No description available.

We are interested in solving the time-harmonic inverse acoustic scattering problem for planar sound-soft obstacles. In this work, we introduce four methods for solving inverse scattering problems. The first method is a variation of the method introduced by Johansson and Sleeman. This method solves the inverse problem when we have the far field pattern given for only one incident wave. It is an iterative method based on a pair of integral equations used to obtain the far field pattern of a known single object. The method proposed in this thesis has a better computational performance than the method of Johansson and Sleeman. The second method we present is a multi-frequency method called the recursive linearization algorithm. This method solves the inverse problem when the far field pattern is given for multiple frequencies. The idea of this method is that from an initial guess, we solve the single frequency inverse problem for the lowest frequency. We use the result obtained as the initial guess to solve the problem for the next highest frequency. We repeat this process until we use the data from all frequencies. To solve the problem at each frequency, we use the first method proposed. To improve the quality of the reconstruction of the shadowed part of the object, we solve the inverse scattering problem of reconstructing an unknown sound-soft obstacle in the presence of known scatterers. We show that depending on the position of the scatterers, we may be able to obtain very accurate reconstructions of the entire unknown object. Next, we introduce a method for solving the inverse problem of reconstructing a convex sound-soft obstacle, given measures of the far field pattern at two frequencies that are not in the resonance region of the object. This method is based on the use of an approximation formula for the far field pattern using geometric optics. We are able to prove that for the reconstruction of the circle of radius $R$ and center at the origin, the size of the interval of convergence of this method is proportional to the inverse of the wavenumber. This procedure is effective at reconstructing the illuminated part of the object; however, it requires an initial guess close to the object for frequencies out of the resonance region. Finally, we propose a globalization technique to obtain a better initial guess to solve the inverse problem at frequencies out of the resonance region. In this technique, given the far field pattern of a convex object at two frequencies out of the resonance region, we use our extrapolation operator to generate synthetic data for low frequencies. We apply the recursive linearization algorithm, using as a single frequency solver the method that is based on geometric optics. We obtain an approximation of the object that can be used as the initial guess to apply the recursive linearization algorithm using the first method introduced as the single frequency solver.

M.Sc. / A class of numerical methods for solving nonstiff initial value problems in ordinary differential equations has been developed. These methods, designated RKrGLn, are based on a Runge-Kutta method of order r (RKr), and Gauss-Legendre integration over n+ 1 nodes. The interval of integration for the initial value problem is subdivided into an integer number of subintervals. On each of these n + 1 nodes are defined in accordance with the zeros of the Legendre polynomial of degree n. The Runge-Kutta method is used to find an approximate solution at each of these nodes; Gauss-Legendre integration is used to find the solution at the endpoint of the subinterval. The process then carries over to the next subinterval. We find that for a suitable choice of n, the order of the local error of the Runge- Kutta method (r + 1) is preserved in the global error of RKrGLn. However, a poor choice of n can actually limit the order of RKrGLn, irrespective of the choice of r. What is more, the inclusion of Gauss-Legendre integration slightly reduces the number of arithmetical operations required to find a solution, in comparison with RKr at the same number of nodes. These two factors combine to ensure that RKrGLn is considerably more efficient than RKr, particularly when very accurate solutions are sought. Attempts to control the error in RKrGLn have been made. The local error has been successfully controlled using a variable stepsize strategy, similar to that generally used in RK methods. The difference lies in that it is the size of each subinterval that is controlled in RKrGLn, rather than each individual stepsize. Nevertheless, local error has been successfully controlled for relative tolerances ranging from 10 -4 to 10-10 . We have also developed algorithms for estimating and controlling the global error. These algorithms require that a complete solution be obtained for a specified distribution of nodes, after which the global error is estimated and then, if necessary, a new node distribution is determined and another solution obtained. The algorithms are based on Richardson extrapolation and the use of low-order and high-order pairs. The algorithms have successfully achieved desired relative global errors as small as 10-1° . We have briefly studied how RKrGLn may be used to solve stiff systems. We have determined the intervals of stability for several RKrGLn methods on the real line, and used this to develop an algorithm to solve a stiff problem. The algorithm is based on the idea of stepsize/subinterval adjustment, and has been used to successfully solve the van der Pol system. Lagrange interpolation on each subinterval has been implemented to obtain a piecewise continuous polynomial approximation to the numerical solution, with same order error, which can be used to find the solution at arbitrary nodes.

The development of efficient mathematical models and numerical methods for the study of haemodynamics is becoming increasingly prominent in the analysis of pathological states of the cardiovascular system. Computational models contribute to medical diagnosis processes, reducing the need of classical invasive medical techniques, which are not risk-free for patients and generate high healthcare costs. The first part of this thesis focuses on the modeling and simulation of coronary blood flow, with emphasis on stable Coronary Artery Disease (CAD), a pathological condition that occurs when an abnormal narrowing builds inside coronary vessel walls. Our goal is to develop a CCTA-based Fractional Flow Reserve (FFR) model which incorporates clinical imaging and patient-specific characteristics to predict the haemodynamic behavior and properties of individuals, reducing the need for invasive measurements. A novel aspect of the proposed methodology is the inclusion of the pressure guidewire, used in clinical settings, and the assessment of its impact on local fluid dynamics and FFR predictions. Thereafter, the second part of this dissertation is devoted to the development of numerical methods for the simulation of incompressible flows with particular emphasis on the simulation of cardiovascular haemodynamics. A novel implicit hybrid finite volume/finite element methodology for the efficient simulation of blood flow is proposed and validated. The implicit discretization of the transport-diffusion equations making use of an inexact Newton-Krylov method with an SGS preconditioner yields to an efficient scheme avoiding the severe CFL condition arising in explicit or semi-implicit methods for blood flow dynamics. Besides, the Ducros flux function employed for the nonlinear convective terms leads to a provably kinetic energy stable scheme of the advection terms. In addition, a staggered semi-implicit method for the simulation of incompressible flows in one-dimensional elastic and viscoelastic vessels is proposed. The convective stage is treated explicitly in time, while the diffusive and pressure stage are handled implicitly to avoid strict bounds on the time steps. The one-dimensional methodology is then extended to networks of vessels by introducing a local three-dimensional representation of the junction.

Neurostimulation continues to demonstrate tremendous success as an intervention for neurodegenerative diseases, including Parkinson's disease, in addition to a range of other neurological and psychiatric disorders. In an effort to enhance the medical efficacy and comprehension of this form of brain therapy, modeling and computational simulation are regarded as valuable tools that enable in silico experiments for a range of neurostimulation research endeavours. To fully realize the capacities of neurostimulation simulations, several areas within computation and numerics need to be considered and addressed. Specifically, simulations of neurostimulation that incorporate (i) computational efficiency, (ii) application versatility, and (iii) characterizations of cellular-level electrophysiology would be highly propitious in supporting advancements in this medical treatment. The focus of this dissertation is on these specific areas. First, preconditioners and iterative methods for solving the linear system of equations resulting from finite element discretizations of partial differential equation based transcranial electrical stimulation models are compared. Second, a software framework designed to efficiently support the range of clinical, biomedical, and numerical simulations utilized within the neurostimulation community is presented. Third, a multiscale model that couples transcranial direct current stimulation administrations to neuronal transmembrane voltage depolarization is presented. Fourth, numerical solvers for solving ordinary differential equation based ligand-gated neurotransmitter receptor models are analyzed. A fundamental objective of this research has been to accurately emulate the unique medical characteristics of neurostimulation treatments, with minimal simplification, thereby providing optimal utility to the scientific research and medical communities. To accomplish this, numerical simulations incorporate high-resolution, MRI-derived three-dimensional head models, real-world electrode configurations and stimulation parameters, physiologically-based inhomogeneous and anisotropic tissue conductivities, and mathematical models accepted by the brain modeling community. It is my hope that this work facilitates advancements in neurostimulation simulation capabilities, and ultimately helps improve the understanding and treatment of brain disease. / Ph. D.

No description available.

Implicit large eddy simulation (ILES) has been shown, in the literature, to have some success for three-dimensional flow (e.g. see [Grinstein, F.F., Margolin, L.G. and Rider, W. Implicit Large Eddy Simulation. Cambridge, 2007]), but it has not previously been examined for two-dimensional flow. This thesis investigates whether ILES can be applied successfully to two-dimensional flow. Modified equation analysis is used to demonstrate the similarities between the truncation errors of certain numerical schemes and the subgrid terms of the barotropic vorticity equation (BVE). This presents a theoretical motivation for the numerical testing. Burgers equation is first used as a model problem to develop the ideas and methodology. Numerical schemes that are known to model Burgers equation well (shock capturing schemes) are shown to be implicitly capturing the subgrid terms of the one-dimensional inviscid Burgers equation through their truncation errors. Numerical tests are performed on three equation sets (BVE, Euler equations and the quasi-geostrophic potential vorticity equation) to assess the application of ILES to two-dimensional flow. The results for each of these equation sets show that the schemes considered for ILES are able to capture some of the subgrid terms through their truncation errors. In terms of accuracy, the ILES schemes are comparable (or outperform) schemes with simple explicit subgrid models when comparing vorticity solutions with a high resolution reference vorticity solution. The results suggest that conservation of vorticity is important to the successful application of ILES to two-dimensional flow, whereas conservation of momentum is not. The schemes considered for ILES are able to successfully model the downscale enstrophy transfer, but none of the schemes considered for ILES (or the schemes with simple subgrid models) can model the correct upscale energy transfer from the subgrid to the resolved scales. Energy backscatter models are considered and are used with the ILES schemes. It is shown that it is possible to create an energy conserving and enstrophy dissipating scheme, composed of an ILES scheme and a backscatter model, that improves the accuracy of the vorticity solution (when compared with the corresponding ILES scheme without backscatter).

This paper present three kinds of nonlinear numerical methods to search for perfect complementary codes, include Newtonian Methods¡BLevenberg-Marquardt Algorithm and Trust-Regions. By searching for the solution of theses nonlinear equations, we can get complementary codes when setting for the length of element codes and the flock size. These search results is very generous. Complete complementary codes¡Bsuper complementary code and poly-phase complementary code are subsets of these searching results¡C These nonlinear equations are set to have ideal auto-correlation and cross-correlation properties, so the searching results of these nonlinear equations are still have perfect orthogonal complementary properties. Because the orthogonal complementary code is obtained via these nonlinear equations, the results are the most generous. So nonlinear numerical method is a good choice to search for another complementary code we don¡¦t know.

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In this work there are analized the realization of the most important numerical methods in computer algebra systems. There are analyzed and comparised the opportiunities of three CAS – Maple, Matlab and Mathcad – for the reason how to solve the mathematical tasks by using the numerical methods. Therefore are given the discription of the methods for solving systems of linear equations, function interpolation, numerical integration and for solving the first order differential equations. There is analyzed how those methods are realized in the CAS. In the last unit the numerical methods of those systems are compared by their complication, convinience for the user and the variety of functions of the numerical methods. After the description of every method there are proposed the examples of the task solutions by using CAS.