As simulation plays an increasingly central role in modern science and engineering research, by supplementing experiments, aiding in the prototyping of engineering systems or informing decisions on safety and reliability, the need to quantify uncertainty in model outputs due to uncertainties in the model parameters becomes critical. However, the statistical characterization of the model parameters is rarely known. In this thesis, we propose a variational approach to solve the stochastic inverse problem of obtaining a statistical description of the diffusion coefficient in an elliptic partial differential equation, based noisy measurements of the model output. We formulate the parameter identification problem as an infinite dimensional constrained optimization problem for which we establish existence of minimizers as well as first order necessary conditions. A spectral approximation of the uncertain observations (via a truncated Karhunen-Loeve expansion) allows us to estimate the infinite dimensional problem by a smooth, albeit high dimensional, deterministic optimization problem, the so-called 'finite noise' problem, in the space of functions with bounded mixed derivatives. We prove convergence of 'finite noise' minimizers to the appropriate infinite dimensional ones, and devise a gradient based, as well as a sampling based strategy for locating these numerically. Lastly, we illustrate our methods by means of numerical examples. / Ph. D.

Two basic approaches to reduce computational requirements for solving distillation problems have been studied: simplifications of the model based on physical approximations and order reduction techniques based on numerical approximations. Several problems have been studied using full and reduced-order techniques along with the following distillation models: Constant Molar Overflow, Constant Molar Holdup and Time-Dependent Molar Holdup. Steady-state results show excellent agreement in the profiles obtained using orthogonal collocation and demonstrate that with an order reduction of up to 54%, reduced-order models yield better results than physically simpler models. Step responses demonstrate that with a reduction in computing time of the order of 60% the method still provides better dynamic simulations than those obtained using physical simplifications. Frequency response data obtained from pulse tests has been used to verify that reduced-order solutions preserve the dynamic characteristics of the original full-order system while physical simplifications do not. The orthogonal collocation technique is also applied to a coupled columns scheme with good results. / Graduation date: 1992

We consider a collocation method for Cauchy singular integral equations on the interval based on weighted Chebyshev polynomials , where the coefficients of the operator are piecewise continuous. Stability conditions are derived using Banach algebra methods, and numerical results are given.

A collocation technique in non-polynomial spline space is presented to approximate solutions of singular neutral functional differential equations (SNFDEs). Using solution representations and general well-posedness results for SNFDEs convergence of the method is shown for a large class of initial data including the case of discontinuous initial function. Using this technique, an identification problem is solved for a particular SNFDE. The technique is also applied to other different examples. Even for the special case in which the initial data is a discontinuous function the identification problem is successfully solved. / Master of Science

L'objectif de ce travail est la résolution des équations intégrales singulières à noyau Cauchy. On y traite les équations singulières de Cauchy de première espèce par la méthode des approximations successives. On s'intéresse aussi aux équations intégrales à noyau de Cauchy de seconde espèce, en utilisant les polynômes trigonométriques et les techniques de Fourier. Dans la même perspective, on utilise les polynômes de Tchebychev de quatrième degré pour résoudre une équation intégro différentielle à noyau de Cauchy. Ensuite, on s'intéresse à une autre équation intégro-différentielle à noyau de Cauchy, en utilisant les polynômes de Legendre, ce qui a donné lieu à développer deux méthodes basées sur une suite de projections qui converge simplement vers l'identité. En outre, on exploite les méthodes de projection pour les équations intégrales avec des opérateurs intégraux bornés non compacts et on a appliqué ces méthodes à l'équation intégrale singulière à noyau de Cauchy de deuxième espèce / The purpose of this thesis is to develop and illustrate various new methods for solving many classes of Cauchy singular integral and integro-differential equations. We study the successive approximation method for solving Cauchy singular integral equations of the first kind in the general case, then we develop a collocation method based on trigonometric polynomials combined with a regularization procedure, for solving Cauchy integral equations of the second kind. In the same perspective, we use a projection method for solving operator equation with bounded noncompact operators in Hilbert spaces. We apply a collocation and projection methods for solving Cauchy integro-differential equations, using airfoil and Legendre polynomials

We consider a collocation method for Cauchy singular integral equations on the interval based on weighted Chebyshev polynomials , where the coefficients of the operator are piecewise continuous. Stability conditions are derived using Banach algebra methods, and numerical results are given.

Esta tese analisa o método de colocação polinomial, para uma classe de equações integro-diferenciais singulares em espaços ponderados de funções contínuas e condições de fronteira não nulas. A convergência do método numérico em espaços com norma uniforme ponderada, é demonstrada, e taxas de convergências são determinadas, usando a suavidade dos dados das funções envolvidas no problema. Exemplos numéricos confirmam as estimativas / This thesis analyses the polynomial collocation method, for a class of singular integro-differential equations in weighted spaces of continuous functions, and non-homogeneous boundary conditions. Convergence of the numerical method, in weighted uniform norm spaces, is demonstrated and convergence rates are determined using the smoothness of the data functions involved in problem. Numerical examples confirm the estimates

In this thesis, a survey on pseudospectral methods for differential equations is presented. Properties of the classical orthogonal polynomials required in this context are reviewed. Differentiation matrices corresponding to Jacobi, Laguerre,and Hermite cases are constructed. A fairly detailed investigation is made for the Hermite spectral methods, which is applied to the Schr&ouml / dinger eigenvalue equation with several potentials. A discussion of the numerical results and comparison with other methods are then introduced to deduce the effciency of the method.

At low Reynolds number flows, the effect of inertia becomes negligible and the fluid motion is dominated by the effect of viscous forces. Understanding of the behaviour of low Reynolds number flows underpins the prediction of the motion of microorganisms and particle sedimentation as well as the development of micro-robots that could potentially swim inside the human body to perform targeted drug/cell delivery and non-invasive microsurgery. The work in this thesis focuses on developing an understanding in the mathematical analysis of objects moving at low Reynolds numbers. A boundary element implementation of the Method of regularized Stokeslets (MRS) is applied to analyse the low Reynolds number flow field around an object of simple shape (sphere and cube). It also showed that the results obtained by a boundary element implementation for an unbounded cube, where singularities are presented in the corners of the cube, agrees with more complex solutions methods such as a GBEM and FEM.A methodology for analysing the effect of walls by locating collocation points on the surface of the walls and the object is presented. First at all, this methodology is validated with a boundary element implementation of the method of images for a sphere at different locations. Then, the method is extended when more than one wall is presented. This methodology is applied to predict the velocity filed of a cube moving in a tow tank at low Reynolds numbers for two different cases with a supporting rod similar to an experimental set-up, and without the supporting rod as in the CFD simulations based on the FVM. The results indicate a good match between CFD and the MRS, and an excellent approximation between the MRS and experimental data from PIV measurements. The drag, thrust and torque generated by helices moving at low Reynolds numbers in an unbounded medium is analysed by the resistive force theory, a slender body theory, and a boundary element method of the MRS. The results show that the resistive force theory predict accurately the drag, thrust and torque of moving helices when the resistive force coefficients are calculated from a slender body theory approximation by calculating independently the resistive force coefficients for translation and rotation, because it is observed that the resistive force coefficients depend also of the nature of motion. Moreover, the thrust generated by helices of different pitch angles is analysed calculated by a CFD numerical simulation based on the FVM and a boundary element implementation, an compared with experimental data. The results also show an excellent prediction between the boundary element implementation, the CFD results and the experimental data. Finally, a boundary element implementation of the MRS is applied to predict swimming of a biomimetic swimmer that mimics the motion of E.coli bacteria in an unbounded medium. The results are compared with the propulsive velocity and induced angular velocity measurement by recording the motion of the biomimetic swimmer in a square tank. It is observed that special care needs to be taken when the biomimetic swimmer is modelled inside the tank, as there is an apparent increment in the calculate thrust propulsion which does not represent a real situation of the biometic swimmer which propels by a power supply. However, this increment does not represent the condition of the biomimetic swimmer and a suggested methodology based on the solution from an unbounded case and when the swimmer is moving inside the tank is presented. In addition, the prediction of the free-swimming velocity for the biomimetic swimmer agrees with the results obtained by the MRS when the resistive force coefficients are calculated from a SBT implementation. The results obtained in this work have showed that a boundary element implementation of the MRS produces results comparable with more complex numerical implementations such as GBEM, FEM, FVM, and also an excellent agreement with results obtained from experimentation. Therefore, it is a suitable and easy to apply methodology to analyse the motion of swimmers at low Reynolds numbers, such as the biomimetic swimmer modelled in this work.

This thesis deals with numerical solution of an integral equation of the second kind with special singular kernel function related to induction heating. The numerical solution is based on collocation and Nyström methods. The idea of collocation methods is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree). The Nyström methods are based on approximation of the integral in equation by numerical integration rule. This thesis describes and gives error estimates of both methods. Error estimates are compared to the exact solutions in simple cases.