It was not until the 20th century that combinatorial design theory was studied as a formal subject. This field has many applications, for example in statistical experimental design, coding theory, authentication codes, and cryptography. Major approaches to the problem of discovering new t-designs rely on (i) the construction of large sets of t designs, (ii) using prescribed automorphism groups, (iii) recursive construction methods. In 2017 and 2018, Tran Van Trung introduced new recursive techniques to construct t – (v, k, λ) designs. These methods are of purely combinatorial nature and require using "ingredient" t-designs or resolutions whose parameters satisfy a system of non-linear equations. Even after restricting the range of parameters in this new method, the task is computationally intractable. In this work, we enhance Tran Van Trung's "Basic Construction" by a robust and efficient hybrid computational apparatus which enables us to construct hundreds of thousands of new t – (v, k, Λ) designs from previously known ingredient designs. Towards the end of the dissertation we also create a new family of 2-resolutions, which will be infinite if there are infinitely many Sophie Germain primes. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection

Two orthonormal bases in the complex space of dimension d, are said to be mutually unbiased if the square of the magnitude of the inner product of any vector from one basis with any vector in other basis is equal to the reciprocal of the dimension d. Mutually unbiased bases are used for optimal state determination of mixed quantum states. It is known that in any dimension d, the number of mutually unbiased bases is at most d+1. Ivanovic found a complete set of mutually unbiased bases for prime dimensions. His construction was generalized by Wootters and Fields for prime power dimensions. There is a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. Based on this connection, there exits a constructive proof of the existence of a complete set of mutually unbiased bases for prime power dimensions. This thesis is an exploration on construction of mutually unbiased bases.

Two orthonormal bases in the complex space of dimension d, are said to be mutually unbiased if the square of the magnitude of the inner product of any vector from one basis with any vector in other basis is equal to the reciprocal of the dimension d. Mutually unbiased bases are used for optimal state determination of mixed quantum states. It is known that in any dimension d, the number of mutually unbiased bases is at most d+1. Ivanovic found a complete set of mutually unbiased bases for prime dimensions. His construction was generalized by Wootters and Fields for prime power dimensions. There is a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. Based on this connection, there exits a constructive proof of the existence of a complete set of mutually unbiased bases for prime power dimensions. This thesis is an exploration on construction of mutually unbiased bases.

We consider combinatorial t-design problems as discrete optimization problems. Our motivation is that only a few studies have been done on the use of exact optimization techniques in designs, and that classical methods in design theory have still left many open existence questions. Roughly defined, t-designs are pairs of discrete sets that are related following some strict properties of size, balance, and replication. These highly structured relationships provide optimal solutions to a variety of problems in computer science like error-correcting codes, secure communications, network interconnection, design of hardware; and are applicable to other areas like statistics, scheduling, games, among others. We give a new approach to combinatorial t-designs that is useful in constructing t-designs by polyhedral methods. The first contribution of our work is a new result of equivalence of t-design problems with a graph theory problem. This equivalence leads to a novel integer programming formulation for t-designs, which we call GDP. We analyze the polyhedral properties of GDP and conclude, among other results, the associated polyhedron dimension. We generate new classes of valid inequalities to aim at approximating this integer program by a linear program that has the same optimal solution. Some new classes of valid inequalities are generated as Chv´atal-Gomory cuts, other classes are generated by graph complements and combinatorial arguments, and others are generated by the use of incidence substructures in a t-design. In particular, we found a class of valid inequalities that we call stable-set class that represents an alternative graph equivalence for the problem of finding a t-design. We analyze and give results on the strength of these new classes of valid inequalities. We propose a separation problem and give its integer programming formulation as a maximum (or minimum) edge-weight biclique subgraph problem. We implement a pure cutting-plane algorithm using one of the stronger classes of valid inequalities derived. Several instances of t-designs were solved efficiently by this algorithm at the root node of the search tree. Also, we implement a branch-and-cut algorithm and solve several instances of 2-designs trying different base formulations. Computational results are included.

We consider combinatorial t-design problems as discrete optimization problems. Our motivation is that only a few studies have been done on the use of exact optimization techniques in designs, and that classical methods in design theory have still left many open existence questions. Roughly defined, t-designs are pairs of discrete sets that are related following some strict properties of size, balance, and replication. These highly structured relationships provide optimal solutions to a variety of problems in computer science like error-correcting codes, secure communications, network interconnection, design of hardware; and are applicable to other areas like statistics, scheduling, games, among others. We give a new approach to combinatorial t-designs that is useful in constructing t-designs by polyhedral methods. The first contribution of our work is a new result of equivalence of t-design problems with a graph theory problem. This equivalence leads to a novel integer programming formulation for t-designs, which we call GDP. We analyze the polyhedral properties of GDP and conclude, among other results, the associated polyhedron dimension. We generate new classes of valid inequalities to aim at approximating this integer program by a linear program that has the same optimal solution. Some new classes of valid inequalities are generated as Chv´atal-Gomory cuts, other classes are generated by graph complements and combinatorial arguments, and others are generated by the use of incidence substructures in a t-design. In particular, we found a class of valid inequalities that we call stable-set class that represents an alternative graph equivalence for the problem of finding a t-design. We analyze and give results on the strength of these new classes of valid inequalities. We propose a separation problem and give its integer programming formulation as a maximum (or minimum) edge-weight biclique subgraph problem. We implement a pure cutting-plane algorithm using one of the stronger classes of valid inequalities derived. Several instances of t-designs were solved efficiently by this algorithm at the root node of the search tree. Also, we implement a branch-and-cut algorithm and solve several instances of 2-designs trying different base formulations. Computational results are included.

We consider the problem of numerical integration, where one aims to approximate an integral of a given continuous function from the function values at given sampling points, also known as quadrature points. A useful framework for such an approximation process is provided by the theory of reproducing kernel Hilbert spaces and the concept of the worst case quadrature error. However, the computation of optimal quadrature points, which minimize the worst case quadrature error, is in general a challenging task and requires efficient algorithms, in particular for large numbers of points. The focus of this thesis is on the efficient computation of optimal quadrature points on the torus T^d, the sphere S^d, and the rotation group SO(3). For that reason we present a general framework for the minimization of the worst case quadrature error on Riemannian manifolds, in order to construct numerically such quadrature points. Therefore, we consider, for N quadrature points on a manifold M, the worst case quadrature error as a function defined on the product manifold M^N. For the optimization on such high dimensional manifolds we make use of the method of steepest descent, the Newton method, and the conjugate gradient method, where we propose two efficient evaluation approaches for the worst case quadrature error and its derivatives. The first evaluation approach follows ideas from computational physics, where we interpret the quadrature error as a pairwise potential energy. These ideas allow us to reduce for certain instances the complexity of the evaluations from O(M^2) to O(M log(M)). For the second evaluation approach we express the worst case quadrature error in Fourier domain. This enables us to utilize the nonequispaced fast Fourier transforms for the torus T^d, the sphere S^2, and the rotation group SO(3), which reduce the computational complexity of the worst case quadrature error for polynomial spaces with degree N from O(N^k M) to O(N^k log^2(N) + M), where k is the dimension of the corresponding manifold. For the usual choice N^k ~ M we achieve the complexity O(M log^2(M)) instead of O(M^2). In conjunction with the proposed conjugate gradient method on Riemannian manifolds we arrive at a particular efficient optimization approach for the computation of optimal quadrature points on the torus T^d, the sphere S^d, and the rotation group SO(3). Finally, with the proposed optimization methods we are able to provide new lists with quadrature formulas for high polynomial degrees N on the sphere S^2, and the rotation group SO(3). Further applications of the proposed optimization framework are found due to the interesting connections between worst case quadrature errors, discrepancies and potential energies. Especially, discrepancies provide us with an intuitive notion for describing the uniformity of point distributions and are of particular importance for high dimensional integration in quasi-Monte Carlo methods. A generalized form of uniform point distributions arises in applications of image processing and computer graphics, where one is concerned with the problem of distributing points in an optimal way accordingly to a prescribed density function. We will show that such problems can be naturally described by the notion of discrepancy, and thus fit perfectly into the proposed framework. A typical application is halftoning of images, where nonuniform distributions of black dots create the illusion of gray toned images. We will see that the proposed optimization methods compete with state-of-the-art halftoning methods.

We consider the problem of numerical integration, where one aims to approximate an integral of a given continuous function from the function values at given sampling points, also known as quadrature points. A useful framework for such an approximation process is provided by the theory of reproducing kernel Hilbert spaces and the concept of the worst case quadrature error. However, the computation of optimal quadrature points, which minimize the worst case quadrature error, is in general a challenging task and requires efficient algorithms, in particular for large numbers of points. The focus of this thesis is on the efficient computation of optimal quadrature points on the torus T^d, the sphere S^d, and the rotation group SO(3). For that reason we present a general framework for the minimization of the worst case quadrature error on Riemannian manifolds, in order to construct numerically such quadrature points. Therefore, we consider, for N quadrature points on a manifold M, the worst case quadrature error as a function defined on the product manifold M^N. For the optimization on such high dimensional manifolds we make use of the method of steepest descent, the Newton method, and the conjugate gradient method, where we propose two efficient evaluation approaches for the worst case quadrature error and its derivatives. The first evaluation approach follows ideas from computational physics, where we interpret the quadrature error as a pairwise potential energy. These ideas allow us to reduce for certain instances the complexity of the evaluations from O(M^2) to O(M log(M)). For the second evaluation approach we express the worst case quadrature error in Fourier domain. This enables us to utilize the nonequispaced fast Fourier transforms for the torus T^d, the sphere S^2, and the rotation group SO(3), which reduce the computational complexity of the worst case quadrature error for polynomial spaces with degree N from O(N^k M) to O(N^k log^2(N) + M), where k is the dimension of the corresponding manifold. For the usual choice N^k ~ M we achieve the complexity O(M log^2(M)) instead of O(M^2). In conjunction with the proposed conjugate gradient method on Riemannian manifolds we arrive at a particular efficient optimization approach for the computation of optimal quadrature points on the torus T^d, the sphere S^d, and the rotation group SO(3). Finally, with the proposed optimization methods we are able to provide new lists with quadrature formulas for high polynomial degrees N on the sphere S^2, and the rotation group SO(3). Further applications of the proposed optimization framework are found due to the interesting connections between worst case quadrature errors, discrepancies and potential energies. Especially, discrepancies provide us with an intuitive notion for describing the uniformity of point distributions and are of particular importance for high dimensional integration in quasi-Monte Carlo methods. A generalized form of uniform point distributions arises in applications of image processing and computer graphics, where one is concerned with the problem of distributing points in an optimal way accordingly to a prescribed density function. We will show that such problems can be naturally described by the notion of discrepancy, and thus fit perfectly into the proposed framework. A typical application is halftoning of images, where nonuniform distributions of black dots create the illusion of gray toned images. We will see that the proposed optimization methods compete with state-of-the-art halftoning methods.