Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms

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Springer Science & Business Media, Jan 13, 2005 - Mathematics - 424 pages

This book deals with the efficient numerical solution of challenging nonlinear problems in science and engineering, both in finite and in infinite dimension. Its focus is on local and global Newton methods for direct problems or Gauss-Newton methods for inverse problems. Lots of numerical illustrations, comparison tables, and exercises make the text useful in computational mathematics classes. At the same time, the book opens many directions for possible future research.

 

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Contents

Introduction
7
12 Newtons Method for General Nonlinear Problems
11
122 Affine invariance and Lipschitz conditions
13
123 The algorithmic paradigm
20
13 A Roadmap of Newtontype Methods
21
14 Adaptive Inner Solvers for Inexact Newton Methods
26
GMRES
28
PCG
30
43 Error Oriented Algorithms
193
431 Local convergence results
194
432 Local GaussNewton algorithms
197
433 Global convergence results
205
434 Adaptive trust region strategies
212
435 Adaptive rank strategies
215
44 Underdetermined Systems of Equations
221
442 Global GaussNewton method
225

CGNE
32
GBIT
35
145 Linear multigrid methods
38
Exercises
40
ALGEBRAIC EQUATIONS
43
Systems of Equations Local Newton Methods
45
212 Simplified Newton method
52
213 Newtonlike methods
56
214 Broydens good rank1 updates
58
215 Inexact NewtonERR methods
67
22 Residual Based Algorithms
76
222 Simplified Newton method
79
223 Broydens bad rank1 updates
81
224 Inexact NewtonRES method
90
23 Convex Optimization
94
232 Simplified Newton method
97
233 Inexact NewtonPCG method
98
Exercises
104
Systems of Equations Global Newton Methods
109
31 Globalization Concepts
110
311 Componentwise convex mappings
111
312 Steepest descent methods
114
313 Trust region concepts
117
314 Newton path
121
32 Residual Based Descent
125
321 Affine contravariant convergence analysis
126
322 Adaptive trust region strategies
128
323 Inexact NewtonRES method
131
33 Error Oriented Descent
134
331 General level functions
135
332 Natural level function
138
333 Adaptive trust region strategies
145
334 Inexact NewtonERR methods
152
34 Convex Functional Descent
161
341 Affine conjugate convergence analysis
162
342 Adaptive trust region strategies
165
343 Inexact NewtonPCG method
168
Exercises
170
Least Squares Problems GaussNewton Methods
173
41 Linear Least Squares Problems
175
412 Equality constrained problems
178
42 Residual Based Algorithms
182
421 Local GaussNewton methods
183
422 Global GaussNewton methods
188
423 Adaptive trust region strategy
190
Exercises
228
Parameter Dependent Systems Continuation Methods
233
51 Newton Continuation Methods
237
512 Affine covariant feasible stepsizes
242
513 Adaptive pathfollowing algorithms
245
52 GaussNewton Continuation Method
250
522 Affine covariant feasible stepsizes
253
523 Adaptive stepsize control
257
53 Computation of Simple Bifurcations
263
532 Newtonlike algorithm for simple bifurcations
271
533 Branchingoff algorithm
277
Exercises
280
DIFFERENTIAL EQUATIONS
283
Stiff ODE Initial Value Problems
285
62 Nonstiff versus Stiff Initial Value Problems
288
622 Newtontype uniqueness theorems
291
63 Uniqueness Theorems for Implicit Onestep Methods
295
64 Pseudotransient Continuation for Steady State Problems
298
641 Exact pseudotransient continuation
300
642 Inexact pseudotransient continuation
306
Exercises
312
ODE Boundary Value Problems
315
71 Multiple Shooting for Timelike BVPs
316
711 Cyclic linear systems
318
712 Realization of Newton methods
324
713 Realization of continuation methods
327
72 Parameter Identification in ODEs
333
73 Periodic Orbit Computation
337
732 Orbit continuation methods
340
733 Fourier collocation method
344
74 Polynomial Collocation for Spacelike BVPs
348
741 Discrete versus continuous solutions
350
742 Quasilinearization as inexact Newton method
356
Exercises
366
PDE Boundary Value Problems
369
82 Global Discrete Newton Methods
378
822 Elliptic PDEs
385
83 Inexact Newton Multilevel FEM for Elliptic PDEs
389
831 Local NewtonGalerkin methods
391
832 Global NewtonGalerkin methods
397
Exercises
403
References
405
Software
416
Index
419
Copyright

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About the author (2005)

Peter Deuflhard is founder and head of the internationally renowned Zuse Institute Berlin (ZIB) and full professor of Numerical Analysis and Scientific Computing at the Free University of Berlin. He is a regular invited speaker at international conferences and universities as well as industry places all over the world.