Lecture Notes in Mathematics

By: Karl W. Breitung

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Paperback | 16 December 1994

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164 Pages

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23.39 x 15.6 x 0.89

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This book gives a self-contained introduction to the subject of asymptotic approximation for multivariate integrals for both mathematicians and applied scientists. A collection of results of the Laplace methods is given. Such methods are useful for example in reliability, statistics, theoretical physics and information theory. An important special case is the approximation of multidimensional normal integrals. Here the relation between the differential geometry of the boundary of the integration domain and the asymptotic probability content is derived. One of the most important applications of these methods is in structural reliability. Engineers working in this field will find here a complete outline of asymptotic approximation methods for failure probability integrals.

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Notation
Introduction	p. 1
The Evaluation of Multivariate Integrals	p. 1
Structural Reliability	p. 2
Stochastic Optimization	p. 6
Large Deviations and Extreme Values	p. 7
Mathematical Statistics	p. 7
Mathematical Preliminaries	p. 9
Results from Linear Algebra	p. 9
Results from Analysis	p. 11
Manifolds	p. 14
Extrema under Constraints	p. 16
Parameter Dependent Integrals	p. 21
Probability Distributions	p. 28
Convergence of Probability Distributions	p. 31
Asymptotic Analysis	p. 34
The Topic of Asymptotic Analysis	p. 34
The Comparison of Functions	p. 35
Asymptotic Power Series and Scales	p. 40
Deriving Asymptotic Expansions	p. 44
Univariate Integrals	p. 45
Watson's Lemma	p. 45
The Laplace Method for Univariate Functions	p. 47
Multivariate Laplace Type Integrals	p. 51
Basic Results	p. 52
Interior Maxima	p. 55
Boundary Maxima	p. 65
Approximations for Normal Integrals	p. 85
Time-Invariant Reliability Problems	p. 85
Linear Approximations (FORM Concepts)	p. 86
SORM Concepts	p. 90
Asymptotic SORM for Normal Integrals	p. 91
The Asymptotic Distribution in the Failure Domain	p. 99
Numerical Procedures and Improvements	p. 102
Examples	p. 103
Arbitrary Probability Integrals	p. 106
Problems of the Transformation Method	p. 106
Asymptotic Approximations in the Original Space	p. 109
Sensitivity analysis	p. 115
Crossing Rates of Stochastic Processes	p. 121
Definition and Properties of Stochastic Processes	p. 121
Maxima and Crossing of a Stochastic Process	p. 124
Crossings through an Hypersurface	p. 127
A Bibliography	p. 135
B Index	p. 145
Table of Contents provided by Blackwell. All Rights Reserved.

Lecture Notes in Mathematics

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Paperback

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