By Martin Simon

ISBN-10: 3658109920

ISBN-13: 9783658109929

ISBN-10: 3658109939

ISBN-13: 9783658109936

This monograph is worried with the research and numerical answer of a stochastic inverse anomaly detection challenge in electric impedance tomography (EIT). Martin Simon reviews the matter of detecting a parameterized anomaly in an isotropic, desk bound and ergodic conductivity random box whose realizations are speedily oscillating. For this objective, he derives Feynman-Kac formulae to scrupulously justify stochastic homogenization in terms of the underlying stochastic boundary worth challenge. the writer combines thoughts from the idea of partial differential equations and practical research with probabilistic principles, paving the right way to new mathematical theorems that could be fruitfully utilized in the remedy of the matter handy. additionally, the writer proposes a good numerical approach within the framework of Bayesian inversion for the sensible resolution of the stochastic inverse anomaly detection challenge.

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Extra info for Anomaly Detection in Random Heterogeneous Media: Feynman-Kac Formulae, Stochastic Homogenization and Statistical Inversion

Example text

Closedness is obvious since for all v ∈ H 1 (D) we can find positive constants c1 , c2 such that c1 ||v||22 ≤ E1 (v, v) ≤ c2 ||v||22 . 4). , we construct for each ε > 0 a differentiable function φε : R → R such that φε (t) = t for all t ∈ [0, 1], −ε ≤ φε (t) ≤ 1 + ε for all t ∈ R and 0 ≤ φε (s) − φε (t) ≤ s − t, whenever t < s. 1 Reflecting diffusion processes 25 where the last inequality is a consequence of the property 0 ≤ φε (t) ≤ 1. As E is closed, this is equivalent to the fact that the unit contraction operates on (E, D(E)), cf.

D, denote a vector field. , d. Moreover, we say that φ is divergence-free if for every ψ ∈ Cc∞ (Rd ), d i=1 Rd φi ∂i ψ dx = 0. 1. e. ω ∈ Γ. }. e. e. ω ∈ Γ. 1) defines a stationary random field with respect to the measure P. We call η the potential corresponding to φ. 1. Note that φ ∈ L2pot (Γ) does not imply that {η(x, ω), (x, ω) ∈ Rd × Γ} is a stationary random field with respect to P. In fact, it can be shown that this is not true for d = 1. , [165]. , |ξ| = 1. The so2 called auxiliary problem for the direction ξ reads as follows: Find χξ ∈ Vpot (Γ) ξ 2 such that κ(ξ + χ ) ∈ Lsol (Γ) or equivalently, M{κ(ξ + χξ ) · φ} = 0 2 for all φ ∈ Vpot (Γ).

2 in the next chapter. 12 than to its actual statement. 14. Let f be a bounded Borel function and let α > 0. 5). This solution admits the FeynmanKac representation ∞ u(x) = Ex e−αt f (Xt ) dLt for all x ∈ D. 25) 0 Proof. 12, however, substituting {Tt , t ≥ 0} with the Feynman-Kac semigroup {Tt , t ≥ 0}, Tt v(x) := Ex e−αt v(Xt ). s. for every x ∈ D. 12) is the following theorem. 15. 12). 26) 0 with t eg (t) := exp − g(Xs ) dLs , t ≥ 0. , the oneparameter family of operators {Ttg , t ≥ 0} defined by Ttg v(x) := Ex eg (t)v(Xt ), x ∈ D and t ≥ 0.

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Anomaly Detection in Random Heterogeneous Media: Feynman-Kac Formulae, Stochastic Homogenization and Statistical Inversion by Martin Simon


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