次扩散逆问题的后验一致性
Posterior Consistency for Recovering Initial States in Nonlinear Subdiffusion Equations
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中文总结 AI 辅助
该研究针对半线性时间分数次扩散方程的初始状态，赋予Whittle--Matérn过程重标高斯先验，证明解的正则性与后验收缩率，并构造小波包集合证明极小极大下界。
中文摘要 AI 辅助
我们研究从含噪声的随机时空点观测中，贝叶斯恢复半线性时间分数次扩散方程的初始状态。对未知初始条件赋予基于Whittle--Matérn过程的重标高斯先验。当非线性项在$H^\kappa$范数下满足利普希茨条件时，我们证明解的$H^{2+\kappa}$正则性。随后建立预测误差在$L^2$范数下及参数在索伯列夫范数下的后验收缩率，该率为样本量的多项式函数，指数依赖于先验光滑性与空间维数。此外，通过构造小波包集合并控制Kullback--Leibler散度，我们证明了极小极大下界。
英文摘要
We study the Bayesian recovery of the initial state in a semilinear time-fractional subdiffusion equation from noisy random space-time point observations. A rescaled Gaussian prior based on a Whittle--Matérn process is assigned to the unknown initial condition. We prove the \(H^{2+κ}\)-regularity of the solution when the nonlinearity satisfies a Lipschitz condition in the \(H^κ\)-norm. We then establish posterior contraction rates for the prediction error in the \(L^2\)-norm and for the parameter in Sobolev norms. The rates are polynomial in the sample size, with exponent depending on the prior smoothness and the spatial dimension. Moreover, we prove a minimax lower bound by constructing a wavelet-packing set and controlling the Kullback--Leibler divergences.
