A new active contour model for finding and mapping the outer cortex in brain images is developed. A cross-section of the brain cortex is modeled as a ribbon, and a constant speed mapping of its spine is sought. A variational formulation, an associated force balance condition, and a numerical approach are proposed to achieve this goal. The primary difference between this formulation and that of snakes is in the specification of the external force acting on the active contour. A study of the uniqueness and fidelity of solutions is made through convexity and frequency domain analyses, and a criterion for selection of the regularization coefficient is developed. Examples demonstrating the performance of this method on simulated and real data are provided.

. Based on the first part of this paper 17 , a numerical method is developed for the real-time computation of neighboring optimal feedback controls for constrained optimal control problems. Using the idea of multiple shooting, a numerical method is developed which has the following properties: 1. The method is applicable to optimal control problems with constraints (differential equations, boundary conditions, inequality constraints, problems with discontinuities, etc.). 2. The control variables and the switching points are computed for the remaining time interval of the process. 3. All constraints are checked. 4. The method is appropriate for real-time computations on onboard computers of space vehicles. 5. The scheme is robust in that controllable deviations from a precalculated flight path are much larger than deviations typical for perturbations occurring in space vehicles. The reentry of a space vehicle is investigated as an example. One problem contains a control variable inequ...

We describe a robust, adaptive algorithm for the solution of singularly perturbed twopoint boundary value problems. Many different phenomena can arise in such problems, including boundary layers, dense oscillations, and complicated or ill-conditioned internal transition regions. Working with an integral equation reformulation of the original differential equation, we introduce a method for error analysis which can be used for mesh refinement even when the solution computed on the current mesh is underresolved. Based on this method, we have constructed a black-box code for stiff problems which automatically generates an adaptive mesh resolving all features of the solution. The solver is direct and of arbitrarily high-order accuracy and requires an amount of time proportional to the number of grid points.

The SLEIGN2 code is based on the ideas and methods of the original SLEIGN code of 1979. The main purpose of the SLEIGN2 code is to compute eigenvalues and eigenfunctions of regular and singular Sturm-Liouville problems, with both separated and coupled boundary conditions, and to approximate the continuous spectrum in the singular case. The code uses a number of different algorithms, some of which are new, and has a user-friendly interface. In this paper the algorithms and their implementation are discussed, and the class of problems to which each algorithm applies is identified.

We investigate the convergence properties of the Iterated Defect Correction (IDeC) method based on the implicit Euler rule for the solution of singular initial value problems with a singularity of the first kind. We show that the method retains its classical order of convergence which means that the sequence of approximations obtained during the iteration shows gradually growing order of convergence limited by the smoothness of the data and technical details of the procedure.

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This paper studies numerical methods for linear stability analysis of periodic solutions in codes for bifurcation analysis of small systems of ordinary differential equations (ODEs). Popular techniques in use today (including the AUTO97 method) produce very inaccurate Floquet multipliers if the system has very large or small multipliers. These codes compute the monodromy matrix explicitly or as a matrix pencil of two matrices. The monodromy matrix arises naturally as a product of many matrices in many numerical methods, but this is not exploited. In this case, all Floquet multipliers can be computed with very high precision by using the periodic Schur decomposition and corresponding algorithm [Bojanczyk et al., 1992]. The time discretisation of the periodic orbit becomes the limiting factor for the accuracy. We present just enough of the numerical methods to show how the Floquet multipliers are currently computed and how the periodic Schur decomposition can be fitted into existing codes...

A new asymptotic multiple scale expansion is used to derive envelope equations for localized spatially periodic patterns in the context of the generalized Swift-Hohenberg equation. An analysis of this envelope equation results in parametric conditions for localized patterns. Furthermore, it yields corrections for wave number selection which are an order of magnitude larger for asymmetric nonlinearities than for the symmetric case. The analytical results are compared with numerical computations which demonstrate that the condition for localized patterns coincides with vanishing Hamiltonian and Lagrangian for periodic solutions. One striking feature of the choice of scaling parameters is that the derived condition for localized patterns agrees with the numerical results for a significant range of parameters which are an O(1) distance from the bifurcation, thus providing a novel approach for studying these localized patterns.

Abstract. We study propagation failure using the one-dimensional scalar bistable equation with a passive “gap ” region. By applying comparison principles for this type of equation, the problem of finding conditions for block is reduced to finding conditions for the existence of steady state solutions. We present a geometrical method that allows one to easily compute the critical gap length above which a steady state solution, and thus block, first occurs. The method also helps to uncover the general bifurcation structure of the problem including the stability of the steady state solutions. In obtaining these results, we characterize the relationship between the properties of the system and propagation failure. The method can easily be extended to other gap dynamics. We use it to show that block associated with any local inhomogeneity must be associated with a limit point bifurcation.

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi’s solution of the Hamilton– Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators. We also illustrate these systematic methods for constructing variational integrators with numerical examples.