Diploma thesis as pdf file - Johannes Kepler University, Linz

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2.2 A predictor-corrector approach for the following the central path 17 2.2.1 The corrector step For given κ > 0 we are given an approximation u ∈ V of u κ and we want to find a better approximation u + by means of Newton’s method, i.e. u + = u + δu where the so called Newton corrector δu is a solution of the problem with min φ κ,u(v) (2.1) v∈V φ κ,u (v) = J κ (u) + J κ(u)v ′ + 1 2 J′′ κ(v,v). Here J ′ κ(u) : V → R, and J ′′ κ(u) : V ×V → R are and ∫ J κ(u)v ′ = Ω ∫ v ∇u∇vdω − 〈 f ,v〉 − κ Ω u − ψ dω ∫ ∫ J κ(u)(v,w) ′′ vw = ∇v∇wdω + κ Ω Ω (u − ψ) 2 dω. For ease of presentation we assume that u − ψ ≥ ε > 0 a.e. in Ω. Then J ′ (u) and J ′′ (u) are well defined. Further, it easily follows that φ κ,u is a continuous strictly convex functional on V and a unique minimizer δu of φ κ,u exists and satisfies J ′′ κ(u)(δu,v) = −J ′ κ(u)v, ∀v ∈ V. So at this step we replaced the objective functional J κ by its quadratic approximation φ κ,u . Since we used the logarithmic barrier functions it’s sufficient here to use Taylor expansion up to the second order. Thus we can rely on Newton’s method for solving this problem. Now let us analyze the next corrector step given by the unique solution δu + of the optimization problem min φ κ,u +(v). v∈V
2.2 A predictor-corrector approach for the following the central path 18 Since J ′ κ(u + )v = we obtain ∫ Ω ∫ ∇u + ∇vdω − 〈 f ,v〉 − κ = J ′ κ(u)v + J ′′ κ(u)(δu,v) − κ ∫ = −κ Ω Ω ∫ Ω (δu) 2 v (u + − ψ)(u − ψ) 2 dω v u + − ψ dω ( v u + − ψ − v u − ψ + δu v ) (u − ψ) 2 dω ∫ J κ(u ′′ + )(δu + ,v) = −J κ(u ′ + (δu) 2 v )v = κ Ω (u + dω, ∀v ∈ V. − ψ)(u − ψ) By taking v = δu + it follows that and ∫ Ω ∫ |∇δu + | 2 dω + κ Ω (δu + ) 2 (u + − ψ) 2 dω = κ ∫Ω Using the Cauchy-Schwartz inequality we conclude that ∫ κ Ω (δu + ) 2 (u + − ψ) 2 dω ≤ κ ( ∫ Ω ∫ Ω (δu + ) 2 (u + − ψ) 2 dω ) 1 2 ( ∫ Ω δu + (δu) 2 (u + dω. (2.2) − ψ)(u − ψ) 2 (δu) 4 (u − ψ) 4 dω ) 1 2 (δu + ) 2 (u + − ψ) ∫Ω 2 dω ≤ (δu) 4 dω. (2.3) (u − ψ) 4 Now consider Newton’s step defined by u n+1 = u n +δu n , n = 0,1,2,..., where δu n is the unique minimizer of φ κ,u n(·). Definition 2.2.1 We say the sequence {γ n } converges R-superlinearly to γ if √ n lim ‖γ n − γ‖ = 0. n→∞ Theorem 2.2.2 Let κ > 0 be fixed and assume that there exist some reals C > 0 such that ∥ δu ∥ ∥∥ ∥ u − ψ ≤ C δu ∥ L ∞ (Ω) u − ψ ∥ L 2 (Ω) ∀u > ψ. Then for all starting points u 0 > ψ with ‖ δu0 u 0 −ψ ‖ L 2 (Ω) < C 1 Newton iterations converge to u κ . Moreover, the convergence is R-superlinear.
Page 1 and 2: ÆØÞÛÖĐÙÖÓÖ×ÙÒ¸ÄÖÙ
Page 3 and 4: Contents Acknowledgments iii Introd
Page 5 and 6: Acknowledgments My sincere gratitud
Page 7 and 8: Chapter 1 Mathematical modeling of
Page 9 and 10: 1.2 Existence and uniqueness of the
Page 11 and 12: 1.3 Alternative equivalent formulat
Page 17 and 18: 2.1 The central path 12 Problem 2.1
Page 19 and 20: 2.1 The central path 14 for some c
Page 21: 2.2 A predictor-corrector approach
Page 25 and 26: 2.2 A predictor-corrector approach
Page 27 and 28: 2.2 A predictor-corrector approach
Page 29 and 30: 3.1 Algorithm 24 Figure 3.1: Stayin
Page 31 and 32: 3.2 Finite element discretization 2
Page 33 and 34: 3.2 Finite element discretization 2
Page 35 and 36: 3.3 Error estimate for Finite Eleme
Page 37 and 38: 3.4 Numerical experiments 32 Hence
Page 39 and 40: 3.4 Numerical experiments 34 ations
Page 41 and 42: 3.4 Numerical experiments 36 Table
Page 43 and 44: 3.4 Numerical experiments 38 In the
Page 45 and 46: 3.4 Numerical experiments 40 3.4.4
Page 47 and 48: Conclusion In this work we consider
Page 49 and 50: Bibliography [1] Bonnans, J.F.; Gil
Page 51: Eidesstattliche Erklarung Ich, Serb

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