|Title:||Spherical tε-design and approximation on the sphere : theory and algorithms|
Sphere -- Mathematical models.
Hong Kong Polytechnic University -- Dissertations
|Department:||Department of Applied Mathematics|
|Pages:||xviii, 105 pages : illustrations ; 30 cm|
|Abstract:||This thesis concentrates on the spherical tε-designs on the two-sphere, numerical algorithms for finding spherical tε-designs and numerical approximation on the sphere using spherical tε-designs. A set of points on the unit sphere is called a spherical t-design if the average value of any polynomial of degree at most t over the set is equal to the average value of the polynomial over the sphere. Spherical t-designs have many important applications in geophysics and bioengineering, and provide many challenging problems in computational mathematics. As a generalization of spherical t-design, we define a spherical tε-design with 0 ≤ t < 1 which provides an integration rule with a set of points on the unit sphere min weight and positive weights satisfying (1-ε)² ≤ min weight/max weight ≤ 1. The integration rule also gives the exact integral for any polynomial of degree at most t. Due to the flexibility of choice for the weights, the number of points in the integration rule can be less for making the exact integral for any polynomial of degree at most t. To our knowledge, so far there is no theoretical result which proves the existence of a spherical t-design with (t+1)² points for arbitrary t. In 2010 Chen, Frommer and Lang developed a computation-assist proof for the existence of spherical t-designs for t = 1,..., 100 with (t+1)² points. Based on the algorithm proposed in that paper, a series of interval enclosures for spherical t-design was computed. In this thesis we prove that all the point sets arbitrarily chosen in these interval enclosures are spherical tε-designs and give an upper bound of ε. We then study the variational characterization and the worst-case error of spherical tε-design. Based on the reproducing kernel theory and its relationship with the geodesic distance, we propose a way to compute the worst-case error for numerical integration using spherical tε-design in Sobolev space. Moreover, we propose an approach for finding spherical tε-designs. We show that finding a spherical tε-design can be reformulated as a system of polynomial equations with box constraints. Using the projection operator, the system can be written as a nonsmooth nonconvex least squares problem with zero residual. We propose a smoothing trust region filter algorithm for solving such problems. We present convergence theorems of the proposed algorithm to a Clarke stationary point or a global minimizer of the problem under certain conditions. Preliminary numerical experiments show the efficiency of the proposed algorithm for finding spherical tε-designs. Another contribution in this thesis is the numerical approximation on the sphere using regularized least squares approaches. We consider two regularized least squares problems using spherical tε-designs: regularized polynomial approximation on the sphere, and regularized hybrid approximation on the sphere using both radial basis functions and spherical polynomials. For the first approach we apply the ℓ₂ regularized form and give an approximation quality estimation. For the second approach we study its ℓ₁ regularized form and solve the problem using alternating direction method with multipliers. Numerical experiments are given to demonstrate the effectiveness of these two models.|
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