Multiple parameter continuation: computing implicitly defined \(k\)-manifolds. (English) Zbl 1044.37053
Summary: We present a new continuation method for computing implicitly defined manifolds. The manifold is represented as a set of overlapping neighborhoods, and extended by an added neighborhood of a boundary point. The boundary point is found using an expression for the boundary in terms of the vertices of a set of finite, convex polyhedra. The resulting algorithm is quite simple, allows adaptive spacing of the computed points, and deals with the problems of local and global overlap in a natural way. The algorithm is robust (the new points need only be near the boundary), and is well suited to problems with large embedding dimension, and small to moderate dimension.
MSC:
| 37M99 | Approximation methods and numerical treatment of dynamical systems |
| 65H10 | Numerical computation of solutions to systems of equations |
Keywords:
sets of overlapping neighborhoods; examples; union of balls; continuation method; computing implicitly defined manifolds; algorithmReferences:
| [1] | Aho A. V., Foundations of Computer Science (1992) · Zbl 0816.68001 |
| [2] | DOI: 10.1137/1022003 · Zbl 0432.65027 · doi:10.1137/1022003 |
| [3] | DOI: 10.1137/0722020 · Zbl 0567.65029 · doi:10.1137/0722020 |
| [4] | DOI: 10.1137/0724033 · Zbl 0618.65006 · doi:10.1137/0724033 |
| [5] | DOI: 10.1142/S0218127491000051 · Zbl 0756.34039 · doi:10.1142/S0218127491000051 |
| [6] | DOI: 10.1016/0031-3203(84)90064-5 · Zbl 0539.52008 · doi:10.1016/0031-3203(84)90064-5 |
| [7] | DOI: 10.1137/0216006 · Zbl 0616.52007 · doi:10.1137/0216006 |
| [8] | DOI: 10.1016/0196-6774(88)90035-1 · Zbl 0642.52009 · doi:10.1016/0196-6774(88)90035-1 |
| [9] | DOI: 10.1145/116873.116880 · doi:10.1145/116873.116880 |
| [10] | DOI: 10.1007/PL00009341 · Zbl 0897.52005 · doi:10.1007/PL00009341 |
| [11] | DOI: 10.1016/0167-8396(88)90013-1 · Zbl 0659.65013 · doi:10.1016/0167-8396(88)90013-1 |
| [12] | DOI: 10.1016/0898-1221(94)00175-8 · Zbl 0815.65068 · doi:10.1016/0898-1221(94)00175-8 |
| [13] | DOI: 10.1016/S0898-1221(98)00164-3 · Zbl 0933.65055 · doi:10.1016/S0898-1221(98)00164-3 |
| [14] | DOI: 10.1016/0021-9991(78)90110-9 · Zbl 0392.73097 · doi:10.1016/0021-9991(78)90110-9 |
| [15] | DOI: 10.1016/0167-6377(91)90042-N · Zbl 0774.90055 · doi:10.1016/0167-6377(91)90042-N |
| [16] | DOI: 10.1145/322139.322141 · Zbl 0403.68067 · doi:10.1145/322139.322141 |
| [17] | DOI: 10.1007/978-1-4757-2249-9 · doi:10.1007/978-1-4757-2249-9 |
| [18] | DOI: 10.1145/88560.88575 · doi:10.1145/88560.88575 |
| [19] | DOI: 10.1142/S0218127497001576 · Zbl 0910.65059 · doi:10.1142/S0218127497001576 |
| [20] | DOI: 10.1007/BF02574053 · Zbl 0826.68053 · doi:10.1007/BF02574053 |
| [21] | Garcia C. B., Pathways to Solutions, Fixed Points, and Equilibria (1981) · Zbl 0512.90070 |
| [22] | Gill P. E., Practical Optimization (1981) · Zbl 0503.90062 |
| [23] | DOI: 10.1137/1.9780898719543 · Zbl 0935.37054 · doi:10.1137/1.9780898719543 |
| [24] | Grunbaum B., Convex Polytopes (1967) |
| [25] | DOI: 10.1109/38.62694 · doi:10.1109/38.62694 |
| [26] | DOI: 10.1142/S021812749100004X · Zbl 0754.70016 · doi:10.1142/S021812749100004X |
| [27] | DOI: 10.1137/0214006 · Zbl 0556.68038 · doi:10.1137/0214006 |
| [28] | Jones T. A., Contouring Geologic Surfaces with the Computer (1986) |
| [29] | H. B. Keller, Applications of Bifurcation Theory, ed. P. Rabinowitz (Academic Press, NY, 1977) pp. 359–384. |
| [30] | Klein R., Lecture Notes in Computer Science 400, in: Concrete and Abstract Voronoi Diagrams (1987) · Zbl 0699.68005 |
| [31] | DOI: 10.1145/37402.37422 · doi:10.1145/37402.37422 |
| [32] | DOI: 10.1016/0898-1221(95)00065-7 · Zbl 0840.65037 · doi:10.1016/0898-1221(95)00065-7 |
| [33] | Morgan A., Solving Polynomial Systems using Continuation for Engineering and Scientific Problems (1987) · Zbl 0733.65031 |
| [34] | DOI: 10.1109/38.252552 · doi:10.1109/38.252552 |
| [35] | DOI: 10.1007/978-3-0348-7241-6_27 · doi:10.1007/978-3-0348-7241-6_27 |
| [36] | DOI: 10.1007/BF01395883 · Zbl 0676.65047 · doi:10.1007/BF01395883 |
| [37] | Sabin M. A., Fundamental Algorithms for Computer Graphics (1985) |
| [38] | DOI: 10.1142/S0218127497001564 · Zbl 0909.58044 · doi:10.1142/S0218127497001564 |
| [39] | DOI: 10.1137/1.9781611970265 · doi:10.1137/1.9781611970265 |
| [40] | DOI: 10.1007/PL00009451 · Zbl 0945.52009 · doi:10.1007/PL00009451 |
| [41] | Watson D. F., Contouring: A Guide to the Analysis and Display of Spatial Data (1992) |
| [42] | C. Zahlten and H. Jürgens, EUROGRAPHICS ’91, eds. F. Post and W. Barth (Elsevier Science Publishers, 1973) pp. 5–19. |
| [43] | Zwillinger D., Standard Mathematical Tables and Formulae (1996) |
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