New PDF release: Anomaly Detection in Random Heterogeneous Media: Feynman-Kac

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By Martin Simon

ISBN-10: 3658109920

ISBN-13: 9783658109929

ISBN-10: 3658109939

ISBN-13: 9783658109936

This monograph is anxious with the research and numerical answer of a stochastic inverse anomaly detection challenge in electric impedance tomography (EIT). Martin Simon reports 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 with regards to the underlying stochastic boundary price challenge. the writer combines strategies from the idea of partial differential equations and sensible research with probabilistic principles, paving find out how to new mathematical theorems that could be fruitfully utilized in the remedy of the matter handy. additionally, the writer proposes a good numerical technique within the framework of Bayesian inversion for the sensible answer of the stochastic inverse anomaly detection challenge.

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

Example text

X ∈ Rd P ω ∈ Γ : c−1 |ξ|2 ≤ ξ · κ(x, ω)ξ ≤ c|ξ|2 = 1. (A2) {κ(x, ω), (x, ω) ∈ Rd × Γ} satisfies the spectral gap property, cf. [58]: There exist constants ρ, r > 0 such that for all measurable functions on κ : Rd → {κ0 ∈ Rd×d : |κ0 ξ| ≤ |ξ|, c|ξ|2 ≤ ξ · κ0 ξ for all ξ ∈ Rd } Vφ ≤ 1 M ρ Rd oscκ|B(x,r) φ 2 dx, where we have set κ) : κ ˜ ∈ Ω, κ ˜ |Rd \B(x,r) = κ|Rd \B(x,r) oscκ|B(x,r) φ := sup φ(˜ − inf φ(˜ κ) : κ ˜ ∈ Ω, κ ˜ |Rd \B(x,r) = κ|Rd \B(x,r) . , κε (·, ·) : Rd × Γ → Rd×d , κε (x, ω) := κ(x/ε, ω).

S. e. x ∈ D. s. e. 35) and the Markov property of X. 15. 26). s. 26) is well-defined. s. e. x ∈ D. Note that the second term on the right-hand side is a local Px -martingale and that eg is continuous, adapted to {Ft , t ≥ 0} and of bounded variation. Multiplication by such functions leaves the class of semimartingales invariant. e. , where the second summand on the right-hand side is a local Px martingale. That is, there exists an increasing sequence (τk )k∈N of stopping times which tend to infinity such that for every k ∈ N t∧τk Mt∧τk := eg (s)∇u(Xs ) dMsu 0 is a Px -martingale.

8. 18). , where W is a standard d-dimensional Brownian motion, L0 is the symmetric local time of X at ∂D2 and L is the boundary local time. Proof. s. s. for every x ∈ D, =2 0 implying that t Mtv = 0 where W is a standard d-dimensional Brownian motion. 3 Derivation of the Feynman-Kac formulae 35 By Green’s formula we have for all w ∈ D(E) ∩ C(D) E(v, w) = ∇v · ∇w dx + κ2 κ1 D1 = − ∇v · ∇w dx D2 wκΔv dx − (κ1 − κ2 ) D ∂ν vw dσ(x) ∂D2 ∂ν vw dσ(x). 29 that N φi is associated with the signed Radon measure −(κ1 − κ2 )ν(x) dσ|∂D2 (x) + κ1 ν(x) dσ|∂D (x).

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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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