Content
Acknowledgements v
ABSTRACT vii
摘摘摘要要要 ix
1 Introduction 1
1.1 Convex optimization . . . . 1
1.2 Inverse problem . . . . 4
1.3 Challenges in computational science . . . . 7
1.4 Supercomputers and the domain decomposition method . . . 10
1.5 Contribution of this dissertation . . . 13
2 Proximal gradient algorithms for convex minimization 19 2.1 Primal-dual fixed point algorithms . . . 21
2.1.1 Model problem and derivation of nested algorithms . . . 21
2.1.2 Main theorems . . . 29
2.2 Analysis of convergence. . . 30
2.2.1 Basic lemmas . . . 30
2.2.2 General convergence . . . 32
2.2.3 Linear convergence rates for special case . . . 38
2.3 Numerical experiments . . . 43
3 Explicit/implicit and Crank-Nicolson domain decomposition meth- ods for parabolic partial differential equations 49 3.1 Model problem and DDM finite element schemes. . . 51
3.1.1 Model problem . . . 51
3.1.2 Domain decomposition schemes . . . 52
3.1.3 Main theorems . . . 56
3.2 Analysis of convergence. . . 57
3.2.1 Basic lemmas . . . 58
3.2.2 Proof of Theorem 3.1 . . . 59
3.2.3 Proof of Theorem 3.2 . . . 69
3.3 Numerical experiments . . . 82
4 Explicit/implicit domain decomposition method for optimal control problems 87 4.1 Optimal control problem and optimality condition . . . 89
4.1.1 Model problem . . . 89
4.1.2 Optimality condition . . . 90
4.2 Finite element approximation based on domain decomposition . . . . 91
4.2.1 Discretization . . . 91
4.2.2 Parallel iterative algorithm . . . 93
4.2.3 Main theorems . . . 96
4.3 Analysis of convergence. . . 98
4.3.1 Intial approximation . . . 98
4.3.2 Basic lemmas . . . 103
4.3.3 Existence of discretization and convergence of iterative algo- rithm. . . 110
4.3.4 Proof of a priori estimate. . . 118
4.4 Numerical experiments . . . 125
5 Non-iterative Domain decomposition methods for wave equations127 5.1 Model problem and DDM finite element procedures . . . 128
5.1.1 Model problem . . . 128
5.1.2 Standard finite element procedures . . . 129
5.1.3 Domain decomposition schemes . . . 131
5.2 Analysis of convergence. . . 134
5.2.1 Basic lemmas . . . 134
5.2.2 Proof of Theorem 5.1 . . . 138
5.2.3 Proof of Theorem 5.2 . . . 145
5.3 Numerical experiments . . . 150
6 Conclusion 155
Bibliography 159
List of Publications during Study for the Doctorate 191 xxiv
