Establishing Convergence

Establishing Convergence PowerPoint PPT Presentation


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Presentation Overview. Convergent SequencesMonotonic SequencesCauchy SequencesContraction MappingsPseudo-contraction MappingsBlock ContractionsRound Robin (Gauss-Seidel) Contractions. Convergent Sequence. A sequence in a metric space X with point such that for every , there is an integer N such that implies This can be equivalently written as or .

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Establishing Convergence

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1. Establishing Convergence James Neel June 8, 2004

2. Presentation Overview Convergent Sequences Monotonic Sequences Cauchy Sequences Contraction Mappings Pseudo-contraction Mappings Block Contractions Round Robin (Gauss-Seidel) Contractions

3. Convergent Sequence A sequence in a metric space X with point such that for every , there is an integer N such that implies This can be equivalently written as or

4. Example Convergent Sequence Given ?, choose N=1/ ?, p=0

5. Convergent Sequence Properties

6. Cauchy Sequence A sequence in a metric space X such that for every , there is an integer N such that if

7. Example Cauchy Sequence Given ?, choose N=2/?, p=0

8. Cauchy Sequences and Cauchy Sequences (Theorem 3.11 in Rudin) (a) In any metric space X, every convergent sequence is a Cauchy sequence. (b) If X is a compact metric space and if is a Cauchy sequence in X, then converges to some point of X. (c) In , every Cauchy sequence converges.

9. Complete metric spaces A metric space in which every Cauchy sequence converges. Examples of complete metric spaces: All compact metric spaces All Euclidean spaces All closed subsets of complete metric spaces.

10. Monotonic Sequences A sequence is monotonically increasing if . A sequence is monotonically decreasing if (Note: some authors use the inclusion of the equals condition to define a sequence to be respectively monotonically nondecreasing or monotonically nonincreasing.). A sequence which is either monotonically increasing or monotonically decreasing is said to be monotonic.

11. Convergent Monotonic Sequences Suppose is a monotonic in X. Then converges iff X is bounded. Note that also converges if X is compact.

12. Set Diameter Set Diameter Let X be a metric space and . The diameter of E is given by Note that if , is Cauchy iff

13. Example Set Diameters

14. Sequence of compact subsets Theorem 3.10 (b) Rudin If Kn is a sequence of compact sets in X such that and if then consists of exactly one point. Note that this can be used to establish both the existence and uniqueness of a fixed point if is generated by iterative evaluation of a function of the form

15. Contraction Mapping Let be a metric space, is a contraction if there is a such that

16. Contraction Mapping Lemma (Blackwell’s Conditions) Let be any map that satisfies (Monotonicity) (Discounting) There is a such that for all Then L is a contraction.

17. Another Common Contraction Mapping Condition Recall If

18. Pseudo-contraction Let be a metric space and with fixed point . f is a pseudo-contraction if there is such that

19. Pseudo-Contraction Lemma Let be a complete metric space, and be a pseudo-contraction. Then f has a unique fixed point. (Proof in Bertsekas, proposition 1.2) Note: not every pseudo-contraction is a contraction.

20. Convergence Rate Given a contraction or pseudo-contraction with modulus ?, where . then

21. Cartesian Product Contractions Define X as where and are given ||xi||i for each Xi. Define ||xi|| = max ||xi||i (over i). This is the block-maximum norm. f: X?X is a block contraction if f is a contraction on the block maximum norm. f: X?X is a block pseudo-contraction if f is a pseudo-contraction on the block maximum norm.

22. Round Robin Updating Also called Gauss-Seidel techniques Define Then define

23. Round Robin Block Contraction If f is a block contraction, then the round robin mapping, S, is also a block contraction. Further S converges to the same fixed point as f if X is closed (Bertsekas 3.1.4) If f is a block pseudo-contraction, then the round robin mapping, S, is also a block pseudo-contraction. Further S converges to the same fixed point as f if X is closed (Bertsekas 3.1.5)

24. Block Contraction Rate If f is a block contraction and X closed, then the round robin mapping, S, and f converge geometrically (? t )

25. Continuous Differentiable Contractions Suppose X is convex. If f:?n? ?n is continuously differentiable and there exists a scalar ? ?[0,1) such that where Gi is an invertible symmetric matrix and ? is a nonzero scalar, then is a block contraction. Note if ni = 1, then can choose Gi = 1. The bigger ? is, the faster it converges. (Bertsekas 3.1.10)

26. Continuous Variant Suppose ni = 1 for all i, X is convex, f:?n? ?n if a) b) Then T(x)=x-? f(x) is a block contraction provided that 0<?<1/K An application of Bertsekas 3.1.10 Similar to Numerical Analysis’s Fixed Point Iteration Criterion

27. More convergence Descent algorithms Monotone mappings Asynchronous algorithms Next time?

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