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60
General logical metatheorems for functional analysis
, 2008
"... In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds ..."
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Cited by 31 (18 self)
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In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, HölderLipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.
A quantitative version of a theorem due to BorweinReichShafrir
 Numerical Functional Analysis and Optimization
, 2000
"... We give a quantitative analysis of a result due to Borwein, Reich and Shafrir on the asymptotic behaviour of the general KrasnoselskiMann iteration for nonexpansive selfmappings of convex sets in arbitrary normed spaces. Besides providing explicit bounds we also get new qualitative results concerni ..."
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Cited by 19 (13 self)
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We give a quantitative analysis of a result due to Borwein, Reich and Shafrir on the asymptotic behaviour of the general KrasnoselskiMann iteration for nonexpansive selfmappings of convex sets in arbitrary normed spaces. Besides providing explicit bounds we also get new qualitative results concerning the independence of the rate of convergence of the norm of that iteration from various input data. In the special case of bounded convex sets, where by wellknown results of Ishikawa, Edelstein/O'Brian and Goebel/Kirk the norm of the iteration converges to zero, we obtain uniform bounds which do not depend on the starting point of the iteration and the nonexpansive function, but only depend on the error #, an upper bound on the diameter of C and some very general information on the sequence of scalars # k used in the iteration. Only in the special situation, where # k := # is constant, uniform bounds were known in that bounded case. For the unbounded case, no quantitative information was ...
On the computational content of the Krasnoselski and Ishikawa fixed point theorems
, 2000
"... This paper is a case study in proof mining applied to noneffective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general ..."
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Cited by 11 (10 self)
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This paper is a case study in proof mining applied to noneffective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general socalled KrasnoselskiMann iterations. These iterations converge to fixed points of f under certain compactness conditions. But, as we show, already for uniformly convex spaces in general no bound on the rate of convergence can be computed uniformly in f . This is related to the nonuniqueness of fixed points. However, the iterations yield even without any compactness assumption and for arbitrary normed spaces approximate fixed points of arbitrary quality for bounded C (asymptotic regularity, Ishikawa 1976). We apply proof theoretic techniques (developed in previous papers of us) to noneffective proofs of this regularity and extract effective uniform bounds on the rate of the asymptotic re...
Learning Dynamics In Mechanism Design: An Experimental Comparison Of Public Goods Mechanisms
, 2003
"... In a repeatedinteraction public goods economy, dynamic behavior may affect the efficiency of various mechanisms thought to be efficient in oneshot games. Inspired by results obtained in previous experiments, the current paper proposes a simple best response model in which players' beliefs are func ..."
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Cited by 10 (2 self)
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In a repeatedinteraction public goods economy, dynamic behavior may affect the efficiency of various mechanisms thought to be efficient in oneshot games. Inspired by results obtained in previous experiments, the current paper proposes a simple best response model in which players' beliefs are functions of previous strategy profiles. The predictions of the model are found to be highly consistent with new experimental data from five mechanisms with various types of equilibria. Interesting properties of a 2parameter VickreyClarkeGroves mechanism help to draw out this result. The simplicity of the model makes it useful in predicting dynamic stability of other mechanisms.
A quadratic rate of asymptotic regularity for CAT(0)spaces
, 2005
"... In this paper we obtain a quadratic bound on the rate of asymptotic regularity for the KrasnoselskiMann iterations of nonexpansive mappings in CAT(0)spaces, whereas previous results guarantee only exponential bounds. The method we use is to extend to the more general setting of uniformly convex hy ..."
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Cited by 8 (0 self)
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In this paper we obtain a quadratic bound on the rate of asymptotic regularity for the KrasnoselskiMann iterations of nonexpansive mappings in CAT(0)spaces, whereas previous results guarantee only exponential bounds. The method we use is to extend to the more general setting of uniformly convex hyperbolic spaces a quantitative version of a strengthening of Groetsch’s theorem obtained by Kohlenbach using methods from mathematical logic (socalled “proof mining”).
Nonlinear Hybrid Procedures and Fixed Point Iterations
, 1998
"... Let (x n ) and (x n ) be two vector sequences converging to a common limit. First, we shall define nonlinear hybrid procedures which consist of constructing a new vector sequence (y n ) with better convergence properties than (x n ) and (x n ). Then, this procedure is used for accelerating the conve ..."
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Cited by 8 (6 self)
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Let (x n ) and (x n ) be two vector sequences converging to a common limit. First, we shall define nonlinear hybrid procedures which consist of constructing a new vector sequence (y n ) with better convergence properties than (x n ) and (x n ). Then, this procedure is used for accelerating the convergence of a given sequence and applied to the construction of fixed point methods. New methods are derived. Finally, the connection between fixed point iterations and methods for the numerical integration of differential equations is also exploited. Numerical results are given.
LookAhead In BiCGSTAB And Other Product Methods For Linear Systems
, 1995
"... The Lanczos method for solving Ax = b consists in constructing the sequence of vectors x k such that r k = b \Gamma Ax k = P k (A)r 0 where P k is the orthogonal polynomial of degree at most k with respect to the linear functional c whose moments are c(¸ i ) = c i = (y ..."
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Cited by 6 (4 self)
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The Lanczos method for solving Ax = b consists in constructing the sequence of vectors x k such that r k = b \Gamma Ax k = P k (A)r 0 where P k is the orthogonal polynomial of degree at most k with respect to the linear functional c whose moments are c(¸ i ) = c i = (y