Wavelet (Besov) priors are a promising way of reconstructing indirectly measured fields in a regularized manner. We demonstrate how wavelets can be used as a localized basis for reconstructing permeability fields with sharp interfaces from noisy pointwise pressure field measurements in the context of the elliptic inverse problem. For this we derive the adjoint method of minimizing the Besov-norm-regularized misfit functional (this corresponds to determining the maximum a posteriori point in the Bayesian point of view) in the Haar wavelet setting. As it turns out, choosing a wavelet–based prior allows for accelerated optimization compared to established trigonometrically–based priors.

中文翻译：

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基于小波的先验加速贝叶斯逆问题中的最大后验优化

小波（Besov）先验是一种有规律的以规则方式重建间接测量场的方法。我们演示了如何将小波用作在椭圆逆问题中从嘈杂的点向压力场测量中重建具有尖锐界面的渗透率场的局部基础。为此，我们推导了在Haar小波设置中最小化Besov-范数正则化的拟合函数（这与确定贝叶斯视点中的最大后验点相对应）最小化的伴随方法。事实证明，与基于三角函数的先验相比，选择基于小波的先验可以加速优化。