Ill-conditioning of the truncated singular value decomposition, Tikhonov regularization and their applications to numerical partial differential equations

• Published in: Numerical linear algebra with applications Vol. 18; no. 2; pp. 205 - 221
• Main Authors: Li, Zi-Cai, Huang, Hung-Tsai, Wei, Yimin
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 01.03.2011
• Subjects: collocation Trefftz method; Decomposition; effective condition number; Error analysis; fundamental solutions; ill-conditioning; Mathematical analysis; Mathematical models; Partial differential equations; Regularization; Stiffness matrix; Tikhonov regularization; truncated singular value decomposition; Vectors (mathematics)
• ISSN: 1070-5325; 1099-1506; 1099-1506
• DOI: 10.1002/nla.766

This paper explores some intrinsic characteristics of accuracy and stability for the truncated singular value decomposition (TSVD) and the Tikhonov regularization (TR), which can be applied to numerical solutions of partial differential equations (numerical PDE). The ill‐conditioning is a severe issue for numerical methods, in particular when the minimal singular value sigmamin of the stiffness matrix is close to zero, and when the singular vector umin of σmin is highly oscillating. TSVD and TR can be used as numerical techniques for seeking stable solutions of linear algebraic equations. In this paper, new bounds are derived for the condition number and the effective condition number which can be used to improve ill‐conditioning by TSVD and TR. A brief error analysis of TSVD and TR is also made, since both errors and condition number are essential for the numerical solution of PDE. Numerical experiments are reported for the discrete Laplace operator by the method of fundamental solutions. Copyright © 2011 John Wiley & Sons, Ltd.