Researches On Multivariate Spline, Multivariate Weak Spline And Piecewise Algebraic Variety

In this doctorial dissertation, we mainly study multivariate spline, multivariate weak spline and piecewise algebraic variety. They are important and have been widely applied in many fields such as function approximation, computational geometry, finite element, algebraic geometry and so on. Multivariate spline is mainly studied in Chapter 2 and Chapter 3; multivariate weak spline is mainly investigated in Chapter 4 and Chapter 5; and the last chapter is devoted to piecewise algebraic variety.In Chapter 2, we give a computational method for the generating bases of the prime module determined by the global conformality condition of the spline over star crosscut partition. In 1975, R.H. Wang originally introduced the so-called "smoothing cofactor conformality method" for studying multivariate spline based on the methods of function theory and algebraic geometry. From this point of view, any problem on multivariate spline can be studied by transferring it into an equivalent algebraic problem. The global conformality condition can be regarded as a homogeneous algebraic system of linear equations with the unknowns of the smoothing cofactors, and its solutions form a prime module over R[x, y]. We discuss the generating bases of the prime module and the results can help us to solve the problem of dimension, spline bases and interpolation for the multivariate spline over any crosscut partition.In Chapter 3, we study a special multivariate spline space S₂^1,0(â—‡) whereâ—‡is a new triangulation obtained from a regular quadrangulation refined as the 4th Powell-Sabin refinement. For any quadratic spline in S₂^1,0(â—‡), it is C¹ over the major interior edges, and C⁰ over all the other interior edges which are the minority. We get its dimension, construct its basis splines, and obtain the explicit representations of the basis splines. And the approximation properties of two constructed quasi-interpolation operators are discussed, some supporting numerical results are presented. We also compare our spline with some traditional splines.In Chapter 4, we mainly study the multivariate weak spline space W_κ^μ(I₁â–³)(whereκ≥2μ+ 1) and W₂¹(I₁^*â–³). By using the "smoothing cofactor-conformality method" for studying multivariate weak spline, we calculate the free degrees step by step and obtain the dimensions of multivariate weak spline spaces W_κ^μ(I₁â–³)(κ≥2μ+ 1) and W₂¹(I₁^*â–³), where I₁â–³is an arbitrary regular rectilinear partition and I₁^*â–³is a rectilinear partition with some additional restrictions. The restrictions result from the fact that the degree 2 and the smoothness 1 of quadratic weak spline is too closer. We also present a method to construct the bases for the spaces. The method avoids the difficulty of solving the huge systems of global-conform-equations so it is simple and convenient.In Chapter 5, we discuss the relation between multivariate weak spline space and minimal determining set. By using the "B-net method" for studying multivariate weak spline, we give a method to construct the minimal determining sets for the multivariate weak spline spaces W_κ^μ(I₁â–³)(whereκ≥2μ+ 1) and W₂¹(I₁â–³) over arbitrary triangulation I₁â–³. Based on the property that the dimension of a multivariate weak spline space equals to the cardinality of the minimal determining set, we completely solve the dimension problem for W_κ^μ(I₁â–³)(whereκ≥2μ+ 1) and W₂¹(I₁â–³) over arbitrary triangulation. And we also study the basic theory for constructing minimal determining set. And the local-support properties for the dual bases determined by the points in minimal determining set are also studied.In Chapter 6, we study the Groebner bases intersection points algorithm for two given piecewise algebraic curves and the relation between piecewise algebraic varieties and ideals. In CAGD, it is important to obtain the common intersection points for two piecewise algebraic curves. The Bezout theorem of piecewise algebraic curves only gives us a theoretical upper bound for the number of the common intersection points. In the first part of this chapter, we give the intersection points algorithm for two given piecewise algebraic curves based on Groebner bases. For a given domain D and a partitionâ–³, we present a flow and introduce the truncated signs, and represent the piecewise algebraic curves in the global form. We get their Groebner bases with respect to a lexicographic order and adopt the interval arithmetic in the back-substitution process, which makes the algorithm numerically precise. Some numerical examples are also presented and the results show that the algorithm is feasible and effective. In the second part of this chapter, we mainly study the ideals of C^μspline ring S^μ(â–³) and the C^μspline algebraic varieties inκⁿ. We detailedly explore four operations of the ideals of S^μ(â–³), i.e. addition, multiplication, intersection and division. Some properties of prime ideals and maximal ideals of S^μ(â–³) are presented. The invalidity of Hilbert's Nullstellensatz in piecewise case is pointed out and the ideal-variety correspondence is also studied in this part.