Allgower, E. L, Georg, K.: Introduction to Numerical Continuation Methods. Springer, New York 1990  Home/Search   Document Not in Database   Summary   Related Articles   Check

....segmentation of connected objects [23] Starting with a given seed voxel, this algorithm iteratively accumulates all 26 connected neighbors 2 , whose gray values exceed a given threshold. This method is based on numerical continuation methods for the piecewise linear approximation of surfaces [1]. The set of all accumulated voxels is assumed to represent the intrahepatic vessel systems. In practical use, the user initially has to guess a threshold, to start the algorithm and to control the result. Usually, these steps must be repeated with modified thresholds, until an appropriate result ....

Allgower, E. L, Georg, K.: Introduction to Numerical Continuation Methods. Springer, New York 1990

....JJT regular problems (regular problems in the sense of Jongen, Jonker and Twilt) cf. Definition 2:4) For the analysis with respect to JJT regular problems, we will assume a higher degree of differentiability of the problemfunctions. Let us recall now the well known concept of embedding (cf. e.g. [1], 2] 4] 9] 10] 11] 13] 21] 22] 24] Construct a one parametric optimization problem P (t) minff(y; t)jy 2 M (t)g; t 2 [0; 1] where M(t) fy 2 IR n jh i (y; t) 0; i 2 I; g j (y; t) 0; j 2 Jg n n, J is a finite index set with J J , with at least the ....

Allgower,E.L. and Georg,K., Introduction to numerical continuation methods, Springer-Verlag Berlin, 1990

....problems, when they are interpreted as embeddings. These properties are mainly related with the use of pathfollowing techniques for solving it. The idea of pathfollowing is not new. It is mainly based on the numerical resolution of equality systems depending on one parameter (see for example [1]) It should be mentioned that an embedding of an optimization problem can be also interpreted in other ways. For example, only as a parametric system describing some critical set of the fixed optimization problem in a fixed point of the parameter. This parametric system is not alwys associated to ....

Allgower, E.L. and Georg, K., Introduction to numerical continuation methods, Springer-Verlag, Berlin, 1990.

.... we claim that the observed constraints must have full dimension(cf. 13] M i = cl intM i i = 1; 2: C3) With the condition (C3) we want to exclude cases where M 1 ae (P 3 i=1 G i ) M 2 ae (P G 1 ) Let us recall now the well known concept of embedding (cf. e.g. Allgower [1], Guddat et al. 5] Dentcheva [3] Gfrerer et al. 10] and propose the following embeddings in order to solve (P i ) i = 1; 2: Let x o 2 P be arbitrarily chosen and dene minff(x; t) j x 2 M 1 (t)g t 2 ( Gamma1; 1] P 1 (t) 2 Theoretical Background 4 where f(x; t) x Gamma x o ) ....

Allgower, E. L. and Georg, K.: Introduction to numerical continuation methods. Springer-Verlag, Berlin, Heidelberg, New York 1990

....their potential for automation. A significant shortcoming in certain applications is their topological inflexibility. In this paper we describe a new class of deformable contour models known as topology adaptive snakes, or T snakes. Our approach exploits an affine cell decomposition (Munkres 1984; Allgower and Georg 1990) of the image domain, creating a mathematically sound framework that significantly extends the abilities of standard snake models. The affine cell image decomposition (ACID) divides the image domain into a collection of convex polytopes. We immerse discrete versions of conventional parametric ....

....defined object. A disambiguation scheme consists of a table lookup to identify ambiguous cases followed by adherence to a disambiguation strategy such as preferred polarity always separate the positive vertices (and join the negatives) or vice versa. In a simplicial cell decomposition (Allgower and Georg 1990), also known as a triangulation, space is partitioned into cells defined by open simplices, where an n simplex is the simplest ge11 Figure 2: Freudenthal triangulation. Figure 3: Simplex classification. ometrical object of dimension n: e.g. a triangle in 2D or a ....

Allgower, E.L. and Georg, K. (1990). Introduction to Numerical Continuation Methods. Berlin, Heidelberg: Springer--Verlag.

....will be presented. We use a particular cone construction for handling the homotopy parameter. Special attention is given to convergence results. Numerical details of the algorithms can only be sketched. For a more detailed presentation of such algorithms and bibliographical remarks we refer to [4]. I. Introduction 1 The first and most prominent example of a PL algorithm was designed by Lemke Howson [33] and Lemke [30] to calculate a solution of the linear complementarity problem. This algorithm played a crucial role in the development of subsequent PL algorithms. Linear complementarity ....

....points v 1 ,v 2 , v n 1 #E. These points are the vertices of #. If a PL manifold M of dimension n consists only of simplices, then M is called a pseudo manifold of dimension n. Such manifolds are of special importance, since PL methods simplify considerably, see e.g. 2] 3] [4], 25] 40] 42] In this case, a pivoting step can also be described in a di#erent way: Let # = v 1 ,v 2 , v n 1 ] #M. Then a facet # of # is obtained by deleting one vertex, say the vertex 4 v i for some i # 1, 2, n 1 . Thus we obtain the facet # = v 1 , v i 1 ,v i 1 , v ....

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E. L. Allgower & K. Georg, "Introduction to numerical continuation methods," in preparation.

....see, e.g. Griewank [11, 12] Decker, Keller and Kelley [7] Rall [22] and related references cited therein. Remark 8. The convergence results of this paper can be used to extend the applicability of Newton like methods to numerical continuation (homotopy) methods (see Allgower and Georg [1]) In [1] the authors invoke convergence properties of Newton s method using the Moore Penrose inverse under the assumption that the Jacobian matrix has maximal rank. 4. Examples We conclude this paper by giving three simple examples. The purpose of the first two examples is to show that ....

....see, e.g. Griewank [11, 12] Decker, Keller and Kelley [7] Rall [22] and related references cited therein. Remark 8. The convergence results of this paper can be used to extend the applicability of Newton like methods to numerical continuation (homotopy) methods (see Allgower and Georg [1] In [1] the authors invoke convergence properties of Newton s method using the Moore Penrose inverse under the assumption that the Jacobian matrix has maximal rank. 4. Examples We conclude this paper by giving three simple examples. The purpose of the first two examples is to show that conditions ....

Allgower, E.L. and Georg, K.: Introduction to numerical continuation methods. Berlin, Heidelberg, New York: Springer--Verlag 1989

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