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In this paper, a two-step regularization method is used to solve an ill-posedspherical pseudo-differential equation in the presence of noisy data. For thefirst step of regularization we approximate the data by means of a sphericalpolynomial that minimizes a functional with a penalty term consisting of thesquared norm in a Sobolev space. The second step is a regularized collocationmethod. An error bound is obtained in the uniform norm, which is potentiallysmaller than that for either the noise reduction alone or the regularizedcollocation alone. We discuss an a posteriori parameter choice, and presentsome numerical experiments, which support the claimed superiority of thetwo-step method.

Two-parameter regularization of ill-posed spherical pseudo-differential equations in the space of continuous functions