We propose novel methods to utilize orthogonal polynomial approximation in higher dimension spaces, which enable us to modify classical differential equation solvers to perform high precision, long-term orbit propagation. These methods have immediate application to efficient propagation of catalogs of Resident Space Objects (RSOs) and improved accounting for the uncertainty in the ephemeris of these objects. More fundamentally, the methodology promises to be of broad utility in solving initial and two point boundary value problems from a wide class of mathematical representations of problems arising in engineering, optimal control, physical sciences and applied mathematics. We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel way, through the use of array algebra and Kronecker operations. Approximation of test functions illustrates the resulting algorithms and provides insight into the errors of approximation, as well as the associated errors arising when the approximations are differentiated or integrated. Two sets of applications are considered that are challenges in astrodynamics. The first application addresses local approximation of high degree and order geopotential models, replacing the global spherical harmonic series by a family of locally precise orthogonal polynomial approximations for efficient computation. A method is introduced which adapts the approximation degree radially, compatible with the truth that the highest degree approximations (to ensure maximum acceleration error < 10^−9ms^−2, globally) are required near the Earths surface, whereas lower degree approximations are required as radius increases. We show that a four order of magnitude speedup is feasible, with both speed and storage efficiency op- timized using radial adaptation. The second class of problems addressed includes orbit propagation and solution of associated boundary value problems. The successive Chebyshev-Picard path approximation method is shown well-suited to solving these problems with over an order of magnitude speedup relative to known methods. Furthermore, the approach is parallel-structured so that it is suited for parallel implementation and further speedups. Used in conjunction with orthogonal Finite Element Model (FEM) gravity approximations, the Chebyshev-Picard path approximation enables truly revolutionary speedups in orbit propagation without accuracy loss.

This paper was presented in WCSMO-9, Shizuoka.

Doctor of Philosophy (PhD) / For many practitioners and market participants, the valuation of financial derivatives is considered of very high importance as its uses range from a risk management tool, to a speculative investment strategy or capital enhancement. A developing market requires efficient but accurate methods for valuing financial derivatives such as American options. A closed form analytical solution for American options has been very difficult to obtain due to the different boundary conditions imposed on the valuation problem. Following the method of solving the American option as a free boundary problem in the spirit of the "no-arbitrage" pricing framework of Black-Scholes, the option price and hedging parameters can be represented as an integral equation consisting of the European option value and an early exercise value dependent upon the optimal free boundary. Such methods exist in the literature and along with risk-neutral pricing methods have been implemented in practice. Yet existing methods are accurate but inefficient, or accuracy has been compensated for computational speed. A new numerical approach to the valuation of American options by cubic splines is proposed which is proven to be accurate and efficient when compared to existing option pricing methods. Further comparison is made to the behaviour of the American option's early exercise boundary with other pricing models.

Nós refinamos o planejador introduzindo uma nova forma para o núcleo da equação que permite facilmente lidar com terrenos não-homogêneos. Isto é obtido através de mudanças locais na concavidade/convexidade do potencial, criando regiões com altas ou baixas preferências de navegação. Nós integramos esta nova equação ao planejador hierárquico, surgindo uma ampla variedade de aplicações. Nossa proposta contribui para diversas áreas incluindo a navegação de agentes, pathfinding em jogos, simulação de multidões, e a navegação de robôs. Nossas publicações reforçam a relevância e robustez do método proposto. / In this work, we propose a new solution to agent navigation based upon boundary value problems (BVP), called Configurable Flows, to control steering behaviors of characters in dynamic environments. We use a potential field formalism that allows synthetic actors to move negotiating space, avoiding collisions, and attaining goals while producing very individual paths. The individuality of each character can be set by changing its inner field parameters leading to a broad range of possible behaviors without jeopardizing its performance. BVP Path Planners generate potential fields through a differential equation whose gradient descent represents navigation routes from any point of the environment to a goal position. Resulting paths are smooth and free from local minima. In spite of these advantages, these kind of planners consumes a lot of time to produce a solution. Our approach combines a BVP Path Planner with the Full Multigrid Method, which solves elliptic partial differential equations using a hierarchical strategy. The proposed planner enables real-time performance in large environments. Results show that our proposal spends less than 1% of the time needed to compute a solution using the original BVP planners in several environments. We refine our Path Planner by introducing a new form of the core equation that permits to easily cope with terrain inhomogeneities. This is accomplished by locally changing the concavity/ convexity of the potential, and then creating regions with higher or lower navigation preferences. As the potential field requires several steps to converge, this approach can be expensive computationally. To overcome this problem, we integrate this novel core equation to the hierarchical planner, emerging a wide variety of applications. We believe our proposal can contribute to several areas of research including agent navigation, pathfinding for games, crowd simulation and robotics. Our publications reinforce the relevance of the proposed method.

Nós refinamos o planejador introduzindo uma nova forma para o núcleo da equação que permite facilmente lidar com terrenos não-homogêneos. Isto é obtido através de mudanças locais na concavidade/convexidade do potencial, criando regiões com altas ou baixas preferências de navegação. Nós integramos esta nova equação ao planejador hierárquico, surgindo uma ampla variedade de aplicações. Nossa proposta contribui para diversas áreas incluindo a navegação de agentes, pathfinding em jogos, simulação de multidões, e a navegação de robôs. Nossas publicações reforçam a relevância e robustez do método proposto. / In this work, we propose a new solution to agent navigation based upon boundary value problems (BVP), called Configurable Flows, to control steering behaviors of characters in dynamic environments. We use a potential field formalism that allows synthetic actors to move negotiating space, avoiding collisions, and attaining goals while producing very individual paths. The individuality of each character can be set by changing its inner field parameters leading to a broad range of possible behaviors without jeopardizing its performance. BVP Path Planners generate potential fields through a differential equation whose gradient descent represents navigation routes from any point of the environment to a goal position. Resulting paths are smooth and free from local minima. In spite of these advantages, these kind of planners consumes a lot of time to produce a solution. Our approach combines a BVP Path Planner with the Full Multigrid Method, which solves elliptic partial differential equations using a hierarchical strategy. The proposed planner enables real-time performance in large environments. Results show that our proposal spends less than 1% of the time needed to compute a solution using the original BVP planners in several environments. We refine our Path Planner by introducing a new form of the core equation that permits to easily cope with terrain inhomogeneities. This is accomplished by locally changing the concavity/ convexity of the potential, and then creating regions with higher or lower navigation preferences. As the potential field requires several steps to converge, this approach can be expensive computationally. To overcome this problem, we integrate this novel core equation to the hierarchical planner, emerging a wide variety of applications. We believe our proposal can contribute to several areas of research including agent navigation, pathfinding for games, crowd simulation and robotics. Our publications reinforce the relevance of the proposed method.

Nós refinamos o planejador introduzindo uma nova forma para o núcleo da equação que permite facilmente lidar com terrenos não-homogêneos. Isto é obtido através de mudanças locais na concavidade/convexidade do potencial, criando regiões com altas ou baixas preferências de navegação. Nós integramos esta nova equação ao planejador hierárquico, surgindo uma ampla variedade de aplicações. Nossa proposta contribui para diversas áreas incluindo a navegação de agentes, pathfinding em jogos, simulação de multidões, e a navegação de robôs. Nossas publicações reforçam a relevância e robustez do método proposto. / In this work, we propose a new solution to agent navigation based upon boundary value problems (BVP), called Configurable Flows, to control steering behaviors of characters in dynamic environments. We use a potential field formalism that allows synthetic actors to move negotiating space, avoiding collisions, and attaining goals while producing very individual paths. The individuality of each character can be set by changing its inner field parameters leading to a broad range of possible behaviors without jeopardizing its performance. BVP Path Planners generate potential fields through a differential equation whose gradient descent represents navigation routes from any point of the environment to a goal position. Resulting paths are smooth and free from local minima. In spite of these advantages, these kind of planners consumes a lot of time to produce a solution. Our approach combines a BVP Path Planner with the Full Multigrid Method, which solves elliptic partial differential equations using a hierarchical strategy. The proposed planner enables real-time performance in large environments. Results show that our proposal spends less than 1% of the time needed to compute a solution using the original BVP planners in several environments. We refine our Path Planner by introducing a new form of the core equation that permits to easily cope with terrain inhomogeneities. This is accomplished by locally changing the concavity/ convexity of the potential, and then creating regions with higher or lower navigation preferences. As the potential field requires several steps to converge, this approach can be expensive computationally. To overcome this problem, we integrate this novel core equation to the hierarchical planner, emerging a wide variety of applications. We believe our proposal can contribute to several areas of research including agent navigation, pathfinding for games, crowd simulation and robotics. Our publications reinforce the relevance of the proposed method.

Systems of grid equations that approximate elliptic boundary value problems on locally modified grids are considered. The triangulation, which approximates the boundary with second order of accuracy, is generated from an initial uniform triangulation by shifting nodes near the boundary according to special rules. This "locally modified" grid possesses several significant features: this triangulation has a regular structure, the generation of the triangulation is rather fast, this construction allows to use multilevel preconditioning (BPX-like) methods. The proposed iterative methods for solving elliptic boundary value problems approximately are based on two approaches: The fictitious space method, i.e. the reduction of the original problem to a problem in an auxiliary (fictitious) space, and the multilevel decomposition method, i.e. the construction of preconditioners by decomposing functions on hierarchical grids. The convergence rate of the corresponding iterative process with the preconditioner obtained is independent of the mesh size. The construction of the grid and the preconditioning operator for the three dimensional problem can be done in the same way.

In the paper we construct exact three-point discretizations of linear and nonlinear two-point boundary value problems with boundary conditions of the first kind. The finite element approach uses basis functions defined by the coefficients of the differential equations. All the discretized boundary value problems are of inverse isotone type and so are its exact discretizations which involve tridiagonal M-matrices in the linear case and M-functions in the nonlinear case.

No description available.

In this work we study vertical, acoustic propagation in a non-homogeneous media for a spatially-compact, time-harmonic source. An analytical, 2-layer model is developed representing the acoustic pressure disturbance propagating in the atmosphere. The validity of the model spans the distance from the Earth's surface to 30,000 meters. This includes the troposphere (adiabatic), ozone layer (isothermal), and part of the stratosphere (isothermal). The results of the model derivation in the adiabatic region yield pressure solutions as Bessel functions of the First (J) and Second (Y) Kind of order $-\frac{7}{2}$ with an argument of $2 \Omega \tau$ (where $\Omega$ represents a dimensionless frequency and $\tau$ is a dimensionless vertical height in z (vertical coordinate)). For an added second layer (isothermal region), the pressure solution is a decaying sinusoidal, exponential function above the first layer. In particular, the vertical, acoustic propagation is examined for various configurations. These are divided into 2 basic classes. The first class consists of examining the pressure response function when the source is located on boundary interfaces, while the second class consists of situations where the source is arbitrarily located within a finite layer. In all instances, a time-harmonic, compact source is implicitly understood. However, each class requires a different method of solution. The first class conforms to a general boundary value problem, while the second requires the use of Green's functions method. In investigating problems of the first class, 3 different scenarios are examined. In the first case, we apply our model to a semi-infinite medium with a time-harmonic source ($e^{-i \omega t}$) located on the ground. In the next 2 cases, a semi-infinite medium is overlain on the previous medium at a height of z=13,000 meters. Thus, there exist two boundaries: the ground and the layer interface between the 2 media. Sources placed at these interfaces represent the 2nd and 3rd scenarios, respectively. The solutions to all 3 cases are of the form $A \frac{J_{-\frac{7}{2}}(2 \Omega \tau)}{{\tau}^{-\frac{7}{2}}} + B \frac{Y_{-\frac{7}{2}}(2 \Omega \tau)}{{\tau}^{-\frac{7}{2}}}$, where \textit{A} and \textit{B} are constants determined by the boundary conditions. For the 2nd class, we examine the application to a time-harmonic, compact source placed arbitrarily within the 1st layer. The method of Green's functions is used to obtain a particular solution for the model equations. This result is compared with a Fast Field Program (FFP) which was developed to test these solutions. The results show that the response given by the Green's function compares favorably with that of the FFP. Keywords: Linear Acoustics, Inhomogeneous Medium, Layered Atmosphere, Boundary Value Problem, Green's Function Method