An evolutionary geometry parametrization is established to represent aerodynamic configurations. This geometry parametrization technique is constructed by integrating the classical B-spline formulation with the knot insertion algorithm. It is capable of inserting control points to a given parametrization without modifying its geometry. Taking advantage of this technique, a shape design problem can be solved as a sequence of optimizations from the basic parametrization to more refined parametrizations. Owing to the nature of the B-spline formulation, feasible parametrization refinements are not unique; guidelines based on sensitivity analysis and geometry constraints are developed to assist the automation of the proposed optimization sequence. Test cases involving airfoil optimization and induced drag minimization are solved adopting this method. Its effectiveness is demonstrated through comparisons with optimizations using uniform refined parametrizations.

For modelling curves, B-splines [3] are among the most versatile control schemes. However, scaling this technique to surface patches has proven to be a non-trivial endeavor. While a suitable scheme exists for rectangular patches in the form of tensor product B-splines, techniques involving the triangular domain are much less spectacular. The current cutting edge in triangular B-splines [2] is the DMS-spline. While the resulting surfaces possess high degrees of continuity, the control scheme is awkward and the evaluation is computationally expensive. A more fundamental problem is the construction bears little resemblance to the construction used for the B-Spline. This deficiency leads to the central idea of the thesis; what happens if the simple blending functions found at the heart of the B-Spline construction are used over higher dimension domains? In this thesis I develop a geometric generalization of B-Spline curves over the triangular domain. This construction mimics the control point blending that occurs with uniform B-Splines. The construction preserves the simple control scheme and evaluation of B-Splines, without the immense computational requirements of DMS-splines. The result is a new patch control scheme, the G-Patch, possessing <i>C</i>0 continuity between adjacent patches.

This thesis proposes an optimization formulation to ensure accuracy and stability in the local volatility function calibration. The unknown local volatility function is represented by kernel splines. The proposed optimization formulation minimizes calibration error and an L1 norm of the vector of coefficients for the kernel splines. The L1 norm regularization forces some coefficients to be zero at the termination of optimization. The complexity of local volatility function model is determined by the number of nonzero coefficients. Thus by using a regularization parameter, the proposed formulation balances the calibration accuracy with the model complexity. In the context of the support vector regression for function based on finite observations, this corresponds to balance the generalization error with the number of support vectors. In this thesis we also propose a trust region method to determine the coefficient vector in the proposed optimization formulation. In this algorithm, the main computation of each iteration is reduced to solving a standard trust region subproblem. To deal with the non-differentiable L1 norm in the formulation, a line search technique which allows crossing nondifferentiable hyperplanes is introduced to find the minimum objective value along a direction within a trust region. With the trust region algorithm, we numerically illustrate the ability of proposed approach to reconstruct the local volatility in a synthetic local volatility market. Based on S&P 500 market index option data, we demonstrate that the calibrated local volatility surface is smooth and resembles in shape the observed implied volatility surface. Stability is illustrated by considering calibration using market option data from nearby dates.

For modelling curves, B-splines [3] are among the most versatile control schemes. However, scaling this technique to surface patches has proven to be a non-trivial endeavor. While a suitable scheme exists for rectangular patches in the form of tensor product B-splines, techniques involving the triangular domain are much less spectacular. The current cutting edge in triangular B-splines [2] is the DMS-spline. While the resulting surfaces possess high degrees of continuity, the control scheme is awkward and the evaluation is computationally expensive. A more fundamental problem is the construction bears little resemblance to the construction used for the B-Spline. This deficiency leads to the central idea of the thesis; what happens if the simple blending functions found at the heart of the B-Spline construction are used over higher dimension domains? In this thesis I develop a geometric generalization of B-Spline curves over the triangular domain. This construction mimics the control point blending that occurs with uniform B-Splines. The construction preserves the simple control scheme and evaluation of B-Splines, without the immense computational requirements of DMS-splines. The result is a new patch control scheme, the G-Patch, possessing <i>C</i>0 continuity between adjacent patches.

This thesis proposes an optimization formulation to ensure accuracy and stability in the local volatility function calibration. The unknown local volatility function is represented by kernel splines. The proposed optimization formulation minimizes calibration error and an L1 norm of the vector of coefficients for the kernel splines. The L1 norm regularization forces some coefficients to be zero at the termination of optimization. The complexity of local volatility function model is determined by the number of nonzero coefficients. Thus by using a regularization parameter, the proposed formulation balances the calibration accuracy with the model complexity. In the context of the support vector regression for function based on finite observations, this corresponds to balance the generalization error with the number of support vectors. In this thesis we also propose a trust region method to determine the coefficient vector in the proposed optimization formulation. In this algorithm, the main computation of each iteration is reduced to solving a standard trust region subproblem. To deal with the non-differentiable L1 norm in the formulation, a line search technique which allows crossing nondifferentiable hyperplanes is introduced to find the minimum objective value along a direction within a trust region. With the trust region algorithm, we numerically illustrate the ability of proposed approach to reconstruct the local volatility in a synthetic local volatility market. Based on S&P 500 market index option data, we demonstrate that the calibrated local volatility surface is smooth and resembles in shape the observed implied volatility surface. Stability is illustrated by considering calibration using market option data from nearby dates.

Electroencephalography (EEG) is a valuable tool for clinical and cognitive applications. EEG allows for measuring and imaging of scalp potentials emitted by brain activity and allows researchers to draw conclusions about underlying brain activity and function. However EEG is limited by poor spatial resolution due to various factors. One reason is the fact that EEG electrodes are separated from current sources in the brain by cerebrospinal fluid (CSF), the skull, and the scalp. Unfortunately the conductivities of these tissues are not yet well known which limits the spatial resolution of EEG.Based on prior research, spatial resolution of the EEG can be improved via use of various mathematical techniques that provide increased accuracy of the representation of scalp potentials. One such method is the surface Laplacian. It has been shown to be a direct approach to improving EEG spatial resolution. Yet this approach depends on a geometric head model and much work has been done on assuming the human head to be spherical.In this project, we will develop a mathematical model for ellipsoidal head geometry based on surface Laplacian calculations by Law [1]. The ellipsoidal head model is more realistic to the human head shape and can therefore improve accuracy of the EEG imaging calculations. We will construct a computational program that utilizes the ellipsoidal head geometry in hopes to provide a more accurate representation of data fits compared to the spherical head models. Also, we will demonstrate that the spline surface Laplacian calculations do indeed increase the spatial resolution thereby affording a greater impact to the clinical and cognitive study community involving EEG. / Department of Physics and Astronomy

Although one significant method of interpolation is that of the cubic spline, it has the drawback of occasionally producing undesired inflections in a curve. As a remedy, the spline can mathematically be "stretched" (so to speak) in much the same way that a draftsman's spline could be pulled at its ends while still being anchored at certain points throughout.This thesis will make use of FORTRAN subroutines given in the April, 1974 issue of Communications of the ACM, which have the capability of applying this tension factor to a cublic spline in a graphics package. It will also discuss the necessary modifications which are required before compatibility can be achieved between these subroutines and the Tektronix terminal which is coupled to the DEC-10 here at Ball State University.

This dissertation presents a synthesis of methods for structural optimization of aircraft wing boxes. The optimization problem considered herein is the minimization of structural weight with respect to component sizes, subject to stress constraints. Different aspects of structural optimization methods representing the current state-of-the-art are discussed, including sequential quadratic programming, sensitivity analysis, parameterization of design variables, constraint handling, and multiple load treatment. Shortcomings of the current techniques are identified and a B-spline parameterization representing the structural sizes is proposed to address them. A new configurable B-spline parameterization method for structural optimization of wing boxes is developed that makes it possible to flexibly explore design spaces. An automatic scheme using different levels of B-spline parameterization configurations is also proposed, along with a constraint aggregation method in order to reduce the computational effort. Numerical results are compared to evaluate the effectiveness of the B-spline approach and the constraint aggregation method. To evaluate the new formulations and explore design spaces, the wing box of an airliner is optimized for the minimum weight subject to stress constraints under multiple load conditions. The new approaches are shown to significantly reduce the computational time required to perform structural optimization and to yield designs that are more realistic than existing methods.

Thesis (M.S.)--Ohio University, August, 2000. / Title from PDF t.p.

Chemnitz, Techn. Univ., Diss., 2006.