Abstract
In this paper, we consider the problem of polynomial reconstruction of smooth functions on the sphere from their noisy values at discrete nodes on the two-sphere. The method considered in this paper is a weighted least squares form with a continuous regularization. Preliminary error bounds in terms of regularization parameter, noise scale, and smoothness are proposed under two assumptions: the mesh norm of the data point set and the perturbation bound of the weight. Condition numbers of the linear systems derived by the problem are discussed. We also show that spherical tϵ-designs, which can be seen as a generalization of spherical t-designs, are well applied to this model. Numerical results show that the method has good performance in view of both the computation time and the approximation quality.
Highlights
IntroductionWhere wi, i 1, . . . , N are positive scalars as weights of the data fitting term. For normalization, we can always assume that Ni 1 wi N
N min g∈PL i 1 wi g xi − fδ xi + λ‖g‖2Hs (1)where wi, i 1, . . . , N are positive scalars as weights of the data fitting term
Condition number and error bound of the model are proposed in the situation that the data are noisy and their cardinality is not necessarily larger than the dimension of the polynomial space
Summary
Where wi, i 1, . . . , N are positive scalars as weights of the data fitting term. For normalization, we can always assume that Ni 1 wi N. E points used are positive-weight cubature formula that is exact for all spherical polynomials of degree up to 2L, which will guarantee the uniqueness of the solution and derive its explicit form. One can assume that X and w establish a cubature rule with polynomial exactness so that it can give a fast way to find the solution of the problem By this view, spherical tε-designs which are proposed and studied in [6, 10,11,12,13,14] are a suitable choice in this paper. Erefore, estimate (31) of the condition number κ(T) is sharp It can be directly obtained by the above theorem that κ(T)≍l2s, which induces that s should be chosen properly as a balance of smoothing and solvability of the problem. The choice of s, which may depend on the nature of target function f ∈ C(S2) and the scale of noise, will be an interesting question to be solved in the future
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