Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

We describe a fast method for the evaluation of an arbitrary high-dimensional multivariate algebraic polynomial in Chebyshev form at the nodes of an arbitrary rank-1 Chebyshev lattice. Our main focus is on conditions on rank-1 Chebyshev lattices allowing for the exact reconstruction of such polynomials from samples along such lattices and we present an algorithm for constructing suitable rank-1 Chebyshev lattices based on a component-by-component approach. Moreover, we give a method for the fast, exact and stable reconstruction.

Nous proposons des méthodes simples et efficaces pour manipuler des expressions trigonométriques de la forme $F=\sum_{k} f_k\cos\tfrac{k\pi}{n}, f_k\in Z$ où $d<n$ fixé. Nous utilisons les polynômes unitaires de Chebyshev qui forment une base de $Z[x]$ avec laquelle toutes les opérations arithmétiques peuvent être exécutées aussi rapidement qu'avec la base de monômes, mais également déterminer le signe et une approximation de $F$, calculer le polynôme minimal de $F$. Dans ce cadre nous calculons efficacement le polynôme minimal de $2\cos\frac{\pi}{n}$ et aussi le polynôme cyclotomique $\Phi_n$. Nous appliquons ces méthodes au calcul des diagrammes de nœuds de Chebyshev $C(a,b,c,\varphi) : x=T_a(t), y=T_b(t), z=T_c(t+\varphi)$, ce qui permet de tester si une courbe donnée est un nœud, et aussi lister tous les nœuds de Chebyshev possibles quand un triple $(a,b,c)$ fixé en bonne complexité. / We propose a set of simple and fast algorithms for evaluating and using trigonometric expressions in the form $F=\sum_{k}f_k\cos\frac{k\pi}{n}$, $f_k\in Z$ where $d<n$ fixed. We make use of the monic Chebyshev polynomials as a basis of $Z[x]$. We can perform arithmetic operations (multiplication, division, gcd) on polynomials expressed in a Chebyshev basis (with the same bit-complexity as in the monomial basis), compute the sign of $F$, evaluate it numerically and compute its minimal polynomial in $Q[x]$. We propose simple and efficient algorithms for computing the minimal polynomial of $2\cos\frac{\pi}{n}$ and also the cyclotomic polynomial $\Phi_n$. As an application, we give a method to determine the Chebyshev knot's diagrams $C(a,b,c,\varphi) : x=T_a(t),y=T_b(t), z=T_c(t+\varphi)$ which allows to test if a given curve is a Chebyshev knot, and point out all the possible Chebyshev knots coressponding a fixed triple $(a,b,c)$, all of these computings can be done with a good bit complexity.

Soit p un polynôme d'une variable complexe z. On peut trouver plusieurs inégalités reliant le module maximum de p et une combinaison de ses coefficients. Dans ce mémoire, nous étudierons principalement les preuves connues de l'inégalité de Visser. Nous montrerons aussi quelques généralisations de cette inégalité. Finalement, nous obtiendrons quelques applications de l'inégalité de Visser à l'inégalité de Chebyshev. / Let p be a polynomial in the variable z. There exist several inequalities between the coefficents of p and its maximum modulus. In this work, we shall mainly study known proofs of the Visser inquality together with some extensions. We shall finally apply the inequality of Visser to obtain extensions of the Chebyshev inequality.

Diffusion is a phenomenon in which particles move from regions of higher density to regions of lower density. Many physical systems, in fields as diverse as plant biology and finance, are known to involve diffusion phenomena. Typically, diffusion systems are modeled by partial differential equations (PDEs), which include certain parameters. These parameters characterize a given diffusion system. Therefore, for both modeling and simulation of a diffusion system, one has to either know or determine these parameters. Moreover, as PDEs are infinite order dynamic systems, for computational purposes one has to approximate them by a finite order model. In this thesis, we investigate these two issues of model reduction and parameter estimation by considering certain specific cases of heat diffusion systems. We first address model reduction by considering two specific cases of heat diffusion systems. The first case is a one-dimensional heat diffusion across a homogeneous wall, and the second case is a two-dimensional heat diffusion across a homogeneous rectangular plate. In the one-dimensional case we construct finite order approximations by using some well known PDE solvers and evaluate their effectiveness in approximating the true system. We also construct certain other alternative approximations for the one-dimensional diffusion system by exploiting the different modal structures inherently present in it. For the two-dimensional heat diffusion system, we construct finite order approximations first using the standard finite difference approximation (FD) scheme, and then refine the FD approximation by using its asymptotic limit. As for parameter estimation, we consider the same one-dimensional heat diffusion system, as in model reduction. We estimate the parameters involved, first using the standard batch estimation technique. The convergence of the estimates are investigated both numerically and theoretically. We also estimate the parameters of the one-dimensional heat diffusion system recursively, initially by adopting the standard recursive prediction error method (RPEM), and later by using two different recursive algorithms devised in the frequency domain. The convergence of the frequency domain recursive estimates is also investigated.

Solutions to trajectory optimization problems are carried out by the direct and indirect methods. Under broad heading of these methods, numerous algorithms such as collocation, direct, indirect and multiple shooting methods have been developed and reported in the literature. Each of these algorithms has certain advantages and limitations. For example, direct shooting technique is not suitable when the number of nonlinear programming variables is large. Indirect shooting method requires analytical derivatives of the control and co-states function and a poorly guessed initial condition can result in numerical unstable values of the adjoint variable. Multiple shooting techniques can alleviate some of these difficulties by breaking down the trajectory into several segments that help in reducing the non-linearity effects of early control on later parts of the trajectory. However, multiple shooting methods then have to handle more number of variables and constraints to satisfy the defects at the segment joints. The sie of the nonlinear programming problem in the collocation method is also large and proper locations of grid points are necessary to satisfy all the path constraints. Stochastic methods such as Genetic algorithms, on the other hand, also require large number of function evaluations before convergence. To overcome some of the limitations of the conventional methods, improved solution techniques are developed. Three improved methods are proposed for the solution of trajectory optimization problems. They are • a genetic algorithm employing dominance and diploidy concept. • a collocation method using chebyshev polynomials , and • a hybrid method that combines collocation and direct shooting technique A conventional binary-coded genetic algorithm uses a haploid chromosome, where a single string contains all the variable information in the coded from. A diploid, as the name suggests, uses pair of chromosomes to store the same characteristic feature. The diploid genetic algorithm uses a dominant map for decoding genotype into a stable, consistent phenotype. In dominance, one allele takes precedence over another. Diploidy and dominance helps in retaining the previous best solution discovered and shields them from harmful selection in a changing environment. Hence, diploid and dominance affect a king of long-term memory in the genetic algorithm. They allow alternate solutions to co-exist. One solution is expressed and the other is held in abeyance. In the improved diploid genetic algorithm, dominant and recessive genes are defined based on the fitness evaluation of each string. The genotype of fittest string is declared as the dominant map. The dominant map is dynamic in nature as it is replaced with a better individual in future generations. The concept of diploidy and dominance in the improved method mimics closer to the principles used in human genetics as compared to any such algorithms reported in the literature. It is observed that the improved diploid genetic algorithm is able to locate the optima for a given trajectory optimization problem with 10% lower computational time as compared to the haploid genetic algorithm. A parameter optimization problem arising from an optimal control problem where states and control are approximated by piecewise Chebyshev polynomials is well known. These polynomials are more accurate than the interpolating segments involving equal spaced data. In the collocation method involving Chebyshev polynomials, derivatives of two neighboring polynomials are matched with the dynamics at the nodal points. This leads to a large number of equality constraints in the optimization problem. In the improved method, derivative of the polynomial is also matched with the dynamics at the center of segments. Though is appears the problem size is merely increased, the additional computations improve the accuracy of the polynomial for a larger segment. The implicit integration step size is enhanced and overall size of the problem is brought down to one-fourth of the problem size defined with a conventional collocation method using Chebyshev polynomials. Hybrid method uses both collocation and direct shooting techniques. Advantages of both the methods are combined to give more synergy. Collocation method is used in the starting phase of the hybrid method. The disadvantage of standalone collocation method is that tuning of grid points is required to satisfy the path constraints. Nevertheless, collocation method does give a good guess required for the terminal phase of the hybrid method, which uses a direct shooting approach. Results show nearly 30% reduction in computation time for the hybrid approach as compared to a method in which direct shooting alone is used, for the same initial guess of control. The solutions obtained from the three improved methods are compared with an indirect method. The indirect method requires derivations of the control and adjoint equations, which are difficult and problem specific. Due to sensitivity of the costate variables, it is often difficult to find a solution through the indirect method. Nevertheless, these methods do provide an accurate result, which defines a benchmark for comparing the solutions obtained through the improved methods. Trajectory design and optimization of a RLV(Reusable Launch Vehicle) Demonstrator mission is considered as a test problem for evaluating the performance of the improved methods. The optimization problem is difficult than a conventional launch vehicle trajectory optimization problem because of the following two reasons. • aerodynamic lift forces in the RLV add one more dimension to the already complex launch vehicle optimization problem. • as RLV performs a sub orbital flight, the ascent phase trajectory influences the re-entry trajectory. Both the ascent and re-entry optimization problem of the RLV mission is addressed. It is observed that the hybrid method gives accurate results with least computational effort, as compared with other improved techniques for the trajectory optimization problem of RLV during its ascent flight. Hybrid method is then successfully used during the re-entry phase and in designing the feasible optimal trajectories under the dispersion conditions. Analytical solutions obtained from literature are used to compare the optimized trajectory during the re-entry phase. Trajectory optimization studies are also carried out for the off-nominal performances. Being a thrusting phase, the ascent trajectory is subjected to significant deviations, mainly arising out of solid booster performance dispersions. The performance index during rhe ascent phase is modified in a novel way for handling dispersions. It minimizes the state errors in a least square sense, defined at the burnout conditions ensure possibilities of safe re-entry trajectories. The optimal trajectories under dispersion conditions serve as a benchmark for validating the closed-loop guidance algorithm that is developed for the ascent phase flight. Finally, an on-line trajectory command-reshaping algorithm is developed which meets the flight objectives under the dispersion conditions. The guidance algorithm uses a pre-computed trajectory database along with some real-time measured parameters in generating the optimal steering profiles. The flight objectives are met under the dispersion conditions and the guidance generated steering profiles matches closely with the optimal trajectories.

Soit p un polynôme d'une variable complexe z. On peut trouver plusieurs inégalités reliant le module maximum de p et une combinaison de ses coefficients. Dans ce mémoire, nous étudierons principalement les preuves connues de l'inégalité de Visser. Nous montrerons aussi quelques généralisations de cette inégalité. Finalement, nous obtiendrons quelques applications de l'inégalité de Visser à l'inégalité de Chebyshev. / Let p be a polynomial in the variable z. There exist several inequalities between the coefficents of p and its maximum modulus. In this work, we shall mainly study known proofs of the Visser inquality together with some extensions. We shall finally apply the inequality of Visser to obtain extensions of the Chebyshev inequality.

Magnetorheological (MR) dampers are adaptive devices whose properties can be adjusted through the application of a controlled voltage signal. A semi-active suspension system incorporating MR dampers combines the advantages of both active and passive suspensions. For this reason, there has been a continuous effort to develop control algorithms for MR-damped vehicle suspension systems to meet the requirements of the automotive industry. The overall aims of this thesis are twofold: (i) The investigation of non-parametric techniques for the identification of the nonlinear dynamics of an MR damper. (ii) The implementation of these techniques in the investigation of MR damper control of a vehicle suspension system that makes minimal use of sensors, thereby reducing the implementation cost and increasing system reliability. The novel contributions of this thesis can be listed as follows: 1- Nonparametric identification modelling of an MR damper using Chebyshev polynomials to identify the damping force from both simulated and experimental data. 2- The neural network identification of both the direct and inverse dynamics of an MR damper through an experimental procedure. 3- The experimental evaluation of a neural network MR damper controller relative to previously proposed controllers. 4- The application of the neural-based damper controller trained through experimental data to a semi-active vehicle suspension system. 5- The development and evaluation of an improved control strategy for a semi-active car seat suspension system using an MR damper. Simulated and experimental validation data tests show that Chebyshev polynomials can be used to identify the damper force as an approximate function of the displacement, velocity and input voltage. Feed-forward and recurrent neural networks are used to model both the direct and inverse dynamics of MR dampers. It is shown that these neural networks are superior to Chebyshev polynomials and can reliably represent both the direct and inverse dynamic behaviours of MR dampers. The neural network models are shown to be reasonably robust against significant temperature variation. Experimental tests show that an MR damper controller based a recurrent neural network (RNN) model of its inverse dynamics is superior to conventional controllers in achieving a desired damping force, apart from being more cost-effective. This is confirmed by introducing such a controller into a semi-active suspension, in conjunction with an overall system controller based on the sliding mode control algorithm. Control performance criteria are evaluated in the time and frequency domains in order to quantify the suspension effectiveness under bump and random road excitations. A study using the modified Bouc-Wen model for the MR damper, and another study using an actual damper fitted in a hardware-in-the-loop- simulation (HILS), both show that the inverse RNN damper controller potentially gives significantly superior ride comfort and vehicle stability. It is also shown that a similar control strategy is highly effective when used for a semi-active car seat suspension system incorporating an MR damper.

We describe a fast method for the evaluation of an arbitrary high-dimensional multivariate algebraic polynomial in Chebyshev form at the nodes of an arbitrary rank-1 Chebyshev lattice. Our main focus is on conditions on rank-1 Chebyshev lattices allowing for the exact reconstruction of such polynomials from samples along such lattices and we present an algorithm for constructing suitable rank-1 Chebyshev lattices based on a component-by-component approach. Moreover, we give a method for the fast, exact and stable reconstruction.

In dieser Dissertation werden algorithmische Verbesserungen und Varianten für Simulationen der zwei-Flavor Gitter QCD mit dynamischen Fermionen studiert. Der O(a)-verbesserte Dirac-Wilson-Operator wird im Schrödinger Funktional mit einem Update des Hybrid Monte Carlo (HMC)-Typs verwendet. Sowohl der Hermitische als auch der nicht-Hermitische Operator werden betrachtet. Für den Hermitischen Dirac-Wilson-Operator untersuchen wir die Vorteile des symmetrischen gegenüber dem asymmetrischen Gerade-Ungerade-Präkonditionierens, wie man von einem mehr Zeitskalen-Integrator profitieren kann, sowie die Auswirkungen der kleinsten Eigenwerte auf die Stabilität des HMC Algorithmus. Im Fall des nicht-Hermitischen Operators leiten wir eine (semi)-analytische Schranke für das Spektrum her und zeigen eine Methode, um Informationen über den spektralen Rand zu gewinnen, indem wir komplexe Eigenwerte mit dem Lanczos-Algorithmus abschätzen. Diese spektralen Ränder erlauben es, Vorzüge des symmetrischen Gerade-Ungerade-Präkonditionierens oder den Effekt des Sheikholeslami-Wohlert-Terms für das Spektrum des nicht-Hermitischen Operators zu zeigen. Unter Verwendung der Informationen des spektralen Randes konstruieren wir angepasste, komplexe, skalierte und verschobene Tschebyschow Polynome zur Approximation des inversen Dirac-Wilson-Operators. Basierend auf diesen Polynomen entwickeln wir eine neue HMC-Variante, genannt nicht-Hermitischer polynomialer Hybrid Monte Carlo (NPHMC). Sie erlaubt, vom Importance Sampling unter Kompensation mit einem Gewichtungsfaktor abzuweichen. Zudem wird eine Erweiterung durch Anwendung des Hasenbusch-Tricks abgeleitet. Erste Größen der Leistungsfähigkeit, die die Abhängingkeit von den Eingabeparametern als auch einen Vergleich mit unserem Standard-HMC zeigen, werden präsentiert. Im Vergleich der beiden ein-Pseudofermion-Varianten ist der neue NPHMC etwas besser; eine eindeutige Aussage im Fall der zwei-Pseudofermion-Variante ist noch nicht möglich. / In this thesis algorithmic improvements and variants for two-flavor lattice QCD simulations with dynamical fermions are studied using the O(a)-improved Dirac-Wilson operator in the Schrödinger functional setup and employing a hybrid Monte Carlo-type (HMC) update. Both, the Hermitian and the Non-Hermitian operator are considered. For the Hermitian Dirac-Wilson operator we investigate the advantages of symmetric over asymmetric even-odd preconditioning, how to gain from multiple time scale integration as well as how the smallest eigenvalues affect the stability of the HMC algorithm. In case of the non-Hermitian operator we first derive (semi-)analytical bounds on the spectrum before demonstrating a method to obtain information on the spectral boundary by estimating complex eigenvalues with the Lanzcos algorithm. These spectral boundaries allow to visualize the advantage of symmetric even-odd preconditioning or the effect of the Sheikholeslami-Wohlert term on the spectrum of the non-Hermitian Dirac-Wilson operator. Taking advantage of the information of the spectral boundary we design best-suited, complex, scaled and translated Chebyshev polynomials to approximate the inverse Dirac-Wilson operator. Based on these polynomials we derive a new HMC variant, named non-Hermitian polynomial Hybrid Monte Carlo (NPHMC), which allows to deviate from importance sampling by compensation with a reweighting factor. Furthermore an extension employing the Hasenbusch-trick is derived. First performance figures showing the dependence on the input parameters as well as a comparison to our standard HMC are given. Comparing both algorithms with one pseudo-fermion, we find the new NPHMC to be slightly superior, whereas a clear statement for the two pseudo-fermion variants is yet not possible.