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Tham khảo tài liệu 'numerical_methods for nonlinear variable problems episode 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 3. Relaxation Methods for Convex Functionals Finite-Dimensional Case 147 Since u is the solution of 3.4 and II 11 e K we have cf. 3.5 J o u u l Ù 4- J u 1 J Au 0 which combined with 3.32 implies Jó u 1 u l - u Ji w 1 - Ji u ổM u 1 - nil . 3.33 Relation 3.33 implies J o u 1 un 1 - u J1 u 1 Ji u i Í 1 uVị dVị N AJ E - ui ji 1 -ji Ui j ỖMW i-u 3.34 where ũ 1 w 1 . u 1 ti 1 . . Since UịeKí it follows from 3.8 that V i 1 . N ứl 1 uĩ 1 - H. jM - jfa 0. 3.35 ÕVị Therefore 3.34 and 3.35 show that E - 1 m 1 - Mll n 1 - D- 3-36 ÕVị OVi Since 1 ũ 1II IIun 1 u it follows from 3.31 that V i 1 . IV we have lim m 1 1 0. 3.37 n- co Since J o 6 C Riv J o is uniformly continuous on the bounded subsets of R v. This property combined with 3.37 implies V i 1 . N lim n 00 CVi OV 0. 3.38 Therefore from 3.28 3.36 3.38 and the properties of 0M it follows that lim II w u 0 which completes the proof of the theorem. 3.5. Various remarks Remark 3.3. We assume that K nF and that J Jo i.e. Jj 0 where Jo v Ỉ ĂV v b v where b 6 IR V and A is an N X N symmetrical positive-definite matrix. 148 V Relaxation Methods and Applications The problem 3.4 associated with this choice of J and K obviously has an unique solution characterized cf. 3.5 by Au b. 3.39 If we apply the algorithm 3.6 3.7 to this particular case we obtain u 6 arbitrarily given 3.40 lt 1 bi - 1 _ a Ju 1 N- 3-41 flii j i j i The algorithm 3.40 3.41 is known as the Gauss-Seidel method for solving 3.39 see e.g. Varga 1 and D. Young 1 . Therefore when A is symmetric and positive definite optimization theory yields another proof of the convergence of the Gauss-Seidel method through Theorem 3.1. Remark 3.4. From the above remark it follows that the introduction of over-or under-relaxation parameters could be effective for increasing the speed of convergence. This possibility will be discussed in the sequel of this chapter. Let F V - R. We define D F r re V F r oo . 3.42 If F is convex and proper then D F is a nonempty convex subset of V. .
