A Bayesian Framework for Spectral Reprojection

作     者：Li, Tongtong Gelb, Anne 

作者机构：Univ Maryland Baltimore Cty Dept Math & Stat Baltimore MD 21250 USA Dartmouth Coll Dept Math Hanover NH 03755 USA 

出 版 物：《JOURNAL OF SCIENTIFIC COMPUTING》 (J Sci Comput)

年 卷 期：2025年第102卷第3期

页      面：1-28页

核心收录：

学科分类：08[工学] 0701[理学-数学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

基　　金：Directorate for Mathematical and Physical Sciences 

主　　题：Bayesian inference method Spectral reprojection Noisy Fourier data Gibbs phenomenon Gegenbauer polynomials Uncertainty quantification 

摘      要：Fourier partial sum approximations yield exponential accuracy for smooth and periodic functions, but produce the infamous Gibbs phenomenon for non-periodic ones. Spectral reprojection resolves the Gibbs phenomenon by projecting the Fourier partial sum onto a Gibbs complementary basis, often prescribed as the Gegenbauer polynomials. Noise in the Fourier data and the Runge phenomenon both degrade the quality of the Gegenbauer reconstruction solution, however. Motivated by its theoretical convergence properties, this paper proposes a new Bayesian framework for spectral reprojection, which allows a greater understanding of the impact of noise on the reprojection method from a statistical point of view. We are also able to improve the robustness with respect to the Gegenbauer polynomials parameters. Finally, the framework provides a mechanism to quantify the uncertainty of the solution estimate.