Diploma thesis as pdf file - Johannes Kepler University, Linz

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Chapter 3 Numerical implementation 3.1 Algorithm Interior point method consists of outer cycle : corrector-predictor steps and inner cycle for solely corrector step. As starting point for the approximate solution we can choose u h = 0. As we already noted the principle of the interior point method is to follow the central path. Ideally, we would like the iterations u i corresponding to κ i would lie on the central path, i.e. u i = u κi , where u κi is the exact solution of the corresponding UP. But to achieve high accuracy on each corrector step would cost us a number of iterations. So, instead, we try our approximate solutions in each corrector step to remain in the predefined neighborhood of the exact solution (picture 3.1). ∥ Under the assumption that we have: ∥ δun+1 ∥ u n+1 − ψ ∥ L ∞ (Ω) ∥ δu0 u 0 −ψ ≤ ≤ ≤ ∥ < τ L 2 (Ω) C , from the proof of theorem 2.2.2 C∥ δun+1 ∥ u n+1 − ψ C∥ δun ∥ u n − ψ τ 2n∥ ∥ δu n ∥ u n − ψ ∥ L 2 (Ω) ∥ L ∞ (Ω) ∥ L ∞ (Ω) ∥ δun ∥ u n − ψ ∥ L 2 (Ω) 23
3.1 Algorithm 24 Figure 3.1: Staying close to the central path by single Newton iterations Thus, if we choose at each predictor step decreasing factor ρ of the κ such that the starting point ũ for the next Newton iterations satisfies ∥ ũ−ψ δũ ∥ < τ L 2 (Ω) C and ∥ ∥ define to stop Newton iterates as the condition < τ is satisfied, then ∥ u−ψ δu ∥ L 2 (Ω) we reach this kind of accuracy τ for the corrector step just in single step: ∥ δu1 ∥ ∥∥ u 1 ∥ − ψ ≤ C δu 0 ∥ L ∞ (Ω) u 0 − ψ ∥ L ∞ (Ω) ∥ δu0 ∥ u 0 − ψ ∥ L 2 (Ω) ≤ C2 τ2 C 2 < τ. We also need to determine the constant C for the assumptions of the theorem 2.2.2 to be fulfilled. This constant can be found by updating it at each Newton ∥ ∥ ∥∥ iterate as the maximal value of the old one and the ratio ∥ / δu ∥ . L ∞ (Ω) L 2 (Ω) ∥ u−ψ δu After solving of the elliptic equations for the corrector δu we need to choose the step-size α such that updated value of the approximate solution would be feasible, i.e. it should hold u + αδ > ψ. This is equivalent to α u−ψ δu > −1, and since under the appropriate assumptions ∥ u−ψ δu ∥ ≤ τ holds, for the implementation L ∞ (Ω) ∥ we write it as ∥α δu ∥ < 1. L ∞ (Ω) u−ψ u−ψ
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 and 22: 2.2 A predictor-corrector approach
Page 27: 2.2 A predictor-corrector approach
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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