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A steepest descent method is constructed for the general setting of a linear differential equation paired with uniqueness-inducing conditions which might yield a generally overdetermined system. The method differs from traditional steepest descent methods by considering the conditions when defining the corresponding Sobolev space. The descent method converges to the unique solution to the differential equation so that change in condition values is minimal. The system has a solution if and only if the first iteration of steepest descent satisfies the system. The finite analogue of the descent method is applied to example problems involving finite difference equations. The well-posed problems include a singular ordinary differential equation and Laplace’s equation, each paired with respective Dirichlet-type conditions. The overdetermined problems include a first-order nonsingular ordinary differential equation with Dirichlet-type conditions and the wave equation with both Dirichlet and Neumann conditions. The method is applied in an investigation of the Tricomi equation, a long-studied equation which acts as a prototype of mixed partial differential equations and has application in transonic flow. The Tricomi equation has been studied for at least ninety years, yet necessary and sufficient conditions for existence and uniqueness of solutions on an arbitrary mixed domain remain unknown. The domains of interest are rectangular mixed domains. A new type of conditions is introduced. Ladder conditions take the uncommon approach of specifying information on the interior of a mixed domain. Specifically, function values are specified on the parabolic portion of a mixed domain. The remaining conditions are specified on the boundary. A conjecture is posed and states that ladder conditions are necessary and sufficient for existence and uniqueness of a solution to the Tricomi equation. Numerical experiments, produced by application of the descent method, provide strong evidence in support of the conjecture. Ladder conditions allow for a continuous deformation from Dirichlet conditions to initial-boundary value conditions. Such a deformation is applied to a class of Tricomi-type equations which transition from degenerate elliptic to degenerate hyperbolic. A conjecture is posed and states that each problem is uniquely solvable and the solutions vary continuously as the differential equation and corresponding conditions vary continuously. If the conjecture holds true, the result will provide a method of unifying elliptic Dirichlet problems and hyperbolic initial-boundary value problem. Numerical evidence in support of the conjecture is presented.

This paper contains four main ideas. First, it shows global existence for the steepest descent in the uniformly convex setting. Secondly, it shows existence of critical points for convex functions defined on uniformly convex spaces. Thirdly, it shows an isomorphism between the dual space of H^{1,p}[0,1] and the space H^{1,q}[0,1] where p > 2 and {1/p} + {1/q} = 1. Fourthly, it shows how the Beurling-Denny theorem can be extended to find a useful function from H^{1,p}[0,1] to L_{p}[1,0] where p > 2 and addresses the problem of using that function to establish a relationship between the ordinary and the Sobolev gradients. The paper contains some numerical experiments and two computer codes.

We develop a numerical method for solving singular differential equations and demonstrate the method on a variety of singular problems including first order ordinary differential equations, second order ordinary differential equations which have variational principles, and one partial differential equation.

The method of steepest descent is used to minimize typical functionals from elasticity.

The method of steepest descent is applied to a nonlinearly constrained optimization problem which arises in the study of liquid crystals. Let Ω denote the region bounded by two coaxial cylinders of height 1 with the outer cylinder having radius 1 and the inner having radius ρ. The problem is to find a mapping, u, from Ω into R^3 which agrees with a given function v on the surfaces of the cylinders and minimizes the energy function over the set of functions in the Sobolev space H^(1,2)(Ω; R^3) having norm 1 almost everywhere. In the variational formulation, the norm 1 condition is emulated by a constraint function B. The direction of descent studied here is given by a projected gradient, called a B-gradient, which involves the projection of a Sobolev gradient onto the tangent space for B. A numerical implementation of the algorithm, the results of which agree with the theoretical results and which is independent of any strong properties of the domain, is described. In chapter 2, the Sobolev space setting and a significant projection in the theory of Sobolev gradients are discussed. The variational formulation is introduced in Chapter 3, where the issues of differentiability and existence of gradients are explored. A theorem relating the B-gradient to the theory of Lagrange multipliers is stated as well. Basic theorems regarding the continuous steepest descent given by the Sobolev and B-gradients are stated in Chapter 4, and conditions for convergence in the application to the liquid crystal problem are given as well. Finally, in Chapter 5, the algorithm is described and numerical results are examined.

<p>This thesis presents improvements and extensions of a previously presented threat assessment algorithm. The algorithm uses Monte Carlo simulation to find threats in a road scene. It is shown that, by using a wider sample distribution and only apply the most likely samples from the Monte Carlo simulation, for the threat assessment, improved results are obtained. By using this method more realistic paths will be chosen by the simulated vehicles and more complex traffic situations will be adequately handled.</p><p>An improvement of the dynamic model is also suggested, which improves the realism of the Monte Carlo simulations. Using the new dynamic model less false positive and more valid threats are detected.</p><p>A systematic method to choose parameters in a stochastic space, using optimisation, is suggested. More realistic trajectories can be chosen, by applying this method on the parameters that represents the human behaviour, in the threat assessment algorithm.</p><p>A new definition of obstacles in a road scene is suggested, dividing them into two groups, Hard and Soft obstacles. A change to the resampling step, in the Monte Carlo simulation, using the soft and hard obstacles is also suggested.</p>

No description available.

This thesis presents improvements and extensions of a previously presented threat assessment algorithm. The algorithm uses Monte Carlo simulation to find threats in a road scene. It is shown that, by using a wider sample distribution and only apply the most likely samples from the Monte Carlo simulation, for the threat assessment, improved results are obtained. By using this method more realistic paths will be chosen by the simulated vehicles and more complex traffic situations will be adequately handled. An improvement of the dynamic model is also suggested, which improves the realism of the Monte Carlo simulations. Using the new dynamic model less false positive and more valid threats are detected. A systematic method to choose parameters in a stochastic space, using optimisation, is suggested. More realistic trajectories can be chosen, by applying this method on the parameters that represents the human behaviour, in the threat assessment algorithm. A new definition of obstacles in a road scene is suggested, dividing them into two groups, Hard and Soft obstacles. A change to the resampling step, in the Monte Carlo simulation, using the soft and hard obstacles is also suggested.

Thesis (Ph. D.)--University of Washington, 1996. / Vita. Includes bibliographical references (leaves [127]-133).