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CHAPTER 4. THEORY 107 where A = I −Z ′ . This new estimate involves the X norm of p(A). This is useful since we have shown in the X norm, Z ′ is a compact linear operator. From the spectral theory of compact linear operators [23], Z ′ has a countable number of eigenvalues with 0 as the only accumulation point of the eigenvalues. Therefore, I −Z ′ has a countable number of eigenvalues with 1 as the only accumulation point eigenvalues. This means that the eigenvalues of I − Z ′ are clustered about 1 with only a finite number of them lying far away from 1 in the complex plane. Therefore, we can conclude that when GMRES is applied to the infinite-dimensional problem, the algorithm will rapidly decrease the residual norm. Modifying a theorem from [10], we can be quantify how rapidly GMRES will converge. Let ¯ρ >0 be the radius of the circle about 1 in the complex plane than contains the cluster. Theorem 4.1. Let A = I − Z ′ be such that Z ′ is compact linear map from X → X where X ⊂ L 2 . Choose ¯ρ >0 so that the circle about 1 in the complex plane will contain the cluster of eigenvalues about 1. Then for any right hand side and initial iterate, there exist a C>0 and d>0 such that the k-th GMRES residual rk satisfies where the constant C is independent of k. rd+k2 ≤ C ¯ρ k r0X, Proof. This proof is very simple and is a slight modification of the proof given in [10]. As previously stated in this chapter, we have for any polynomial p satisfying 1. p(0) = 1 2. p has degree k,
CHAPTER 4. THEORY 108 we have at the k-th GMRES that rk2 ≤ p(A)2r02 ≤ √ √ Lp(A)X Lr0X = Lp(A)Xr0X. Since Z ′ is compact in the X norm, we know that A = I −Z ′ has eigenvalues clustered about 1. Then writing p(A) = 1 2πi Γ (zI − A)−1p(z)dz, where Γ is a curve containing all the eigenvalues of A, we can use the exact same proof as given in [10] to bound p(A)X and create the constants C>0andd>0 so that the estimate on the k-th GMRES residual holds. The difference between the proof shown here and the one given in [10] is that in [10], the compact linear operator is assumed to map a Hilbert space back to itself. Here, we are working on a subspace of a Hilbert space. One is free to choose what ¯ρ is, and can therefore, choose the convergence rate of GMRES. This convergence rate is asymptotic though, and GMRES first iterations will have to take care of the outlying eigenvalues before this convergence rate will kick in. This is represented by the constant d, the number of GMRES iterations needed before the asymptotic convergence rate begins. By choosing ¯ρ smaller though, the number of outliers increases, which will in turn increase the number of GMRES iterations needed before the asymptotic convergence rate applies.
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Numerical Methods for the Wigner-Po
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NUMERICAL METHODS FOR THE WIGNER-PO
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Acknowledgments First and foremost,
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Table of Contents List of Tables ix
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4.3 ApplicationofSchauder’sFixedP
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List of Tables 1.1 PhysicalConstant
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3.11 I-V Curve with 30 Percent Redu
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CHAPTER 1. OVERVIEW 2 being heavily
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CHAPTER 1. OVERVIEW 4 Doping Spacer
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CHAPTER 1. OVERVIEW 6 and identfyin
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CHAPTER 1. OVERVIEW 8 for predictin
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CHAPTER 1. OVERVIEW 10 by Schrödin
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CHAPTER 1. OVERVIEW 12 ranging term
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CHAPTER 1. OVERVIEW 14 But because
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CHAPTER 1. OVERVIEW 16 = h 2π ,we
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CHAPTER 1. OVERVIEW 18 Table 1.1: P
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CHAPTER 1. OVERVIEW 20 where q is t
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CHAPTER 1. OVERVIEW 22 get K(f)
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CHAPTER 1. OVERVIEW 24 rule. The pr
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CHAPTER 1. OVERVIEW 26 The weights
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Chapter 2 Temporal Integration 2.1
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CHAPTER 2. TEMPORAL INTEGRATION 30
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Chapter 3 Bifurcation Analysis 3.1
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CHAPTER 3. BIFURCATION ANALYSIS 46
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Page 69 and 70: CHAPTER 3. BIFURCATION ANALYSIS 56
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: CHAPTER 4. THEORY 106 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 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
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