Abstract

More documents

Recommendations

Info

CHAPTER 2. TEMPORAL INTEGRATION 37 2.4 Preconditioning Preconditioning is essential here for reducing the number of iterations GMRES takes. Preconditioning is where the coefficient matrix is multiplied by a preconditioner ˜ M ∈ R M×M so that the new coefficient matrix is easier for GMRES to handle. This works well if ˜ M ≈ D −1 . There are two ways to precondition: left precondtioning and right preconditioning. In left preconditioning, the linear equation is multiplied on both sides by ˜ M to get ˜MDa = ˜ M b So we apply GMRES to the new right hand side ˜ M b and the new coefficient matrix ˜MD. Now, GMRES terminates iterating when ˜ M b− ˜ MDa j 2 is small, not when the residual of our original problem b−Da j is small. Though if ˜ MD is well-conditioned, it can be shown [19] that the terminating on small relative residuals (as one does with GMRES) will result in small relative error in the numerical solution of the original problem. In right preconditioning, the coefficient matrix D is multiplied on the right by ˜ M, and we use GMRES to solve for s ∈ R M D ˜ Ms = b Once we have s, then the solution is a = ˜ Ms. GMRES terminates when b − Da n 2 is small, which is the original residual. So right preconditioning allows the user to
CHAPTER 2. TEMPORAL INTEGRATION 38 keep the same termination criterion. 2.5 Global Convergence One problem with the convergence theory of Newton’s method is the requirement for a close initial iterate. This is sometimes infeasable. A globally convergent Newton method can be developed so that starting at any initial iterate z0, the Newton iteration converges to a root of F or with fail in a finite number of ways. NITSOL uses a line search method to get global convergence. Newton-Armijo [3], one line search method, takes the computed Newton step sn and searches along it to find sufficient decrease in the norm of the function F , that is for a fixed ξ ∈ (0, 1) the algorithms finds a σ ∈ (0, 1] such that F (zm + σsm)2 < (1 − σξ)F (zm)2 In our case, since we are using an iterative method to solve for the Newton step, the step will satisfy the inexact-Newton condition (2.3). The inexact-Newton-Armijo algorithm can be summarized as: 1. Use an iterative method to find sm ∈ R M such that (2.3) is met. 2. Test if zm + sm (i.e. σ = 1) meets sufficient decrease. If so, terminate. 3. If σ = 1 fails, reduce σ is a safe way (such as σ → σ) until sufficient decrease is 2
Page 1 and 2: Numerical Methods for the Wigner-Po
Page 3 and 4: NUMERICAL METHODS FOR THE WIGNER-PO
Page 5 and 6: Acknowledgments First and foremost,
Page 7 and 8: Table of Contents List of Tables ix
Page 9 and 10: 4.3 ApplicationofSchauder’sFixedP
Page 11 and 12: List of Tables 1.1 PhysicalConstant
Page 13 and 14: 3.11 I-V Curve with 30 Percent Redu
Page 15 and 16: CHAPTER 1. OVERVIEW 2 being heavily
Page 17 and 18: CHAPTER 1. OVERVIEW 4 Doping Spacer
Page 19 and 20: CHAPTER 1. OVERVIEW 6 and identfyin
Page 21 and 22: CHAPTER 1. OVERVIEW 8 for predictin
Page 23 and 24: CHAPTER 1. OVERVIEW 10 by Schrödin
Page 25 and 26: CHAPTER 1. OVERVIEW 12 ranging term
Page 27 and 28: CHAPTER 1. OVERVIEW 14 But because
Page 29 and 30: CHAPTER 1. OVERVIEW 16 = h 2π ,we
Page 31 and 32: CHAPTER 1. OVERVIEW 18 Table 1.1: P
Page 33 and 34: CHAPTER 1. OVERVIEW 20 where q is t
Page 35 and 36: CHAPTER 1. OVERVIEW 22 get K(f)
Page 37 and 38: CHAPTER 1. OVERVIEW 24 rule. The pr
Page 39 and 40: CHAPTER 1. OVERVIEW 26 The weights
Page 41 and 42: Chapter 2 Temporal Integration 2.1
Page 43 and 44: CHAPTER 2. TEMPORAL INTEGRATION 30
Page 49: CHAPTER 2. TEMPORAL INTEGRATION 36
Page 57 and 58: Chapter 3 Bifurcation Analysis 3.1
Page 59 and 60: CHAPTER 3. BIFURCATION ANALYSIS 46
Page 87 and 88: Chapter 4 Theory 4.1 Steady-State T
Page 89 and 90: CHAPTER 4. THEORY 76 If we write ou
Page 91 and 92: CHAPTER 4. THEORY 78 So if we write
Page 93 and 94: CHAPTER 4. THEORY 80 Combining Equa
Page 95 and 96: CHAPTER 4. THEORY 82 We now estimat
Page 97 and 98: CHAPTER 4. THEORY 84 pendent of x,
Page 99 and 100: CHAPTER 4. THEORY 86 interval we ha
Page 101 and 102:
CHAPTER 4. THEORY 88 So an estimate
Page 103 and 104:
CHAPTER 4. THEORY 90 We want to cho
Page 105 and 106:
CHAPTER 4. THEORY 92 The exact same
Page 107 and 108:
CHAPTER 4. THEORY 94 Since un → u
Page 109 and 110:
CHAPTER 4. THEORY 96 thereexistsaco
Page 111 and 112:
CHAPTER 4. THEORY 98 For 0
Page 113 and 114:
CHAPTER 4. THEORY 100 denote this d
Page 115 and 116:
CHAPTER 4. THEORY 102 Now, we can t
Page 117 and 118:
CHAPTER 4. THEORY 104 operator. 4.2
Page 119 and 120:
CHAPTER 4. THEORY 106 we have at th
Page 121 and 122:
CHAPTER 4. THEORY 108 we have at th
Page 123 and 124:
CHAPTER 4. THEORY 110 4.3 Applicati
Page 125 and 126:
CHAPTER 4. THEORY 112 We need to fi
Page 127 and 128:
Chapter 5 Conclusion In this work,
Page 129 and 130:
CHAPTER 5. CONCLUSION 116 • Findi
Page 131 and 132:
REFERENCES 118 ODE solver. Technica
Page 133 and 134:
REFERENCES 120 [21] C. T. Kelley an
Page 135 and 136:
REFERENCES 122 [37] Homer F. Walker
Page 137 and 138:
APPENDIX A. NOTATION 124 ˜B Genera
Page 139 and 140:
APPENDIX A. NOTATION 126 k Wave num
Page 141 and 142:
APPENDIX A. NOTATION 128 ∆t Incre
Page 143 and 144:
APPENDIX A. NOTATION 130 ɛc ɛk ɛ
Page 145 and 146:
APPENDIX B. TRILINOS 132 user-given
Page 147 and 148:
APPENDIX B. TRILINOS 134 Trilinos,
Page 149 and 150:
APPENDIX B. TRILINOS 136 Minimum Re
Page 151 and 152:
APPENDIX B. TRILINOS 138 A very use
Page 153 and 154:
Appendix C Guide to RTD Simulation
Page 155 and 156:
APPENDIX C. GUIDE TO RTD SIMULATION
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Appendix D GMRES and Arnoldi Iterat
Page 165 and 166:
APPENDIX D. GMRES AND ARNOLDI ITERA
Page 167 and 168:
APPENDIX D. GMRES AND ARNOLDI ITERA
show all

Abstract

Create successful ePaper yourself

Delete template?

Save as template?