从Feynman-Kac公式到电阻抗成像的数值随机均匀化
From Feynman-Kac Formulae to Numerical Stochastic Homogenization in Electrical Impedance Tomography
摘要：
在本文中，我们使用对称Dirichlet形式的理论，推导出Feynman-Kac公式的电阻抗层析成像的正问题与可能各向异性，只是可测量的电导率对应于不同的电极模型上有界Lipschitz域。随后，我们采用这些Feynman-Kac公式严格证明随机均匀化的情况下所产生的逆异常检测问题的随机边值问题。受此理论结果的启发，我们证明了一个估计的基本过程，这可能是新的可扩展的随机数值均匀化方案的发展的理论基础的投影均方位移的收敛速度。
英文摘要：
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.
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