Corner singularity and its application in regular parameters optimization: technique renovation for Grad-Shafranov reconstruction

TL;DR

This paper introduces a novel method for optimizing regular parameters in Grad-Shafranov reconstruction by addressing corner singularities using advanced mathematical techniques and nonlinear optimization, resulting in improved robustness and efficiency.

Contribution

The study develops a new approach for regular parameter optimization in GS reconstruction, incorporating corner singularity analysis, EHT properties, and a convex objective function solved via constrained nonlinear optimization.

Findings

Successful optimization of regular parameters with improved accuracy.

Demonstrated robustness and efficiency through benchmark tests.

Objective method for selecting optimal parameters established.

Abstract

Further studies on the corner singularity of GS reconstruction are are compiled in this paper. It's focused on solution of the Data Completion (DC) problem with the Extended Hilbert Transform (EHT) over plane rectangular region. Optimal selections of the regular parameters in{\it Tikhonov} solution of corresponding DC problem are developed in this study. The 6-parameter regular solutions and the {\it Jacobian} matrix and one {\it Hessian} tensor to the regular parameters are derived in this work. A concise formula for EHT in the near field of corners, which shows property for EHT near the corner, are also provided. It serves as the additional constraints for our parameter optimization problem (OP). Third, a nonlinear convex function defined by the regular solution and the corner constraints is introduced and is taken as the object function for the OP of the 6 regular parameters on half-space $(\mathbf{p}>0)$. Given an initial guess of $\mathbf{p}_0$, the optimal parameters are solved from the OP through a well known constrained nonlinear optimization method. Last, the benchmark tests to the proposed solution approach are carried out, and detailed results from totally 9 different bench-cases are tabulated. In contrast to solutions with given regular parameters, our bench results demonstrate that an objective way for selection of the optimal $\mathbf{p}$ is successfully laid out here. Robustness and efficiency of the suggested new approach are also highlighted in this study.