FROM FEYNMAN-KAC FORMULAE TO NUMERICAL STOCHASTIC HOMOGENIZATION IN ELECTRICAL IMPEDANCE TOMOGRAPHY

• 成果类型: Article
• 署名作者: Piiroinen, Petteri; Simon, Martin
• 署名单位: University of Helsinki; Johannes Gutenberg University of Mainz
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/15-AAP1168
• 发表日期: 2016
• 页码: 3001-3043
• 关键词: boundary-value-problems; reversible markov-processes; probabilistic approach; effective conductivity; reflecting diffusions; brownian-motion; coefficients; functionals; Operators; domains

摘要

In this paper, we use the theory of symmetric Dirichlet forms to derive Feynman-Kac formulae for the forward problem of electrical impedance tomography with possibly anisotropic, merely measurable conductivities corresponding to different electrode models on bounded Lipschitz domains. Subsequently, we employ these Feynman-Kac formulae to rigorously justify stochastic homogenization in the case of a stochastic boundary value problem arising from an inverse anomaly detection problem. Motivated by this theoretical result, we prove an estimate for the speed of convergence of the projected mean-square displacement of the underlying process which may serve as the theoretical foundation for the development of new scalable stochastic numerical homogenization schemes.