标题： 用高效高阶Hessian近似加速地震反演和不确定性量化 显示英文标题

标题： Accelerating seismic inversion and uncertainty quantification with efficient high-rank Hessian approximations

摘要： 高效的大秩Hessian近似可以加速地震全波形反演（FWI）和不确定性量化（UQ）。 在FWI中，Hessian逆的近似可以用作牛顿型或拟牛顿算法的预条件器，降低计算成本并提高深层地下区域的恢复效果。 在贝叶斯UQ中，Hessian近似能够构建捕捉后验方向缩放的马尔可夫链蒙特卡洛（MCMC）提议，提高MCMC的效率。 对于大规模问题，计算精确的Hessian是不可行的，因为Hessian只能通过矩阵-向量乘积获得，而每次矩阵-向量乘积都需要昂贵的波动方程求解。 此外，Hessian是高秩的，这意味着通常用于大规模反问题的低秩方法效率低下。 我们适应了两种现有的高秩Hessian近似方法——点扩散函数方法和伪微分算子探测方法。 基于这两种方法之间观察到的对偶性，我们开发了一种新方法，统一了它们的互补优势。 我们在一个合成二次模型和Marmousi模型上验证了这些方法。 数值实验表明，这些高秩Hessian近似显著降低了FWI中的计算成本。 在UQ中，使用无Hessian近似或低秩近似的MCMC样本缓慢探索后验分布，在数万次迭代后提供很少有意义的统计信息，并低估方差。 同时，有效样本大小被高估，提供了虚假的置信度。 相比之下，使用高秩Hessian近似的MCMC样本能够提供关于后验的有意义的统计信息，并更准确地评估后验方差。 显示英文摘要

摘要： Efficient high-rank approximations of the Hessian can accelerate seismic full waveform inversion (FWI) and uncertainty quantification (UQ). In FWI, approximations of the inverse of the Hessian may be used as preconditioners for Newton-type or quasi-Newton algorithms, reducing computational costs and improving recovery in deeper subsurface regions. In Bayesian UQ, Hessian approximations enable the construction of Markov chain Monte Carlo (MCMC) proposals that capture the directional scalings of the posterior, enhancing the efficiency of MCMC. Computing the exact Hessian is intractable for large-scale problems because the Hessian is accessible only through matrix-vector products, and performing each matrix-vector product requires costly solution of wave equations. Moreover, the Hessian is high-rank, which means that low-rank methods, often employed in large-scale inverse problems, are inefficient. We adapt two existing high-rank Hessian approximations -- the point spread function method and the pseudo-differential operator probing method. Building on an observed duality between these approaches, we develop a novel method that unifies their complementary strengths. We validate these methods on a synthetic quadratic model and on the Marmousi model. Numerical experiments show that these high-rank Hessian approximations substantially reduce the computational costs in FWI. In UQ, MCMC samples computed using no Hessian approximation or a low-rank approximation explore the posterior slowly, providing little meaningful statistical information after tens of thousands of iterations and underestimating the variance. At the same time, the effective sample size is overestimated, providing false confidence. In contrast, MCMC samples generated using the high-rank Hessian approximations provide meaningful statistical information about the posterior and more accurately assess the posterior variance.