15,120 research outputs found
Boundary knot method: A meshless, exponential convergence, integration-free, and boundary-only RBF technique
Based on the radial basis function (RBF), non-singular general solution and
dual reciprocity principle (DRM), this paper presents an inheretnly meshless,
exponential convergence, integration-free, boundary-only collocation techniques
for numerical solution of general partial differential equation systems. The
basic ideas behind this methodology are very mathematically simple and
generally effective. The RBFs are used in this study to approximate the
inhomogeneous terms of system equations in terms of the DRM, while non-singular
general solution leads to a boundary-only RBF formulation. The present method
is named as the boundary knot method (BKM) to differentiate it from the other
numerical techniques. In particular, due to the use of non-singular general
solutions rather than singular fundamental solutions, the BKM is different from
the method of fundamental solution in that the former does no need to introduce
the artificial boundary and results in the symmetric system equations under
certain conditions. It is also found that the BKM can solve nonlinear partial
differential equations one-step without iteration if only boundary knots are
used. The efficiency and utility of this new technique are validated through
some typical numerical examples. Some promising developments of the BKM are
also discussed.Comment: 36 pages, 2 figures, Welcome to contact me on this paper: Email:
[email protected] or [email protected]
A multigrid continuation method for elliptic problems with folds
We introduce a new multigrid continuation method for computing solutions of nonlinear elliptic eigenvalue problems which contain limit points (also called turning points or folds). Our method combines the frozen tau technique of Brandt with pseudo-arc length continuation and correction of the parameter on the coarsest grid. This produces considerable storage savings over direct continuation methods,as well as better initial coarse grid approximations, and avoids complicated algorithms for determining the parameter on finer grids. We provide numerical results for second, fourth and sixth order approximations to the two-parameter, two-dimensional stationary reaction-diffusion problem: Δu+λ exp(u/(1+au)) = 0.
For the higher order interpolations we use bicubic and biquintic splines. The convergence rate is observed to be independent of the occurrence of limit points
Dynamic Estimation of Rigid Motion from Perspective Views via Recursive Identification of Exterior Differential Systems with Parameters on a Topological Manifold
We formulate the problem of estimating the motion of a rigid object viewed under perspective projection as the identification of a dynamic model in Exterior Differential form with parameters on a topological manifold.
We first describe a general method for recursive identification of nonlinear implicit systems using prediction error criteria. The parameters are allowed to move slowly on some topological (not necessarily smooth) manifold. The basic recursion is solved in two different ways: one is based on a simple extension of the traditional Kalman Filter to nonlinear and implicit measurement constraints, the other may be regarded as a generalized "Gauss-Newton" iteration, akin to traditional Recursive Prediction Error Method techniques in linear identification. A derivation of the "Implicit Extended Kalman Filter" (IEKF) is reported in the appendix.
The ID framework is then applied to solving the visual motion problem: it indeed is possible to characterize it in terms of identification of an Exterior Differential System with parameters living on a C0 topological manifold, called the "essential manifold". We consider two alternative estimation paradigms. The first is in the local coordinates of the essential manifold: we estimate the state of a nonlinear implicit model on a linear space. The second is obtained by a linear update on the (linear) embedding space followed by a projection onto the essential manifold. These schemes proved successful in performing the motion estimation task, as we show in experiments on real and noisy synthetic image sequences
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
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