We prove convergence of the Adomian Decomposition Method (ADM) by using the Cauchy-Kovalevskaya theorem for differential equations with analytic vector fields, and obtain a new result on the convergence rate of the ADM. Picard's iterative method is considered for the same class of equations in comparison with the decomposition method. We outline some substantial differences between the two methods and show that the decomposition method converges faster than the Picard method. Several nonlinear differential equations are considered for illustrative purposes and the numerical approximations of their solutions are obtained using MATLAB. The numerical results show how the decomposition method is more effective than the standard ODE solvers. Moreover, we prove convergence of the ADM for the partial differential equations and apply it to the cubic nonlinear Schrodinger equation with a localized potential. / Thesis / Master of Science (MSc)

The new modeling method presented here is classified as a substructure synthesis (SS) technique. The distinction between the new SS method and the component mode synthesis formulation is that no transformation between local coordinates and generalized coordinates occurs in the new SS method. The advantage of this is a retention of physical insight into the model and the ability to form equations of motions directly with generalized coordinates. The new formulation differs from other substructure synthesis formulation because it satisfies geometric, natural, displacement and force constraints between substructures into one mathematical process, instead of using both kinematic chains and boundary condition approximation methods. This has the advantage of reducing the complexity of the integrals that are required in the computation. The new formulation also results in global eigenfunction approximations and global generalized coordinates, which eventually satisfies the inclusion principle which means eigenvalue estimates converge from above their actual values. The analysis method also facilitates the examination of boundary conditions in a unique manner. The method is unique because constraints are explicitly examined and selectively satisfied. This allows the identification of extraneous constraints and provides guidance in the selection of admissible functions. The new SS formulation may be divided into two steps. The first step is to satisfy geometric boundary conditions of substructures with appropriate admissible functions. The second step is the modification of these admissible functions to minimally satisfy geometric constraints imposed by the interaction of substructures. Natural constraints can also be satisfied to improve convergence to the exact eigenvalues. The MAF-SS formulation results in explicit knowledge of the constraints coupling substructures. Changing these constraints with active feedback results in a modified structure. The effect of active feedback of terms proportional to the coupling constraints is to lower the stiffness of the structure. This increases the isolation between substructures. The ability to improve isolation using this unique type of feedback is demonstrated. The concept of structural modification through substructure constraint alteration 1s applied to systems using a multivariable feedback method. This is accomplished by combing the MAF-SS method with a standard eigenstructure assignment technique. This method uses the MAF-SS formulation to define a system with substructure constraint eigenstructure properties, the active feedback gain that realizes these systems is calculated with an eigenstructure assignment method. The MAF-SS has application to active control formulation, the result of this control can be an improvement in substructure isolation. / Ph. D.

Nagrinėjant situaciją, kai draudimo kompanija moka dividendus akcininkams pagal barjero strategiją su parametru b, iškyla sunkumų nustatant optimalų dividendų barjerą. Dažnai individualių žalų dydžių skirstinys nežinomas, tačiau galime tikėtis kelių pirmųjų momentų įverčių. Šiame darbe nagrinėjami metodai, kurie leidžia žinant kelis pirmuosius momentus rasti optimalų dividendų barjerą. Šiam tikslui nagrinėjamos De Vylderio aproksimacijos bei difuzinės aproksimacijos. Pasirinktiems skirstiniams pritaikius De Vylderio A, De Vylderio B bei Vynerio, I eilės ir II eilės difuzines aproksimacijas, gaunami optimalūs dividendų barjerai. Gauti rezultatai palyginami su tiksliomis optimalių dividendų barjerų reikšmėmis. / In the financial management of insurance companies and other financial systems an important aspect are dividends. Consider the situation dividends are paid to the shareholders of the insurance company according to barrier strategy with parameter b. In practical situations complete information about the individual claim amount distribution is often not known and the company faces the difficulty in finding the optimal dividend barrier. Model of an insurance company is defined in such way: the premiums of a company are received at rate c, the agregate claims process {S(t)} is a compound Poisson process with Poisson parameter λ and the probability density function of an individual claim amount is denoted by p(y), y>0. In the following, the moment of an individual claim amount distribution of order k will be denoted as pk, k=1, 2, 3,.... Often when complete information about the individual claim amount distribution is not known, estimates for the first few moments of this distribution are available. For such a situation, in this paper methods for estimating the optimal dividend barrier are examined. De Vylder A approximation requires knowledge of p1, p2 and p3. De Vylder B requires knowledge of p1 and p2. Wiener approximation requires knowledge of the same information as De Vylder B, while the diffusion approximation of order k requires knowledge of p1, p2, …, pk+2 . In order to illustrate the approximation methods for several claim amount distributions De Vylder A, De Vylder... [to full text]

The purpose of this paper is to investigate the nature and practical uses of Chebyshev polynomials. Chapter I gives recognition to mathematicians responsible for studies in this area. Chapter II enumerates several mathematical situations in which the polynomials naturally arise and suggests reasons for the pursuance of their study. Chapter III includes: Chebyshev polynomials as related to "best" polynomial approximation, Chebyshev series, and methods of producing polynomial approximations to continuous functions. Chapter IV discusses the use of Chebyshev polynomials to solve certain differential equations and Chebyshev-Gauss quadrature.

In many digital signal processing applications, we often need some special function units that can compute complicated arithmetic functions such as reciprocal, logarithm, power of 2, trigonometric functions, etc. The most popular designs are based on look-up tables with polynomial approximation. However, the table size will increase significantly in accordance with precision. In this thesis, we propose a method called remapping to reduce the table size by using non-uniform segmentation. When we obtain the coefficients for all segments, we do not store them in order. By sorting the coefficients in the ROM ,we design a efficient hardware mapping. The method can reduce the ROM size with lower extra cost spent in address mapping for non-uniform segmentation.

Dans cette thèse, nous donnons des approximations discrètes de quantités lisses associées à certaines courbes planes ou à certaines surfaces de l'espace euclidien de dimension 3. Dans le cas des courbes, le défaut angulaire en un point P de la courbe est une bonne approximation de la courbure de la courbe en ce point. Nous donnons une majoration de l'erreur commise en fonction du jet d'ordre 1 de la courbure, de la géométrie de la courbe et du maximum de la distance entre P et un point variable de la courbe. Dans le cas des surfaces, nous donnons une majoration entre la courbure discrète en un point P d'une surface lisse S et un polynôme homogène en les courbures principales de S en P. Notre majorant dépend du jet d'ordre 1 des courbures de S en P, de l'épaisseur, du nombre de points du maillage et surtout de sa taille. Enfin, nous construisons une classe particulière de maillages qui permet d'avoir des résultats de convergence ponctuels lorsque la taille des maillages tend vers 0.

This report discusses three topics relating to errors of numerical methods and to improvements of numerical approximations. The introduction connects these topics to the secondary mathematics curriculum. The three chapters which follow develop the three selected topics: improving approximations of irrational numbers, error analysis of numerical integration methods, and discretization versus rounding error in Euler’s Method for solving ordinary differential equations. The conclusion describes specific national secondary mathematical standards and classroom activities relevant to numerical approximations and error analysis. / text

Third generation full-waveform (FW) LIDAR systems image an entire scene by emitting laser pulses in particular directions and measuring the echoes. Each of these echoes provides range measurements about the objects intercepted by the laser pulse along a specified direction. By scanning through a specified region using a series of emitted pulses and observing their echoes, connected 1D profiles of 3D scenes can be readily obtained. This extra information has proven helpful in providing additional insight into the scene structure which can be used to construct effective characterizations and classifications. Unfortunately, massive amounts of data are typically collected which impose storage, processing and transmission limitations. To address these problems, a number of compression approaches have been developed in the literature. These, however, generally require the initial acquisition of large amounts of data only to later discard most of it by exploiting redundancies, thus sampling inefficiently. Based on this, our main goal is to apply efficient and effective LIDAR sampling schemes that achieve acceptable reconstruction quality of the 3D scenes. To achieve this goal, we propose on using compressive sampling by emitting pulses only into random locations within the scene and collecting only the corresponding returned FW signals. Under this framework, the number of emissions would typically be much smaller than what traditional LIDAR systems require. Application of this requires, however, that scenes contain many degrees of freedom. Fortunately, such a requirement is satisfied in most natural and man-made scenes. Here, we propose to use a measure of rank as the measure of degrees of freedom. To recover the connected 1D profiles of the 3D scene, matrix completion is applied to the tensor slices. In this paper, we test our approach by showing that recovery of compressively sampled 1D profiles of actual 3D scenes is possible using only a subset of measurements.

I develop a numerical method that combines functional approximations and dynamic programming to solve high-dimensional discrete-time stochastic control problems under general constraints. The method relies on three building blocks: first, a quasi-random grid and the radial basis function method are used to discretize and interpolate the high-dimensional state space; second, to incorporate constraints, the method of Lagrange multipliers is applied to obtain the first order optimality conditions; third, the conditional expectation of the value function is approximated by a second order polynomial basis, estimated using ordinary least squares regressions. To reduce the approximation error, I introduce the test region iterative contraction (TRIC) method to shrink the approximation region around the optimal solution. I apply the method to two Finance applications: a) dynamic portfolio choice with constraints, a continuous control problem; b) dynamic portfolio choice with capital gain taxation, a high-dimensional singular control problem. / text

One of the central difficulties in many models of glacier and ice sheet flow lies in the prescription of boundary conditions at the bed. Often, processes which occur there dominate the evolution of the ice mass as they control the speed at which the ice is able to slide over the bed. In part I of this thesis, we study two complications to classical models of glacier and ice sheet sliding. First, we focus on the effect of cavity formation on the sliding of a glacier over an undeformable, impermeable bed. Our results do not support the widely used sliding law $u_b = C\tau_b^pN^{-q}$, but indicate that $\tau_b/N$ actually decreases with $u_b/N$ at high values of the latter, as suggested previously for simple periodic beds by Fowler (1986). The second problem studied is that of an ice stream whose motion is controlled by bed obstacles with wavelengths comparable to the thickness of ice. By contrast with classical sliding theory for ice of constant viscosity,the bulk flow velocity does not depend linearly on the driving stress. Indeed, the bulk flow velocity may even be a multi-valued function of driving stress and ice thickness. In the second part of the thesis, attention is turned to the formation of drumlins. The viscous till model of Hindmarsh (1998) and Fowler (2000) is analysed in some detail. It is shown that the model does not predict the formation of three-dimensional drumlins, but only that of two-dimensional features, which may be interpreted as Rogen moraines. A non-linear model allows the simulation of the predicted bedforms at finite amplitude. Results obtained indicate that the growth of bedforms invariably leads to cavitation. A model for travelling waves in the presence of cavitation is also developed, which shows that such travelling waves can indeed exist. Their shape is, however, unlike that of real bedforms, with a steep downstream face and no internal stratification. These results indicate that Hindmarsh and Fowler's model is probably not successful at describing the processes which lead to the formation of streamlined subglacial bedforms.