A Multiscale RBF Method for Severely Ill-Posed Problems on Spheres

Zhong, Min; Gia, Quôc Thông Lê; Sloan, Ian H.

doi:10.1007/s10915-022-02046-9

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2023

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“…We did not touch the topic in this paper about the severely ill-posed problems, but the last example still shows the effectiveness of the method. We recommend a very recent publication [31], where the multiscale radial basis kernels were utilized, and a kind of specific severely ill-posed problem was discussed in a unit sphere. Extending the algorithm to a multiscale scheme and consider the a posterior choice strategy for regularization parameter would be very attractive as well.…”

Section: Discussionmentioning

confidence: 99%

Semi-discrete Tikhonov regularization in RKHS with large randomly distributed noise

Zhong

2023

Inverse Problems

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The concern of this paper is the numerical solution of the moder-ately ill-posed first kind integral equations in reproducing kernel Hilbert spaces (RKHS). Different with previous works in [9, 29], the amount of observation data is large, and corrupted by randomly distributed noise with large variance. In this paper, a purely data driven semi-discrete Tikhonov regularization method is proposed, effectively reconstructing the sought solution. The choice rules of parameters are provided and the rigorous upper bound estimation of confidence interval of the error in L2norm is established. Some numerical examples are provided to illustrate the appropriateness of the parameter choice and show the computational performances of the method.

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Section: Discussionmentioning

confidence: 99%