Incremental algorithm is one of the most popular procedures for constructing Delaunay triangulations （DTs）. However, the point insertion sequence has a great impact on the amount of work needed for the construction ...Incremental algorithm is one of the most popular procedures for constructing Delaunay triangulations （DTs）. However, the point insertion sequence has a great impact on the amount of work needed for the construction of DTs. It affects the time for both point location and structure update, and hence the overall computational time of the triangulation algorithm. In this paper, a simple deterministic insertion sequence is proposed based on the breadth-first-search on a Kd-tree with some minor modifications for better performance. Using parent nodes as search-hints, the proposed insertion sequence proves to be faster and more stable than the Hilbert curve order and biased randomized insertion order （BRIO）, especially for non-uniform point distributions over a wide range of benchmark examples.展开更多
In this paper a triangulation of continuous and arbitrary refinement of grid sizes is proposed for simplicial homotopy algorithms to compute zero points on a polytope P. The proposed algorithm generates a piecewise li...In this paper a triangulation of continuous and arbitrary refinement of grid sizes is proposed for simplicial homotopy algorithms to compute zero points on a polytope P. The proposed algorithm generates a piecewise linear path in P × [1,∞) from any chosen interior point x0 of P on level {1} to a solution of the underlying problem. The path is followed by making linear programming pivot steps in a linear system and replacement steps in the triangnlation.The starting point x0 is left in a direction to one vertex of P. The direction in which x0 leaves depends on the function value at x0 and the polytope P. Moreover, we also give a new equivalent form of the Brouwer fixed point theorem on polytopes. This form has many important applications in mathematical programming and the theory of differential equations.展开更多
基金supported by the National Natural Science Foundation of China (10972006 and 11172005)the National Basic Research Program of China (2010CB832701)
文摘Incremental algorithm is one of the most popular procedures for constructing Delaunay triangulations （DTs）. However, the point insertion sequence has a great impact on the amount of work needed for the construction of DTs. It affects the time for both point location and structure update, and hence the overall computational time of the triangulation algorithm. In this paper, a simple deterministic insertion sequence is proposed based on the breadth-first-search on a Kd-tree with some minor modifications for better performance. Using parent nodes as search-hints, the proposed insertion sequence proves to be faster and more stable than the Hilbert curve order and biased randomized insertion order （BRIO）, especially for non-uniform point distributions over a wide range of benchmark examples.
文摘In this paper a triangulation of continuous and arbitrary refinement of grid sizes is proposed for simplicial homotopy algorithms to compute zero points on a polytope P. The proposed algorithm generates a piecewise linear path in P × [1,∞) from any chosen interior point x0 of P on level {1} to a solution of the underlying problem. The path is followed by making linear programming pivot steps in a linear system and replacement steps in the triangnlation.The starting point x0 is left in a direction to one vertex of P. The direction in which x0 leaves depends on the function value at x0 and the polytope P. Moreover, we also give a new equivalent form of the Brouwer fixed point theorem on polytopes. This form has many important applications in mathematical programming and the theory of differential equations.
