Uncertainty Quantification and Stochastic Modeling with Matlab
Language: English
Pages: 456
ISBN: 1785480057
Format: PDF / Kindle (mobi) / ePub
Uncertainty Quantification (UQ) is a relatively new research area which describes the methods and approaches used to supply quantitative descriptions of the effects of uncertainty, variability and errors in simulation problems and models. It is rapidly becoming a field of increasing importance, with many real-world applications within statistics, mathematics, probability and engineering, but also within the natural sciences.
Literature on the topic has up until now been largely based on polynomial chaos, which raises difficulties when considering different types of approximation and does not lead to a unified presentation of the methods. Moreover, this description does not consider either deterministic problems or infinite dimensional ones.
This book gives a unified, practical and comprehensive presentation of the main techniques used for the characterization of the effect of uncertainty on numerical models and on their exploitation in numerical problems. In particular, applications to linear and nonlinear systems of equations, differential equations, optimization and reliability are presented. Applications of stochastic methods to deal with deterministic numerical problems are also discussed. Matlab® illustrates the implementation of these methods and makes the book suitable as a textbook and for self-study.
- Discusses the main ideas of Stochastic Modeling and Uncertainty Quantification using Functional Analysis
- Details listings of Matlab® programs implementing the main methods which complete the methodological presentation by a practical implementation
- Construct your own implementations from provided worked examples
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that X is a 74 Uncertainty Quantification and Stochastic Modeling with Matlab® scalar random variable having density f , supported by (a, b), we may use the following code: Listing 1.17. Evaluation of statistics f u n c t i o n [mX, vX , sdX ] = s t a t s f ( f , a , b ) % % e v a l u a t i o n o f t h e mean % f 1 = @( x ) x * f ( x ) ; mX = n u m i n t ( f1 , a , b ) ; % % e v a l u a t i o n o f t h e s e c o n d moment % f 2 = @( x ) x ^2 * f ( x ) ; m2X = n u m i n t ( f2 , a , b ) ; % %
(t) dt for δ (t) −→ 0, a i.e., D EFINITION 1.29.– Let X ∈ L2 (a, b) , L2 (Ω, P ) . We say that b X (t) dt = Y ∈ L2 (Ω, P ) if and only if, for any ε > 0, there exists a η (ε) > 0 such that ∀ t ∈ Part ((a, b)) : δ (t) ≤ η (ε) =⇒ Y − I (X, t) L2 (Ω, P ) ≤ ε. Uncertainty Quantification and Stochastic Modeling with Matlab® 116 Let us recall that the scalar product and the norm of L2 (Ω, P ) are given by (X, Y )L2 (Ω, P ) = E (XY ) ; Z L2 (Ω, P ) = E Z2 . Let n > 0 and consider n X (a +
their quality. Maximum Entropy and Information 161 In [KNU 98], the author describes his personal attempt to build a random generator of random variables. He proposed an algorithm that worked with 10-digit decimal numbers, generating a sequence of values {xi }i ∈ N from an initial value. Despite the apparent complication, Knuth found that his algorithm could quickly converge to a fixed point. In addition, the tests showed that, even when using different initial values, the sequence of numbers
of Rk as column vectors, i.e. elements of M (k, 1). This allow us to write the product of an m × k matrix A ∈ M (m, k) and an element of Rk , considered as an element of x ∈ Rk ≡ M (k, 1): y = Ax ∈ M (m, 1) ≡ Rm . As usual, we have y = (y1 , ..., ym ), with k yi = Aij xj . j=1 Elements of Probability Theory and Stochastic Processes 3 Sometimes, in order to simplify the expressions, we use the convention on the implicit sum on repeated indexes (called Einstein’s convention) by simply writing
2 3 0 Ω and 1 X(ω)P (dω) = E (X) = 1 ω 2 dω = . 3 0 Ω The variances are: 1 2 (U (ω) − E (U )) P (dω) = V (U ) = √ ω− 2 3 2 ω2 − 1 3 2 dω = 1 18 dω = 4 . 45 0 Ω and 1 2 (X(ω) − E (X)) P (dω) = V (X) = 0 Ω The result may also be obtained by using the probability density of the variable U. For instance, the cumulative function of U is: ⎧ ⎨ 0, if u ≤ 0 FU (u) = P (U < u) = u2 , if 0 < u ≤ 1 , ⎩ 1, if u > 1 so that its probability density is: 0, if u < 0 or u > 1 . 2u, if
