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Effective Condition Number for Numerical Partial Differential Equations [Kõva köide]

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  • Formaat: Hardback, 286 pages, kõrgus x laius: 240x160 mm, kaal: 515 g
  • Ilmumisaeg: 30-Jan-2014
  • Kirjastus: Alpha Science International Ltd
  • ISBN-10: 1842659138
  • ISBN-13: 9781842659137
Teised raamatud teemal:
Effective Condition Number for Numerical Partial Differential EquationsSuurem pilt
  • Formaat: Hardback, 286 pages, kõrgus x laius: 240x160 mm, kaal: 515 g
  • Ilmumisaeg: 30-Jan-2014
  • Kirjastus: Alpha Science International Ltd
  • ISBN-10: 1842659138
  • ISBN-13: 9781842659137
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781842659137.html
For a numerical method, its stability is a crucial issue in the sense that an unstable algorithm can render it useless in practical computation. For an over-determined linear algebraic equation , its stability is typically evaluated by using the traditional condition number, Cond= , where and are the maximal and the minimal singular values of matrix, respectively.

In this book, the concept of an effective condition number, Cond_eff= , is introduced. Cond_eff is smaller, and generally much smaller, than Cond, and is a better stability criterion. The Cond_eff can be used as an estimation on stability for the numerical solutions of partial differential equations (PDEs) using algorithms, such as the collocation Trefftz method, the spectral method, the finite difference method, and the finite element method. An analysis of stability integrated with errors is the focus of this book.

For a numerical method, its stability is a crucial issue in the sense that an unstable algorithm can render it useless in practical computation. For an over-determined linear algebraic equation , its stability is typically evaluated by using the traditional condition number, Cond= , where and are the maximal and the minimal singular values of matrix, respectively.

In this book, the concept of an effective condition number, Cond_eff= , is introduced. Cond_eff is smaller, and generally much smaller, than Cond, and is a better stability criterion. The Cond_eff can be used as an estimation on stability for the numerical solutions of partial differential equations (PDEs) using algorithms, such as the collocation Trefftz method, the spectral method, the finite difference method, and the finite element method. An analysis of stability integrated with errors is the focus of this book.

Introduces the concept of an effective condition number, Cond_eff. Cond_eff is smaller, and generally much smaller, than Cond, and is a better stability criterion. The Cond_eff can be used as an estimation on stability for the numerical solutions of partial differential equations.
Preface
Acknowledgments
Chapter 1 Effective Condition Number
1(46)
1.1 Introduction
1(2)
1.2 Preliminary
3(2)
1.3 Symmetric Matrices
5(4)
1.3.1 Definitions of effective condition numbers
6(2)
1.3.2 A posteriori computation
8(1)
1.4 Overdetermined Systems
9(12)
1.4.1 Basic algorithms
9(4)
1.4.2 Refinements of (1.4.10)
13(3)
1.4.3 Criteria
16(1)
1.4.4 Advanced refinements
17(2)
1.4.5 Effective condition number in p-norms
19(2)
1.5 Linear Algebraic Equations by GE or QR
21(2)
1.6 Application to Numerical PDE
23(10)
1.7 Application to Boundary Integral Equations
33(7)
1.8 Weighted Linear Least Squares Problems
40(7)
1.8.1 Effective condition number
41(3)
1.8.2 Perturbation bounds
44(1)
1.8.3 Applications and comparisons
45(2)
Chapter 2 Collocation Trefftz Methods
47(28)
2.1 Introduction
47(1)
2.2 CTM for Motz's Problem
48(3)
2.3 Bounds of Effective Condition Number
51(4)
2.4 Stability for CTM of Rp = 1
55(2)
2.5 Numerical Experiments
57(3)
2.5.1 Choice of Rp
57(2)
2.5.2 Extreme accuracy of D0
59(1)
2.6 GCTM Using Piecewise Particular Solutions
60(3)
2.7 Stability Analysis of GCTM
63(4)
2.7.1 Trefftz methods
63(3)
2.7.2 Collocation Trefftz methods
66(1)
2.8 Method of Fundamental Solutions
67(3)
2.9 Collocation Methods Using RBF
70(2)
2.10 Comparisons Between Cond_eff and Cond
72(1)
2.10.1 CTM using particular solutions for Motz's problem
72(1)
2.10.2 MFS and CM-RBF
73(1)
2.11 A Few Remarks
73(2)
Chapter 3 Simplified Hybrid Trefftz Methods
75(12)
3.1 The Simplified Hybrid TM
75(5)
3.1.1 Algorithms
75(4)
3.1.2 Error analysis
79(1)
3.1.3 Integration approximation
79(1)
3.2 Stability Analysis for Simplified Hybrid TM
80(7)
Chapter 4 Penalty Trefftz Method Coupled with FEM
87(20)
4.1 Introduction
87(2)
4.2 Combinations of TM and Adini's Elements
89(9)
4.2.1 Algorithms
89(3)
4.2.2 Basic theorem
92(1)
4.2.3 Global superconvergence
93(5)
4.3 Bounds of Cond_eff for Motz's Problem
98(6)
4.4 Effective Condition Number of One and Infinity Norms
104(1)
4.5 Concluding Remarks
105(2)
Chapter 5 Trefftz Methods for Biharmonic Equations with Crack Singularities
107(26)
5.1 Introduction
107(1)
5.2 Collocation Trefftz Methods
107(5)
5.2.1 Three crack models
107(3)
5.2.2 Description of the method
110(1)
5.2.3 Error bounds
111(1)
5.3 Stability Analysis
112(5)
5.3.1 Upper bound for σmax(F)
112(1)
5.3.2 Lower bound for σmin(F)
113(3)
5.3.3 Upper bound for Cond_eff and Cond
116(1)
5.4 Proofs of Important Results Used in Section 5.3
117(11)
5.4.1 Basic theorem
117(4)
5.4.2 Proof of Lemma 5.4.3
121(2)
5.4.3 Proof of Lemma 5.4.4
123(5)
5.5 Numerical Experiments
128(3)
5.6 Concluding Remarks
131(2)
Chapter 6 Finite Difference Method
133(12)
6.1 Introduction
133(1)
6.2 Shortley-Weller Difference Approximation
133(12)
6.2.1 A Lemma
135(2)
6.2.2 Bounds for Cond_EE
137(4)
6.2.3 Bounds for Cond_eff
141(4)
Chapter 7 Boundary Penalty Techniques of FDM
145(17)
7.1 Introduction
145(1)
7.2 Finite Difference Method
146(3)
7.2.1 Shortley-Weller difference approximation
147(1)
7.2.2 Superconvergence of solution derivatives
147(1)
7.2.3 Bounds for Cond_eff
148(1)
7.3 Penalty-Integral Techniques
149(5)
7.4 Penalty-Collocation Techniques
154(6)
7.5 Relations Between Penalty-Integral and Penalty-Collocation Techniques
160(1)
7.6 Concluding Remarks
161(1)
Chapter 8 Boundary Singularly Problems by FDM
162(23)
8.1 Introduction
162(1)
8.2 Finite Difference Method
163(1)
8.3 Local Refinements of Difference Grids
164(13)
8.3.1 Basic results
165(6)
8.3.2 Nonhomogeneous Dirichlet and Neumann boundary conditions
171(2)
8.3.3 A remark
173(3)
8.3.4 A view on assumptions A1-A4
176(1)
8.3.5 Discussions and comparisons
177(1)
8.4 Numerical Experiments
177(6)
8.5 Concluding Remarks
183(2)
Chapter 9 Finite Element Method Using Local Mesh Refinements
185(23)
9.1 Introduction
185(1)
9.2 Optimal Convergence Rates
186(5)
9.3 Homogeneous Boundary Conditions
191(5)
9.4 Nonhomogeneous Boundary Conditions
196(4)
9.5 Intrinsic View of Assumption A2 and Improvements of Theorem 9.4.1
200(3)
9.5.1 Intrinsic view of assumption A2
200(1)
9.5.2 Improvements of Theorem 9.4.1
201(2)
9.6 Numerical Experiments
203(5)
Chapter 10 Hermite FEM for Biharmonic Equations
208(13)
10.1 Introduction
208(1)
10.2 Description of Numerical Methods
209(2)
10.3 Stability Analysis
211(4)
10.3.1 Bounds of Cond
211(1)
10.3.2 Bounds of Cond_eff
211(4)
10.4 Numerical Experiments
215(6)
Chapter 11 Truncated SVD and Tikhonov Regularization
221(15)
11.1 Introduction
221(3)
11.2 Algorithms of Regularization
224(1)
11.3 New Estimates of Cond and Cond_eff
225(6)
11.4 Brief Error Analysis
231(5)
Appendix Definitions and Formulas
236(11)
A.1 Square Systems
236(4)
A.1.1 Symmetric and positive definite matrices
237(2)
A.1.2 Symmetric and nonsingular matrices
239(1)
A.1.3 Nonsingular matrices
239(1)
A.2 Overdetermined Systems
240(1)
A.3 Underdetermined Systems
241(1)
A.4 Method of Fundamental Solutions
242(1)
A.5 Regularization
243(2)
A.5.1 Truncated singular value decomposition
244(1)
A.5.2 Tikhonov regularization
244(1)
A.6 p-Norms
245(1)
A.7 Conclusions
246(1)
Epilogue 247(2)
Bibliography 249(14)
Index 263