMS that dynamically optimises the operation of stand-alone dc microgrids, consisting of wind, photovoltaic (PV), and
battery branches, and coordinately manage all energy flows in order to achieve four control objectives: i) regulating dc bus voltage level of microgrids; ii) proportional power sharing between generators as a local droop control realization; iii) charging batteries as close to IU regime as possible; and iv) tracking MPPs of wind and PV branches during their normal operations.
Non-linear model predictive control (NMPC) strategies are inherently multivariable and handle constraints and delays. In this thesis, the above mentioned EMS is developed as a NMPC strategy to extract the optimal control signals, which are duty cycles of three DC-DC converters and pitch angle of a wind turbine.
Due to bimodal operation and discontinuous differential states of batteries, microgrids belong to the class of hybrid dynamical systems of non-Filippov type. This
dissertation work involves a mathematical approximation of stand-alone dc microgrids as complementarity systems (CSs) of Filippov type. The proposed model is used to develop NMPC strategies and to simulate microgrids using Modelica.
As part of the modelling efforts, this dissertation work also proposes a novel algorithm to identify an accurate equivalent electrical circuit of PV modules using both
standard test condition (STC) and nominal operating cell temperature (NOCT) information provided by manufacturers. Moreover, two separate stochastic models are presented for hourly wind speed and solar irradiance levels
Mathematical Methods, Modelling and Applications
This volume deals with novel high-quality research results of a wide class of mathematical models with applications in engineering, nature, and social sciences. Analytical and numeric, deterministic and uncertain dimensions are treated. Complex and multidisciplinary models are treated, including novel techniques of obtaining observation data and pattern recognition. Among the examples of treated problems, we encounter problems in engineering, social sciences, physics, biology, and health sciences. The novelty arises with respect to the mathematical treatment of the problem. Mathematical models are built, some of them under a deterministic approach, and other ones taking into account the uncertainty of the data, deriving random models. Several resulting mathematical representations of the models are shown as equations and systems of equations of different types: difference equations, ordinary differential equations, partial differential equations, integral equations, and algebraic equations. Across the chapters of the book, a wide class of approaches can be found to solve the displayed mathematical models, from analytical to numeric techniques, such as finite difference schemes, finite volume methods, iteration schemes, and numerical integration methods
New Directions for Contact Integrators
Contact integrators are a family of geometric numerical schemes which
guarantee the conservation of the contact structure. In this work we review the
construction of both the variational and Hamiltonian versions of these methods.
We illustrate some of the advantages of geometric integration in the
dissipative setting by focusing on models inspired by recent studies in
celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282
Semiannual report, 1 October 1990 - 31 March 1991
Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis, and computer science is summarized