Download e-book for kindle: Anomaly Detection in Random Heterogeneous Media: Feynman-Kac by Martin Simon

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 experiences the matter of detecting a parameterized anomaly in an isotropic, desk bound and ergodic conductivity random box whose realizations are swiftly oscillating. For this function, he derives Feynman-Kac formulae to carefully justify stochastic homogenization with regards to the underlying stochastic boundary worth challenge. the writer combines options from the idea of partial differential equations and sensible research with probabilistic principles, paving easy methods to new mathematical theorems that could be fruitfully utilized in the remedy of the matter to hand. furthermore, the writer proposes a good numerical strategy within the framework of Bayesian inversion for the sensible resolution 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

Fix x ∈ D. Then by the Chapman-Kolmogorov equation p(t, x, x) − |D|−1 p(t/2, x, y)p(t/2, y, x) dy − |D|−1 = D (p(t/2, x, y))2 dy − |D|−1 = D (p(t/2, x, y) − |D|−1 )2 dy ≥ 0, = D where we have used that D p(t, x, y) dy = 1 for all t ≥ 0 and x ∈ D. Moreover, we have by the analyticity of the mapping t → p(t, x, x), cf. 4) and the Poincaré inequality φ − |D|−1 φ(x) dx D 2 ≤ cD ||∇φ||2 for all φ ∈ H 1 (D). , there exist positive constants c1 and c2 such that 0 ≤ p(t, x, x) − |D|−1 ≤ c1 exp(−c2 t) for every t ≥ 0.

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.

5) is the following theorem. 12. Let f be a bounded Borel function satisfying f, 1 ∂D = 0. 5). This solution admits the Feynman-Kac representation t u(x) = lim Ex t→∞ f (Xs ) dLs for all x ∈ D. 22) 0 Proof. 5) is guaranteed by the standard theory of linear elliptic boundary value problems. Let us set t ut (x) := Ex f (Xs ) dLs and u∞ (x) := lim ut (x), t→∞ 0 x ∈ D, respectively. 6), it follows immediately that t (p(s, x, y) − |D|−1 )f (y) dσ(y) ds for all x ∈ D. 9 the convergence towards the stationary distribution is uniform over D, in particular, ∞ (p(s, x, y) − |D|−1 )f (y) dσ(y) ds for all x ∈ D.

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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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