Numer. Math. Theor. Meth. Appl., 2 (2009), pp. 353-376. A Compound Algorithm of Denoising Using Second-Order and Fourth-Order Partial Differential Equations Qianshun Chang 1, Xuecheng Tai 2, Lily Xing 1*1 Institute of Applied Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, China. 2 School of Physical and Mathematical Sciences, Nanyang Technological University, SPMS-04-01, 21 Nanyang Link, Singapore 637371; and Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008 Bergen, Norway. Received 8 February 2009; Accepted (in revised version) 6 June 2009 Abstract In this paper, we propose a compound algorithm for the image restoration. The algorithm is a convex combination of the ROF model and the LLT model with a parameter function $\theta$. The numerical experiments demonstrate that our compound algorithm is efficient and preserves the main advantages of the two models. In particular, the errors of the compound algorithm in $L_2$ norm between the exact images and corresponding restored images are the smallest among the three models. For images with strong noises, the restored images of the compound algorithm are the best in the corresponding restored images. The proposed algorithm combines the fixed point method, an improved AMG method and the Krylov acceleration. It is found that the combination of these methods is efficient and robust in the image restoration. AMS subject classifications: 68U10, 65M55 Key words: Algorithm of denoising, image restoration, total variation, second-order functional. *Corresponding author. Email: qschang@amss.ac.cn (Q. Chang), tai@mi.uib.no (X. Tai), xinglily@amss.ac.cn (L. Xing)