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全波形反演中目标函数的Hessian信息对加速收敛起着重要作用,但直接计算Hessian矩阵及其逆通常是不可行的。有限内存BFGS(limited memory Broyden-Fletcher-Goldfarb-Shannon,L-BFGS)法或Hessian-free不精确牛顿(Hessian-free inexact Newton,HFN)法可以使用近似的Hessian信息,但使用程度有限。这两种方法能够互相提供Hessian信息,因而可以混合迭代。混合方法的性能依赖于二者间的有效转换。本文设计了基于目标函数下降比率(下降百分比)的迭代方法动态转换的新方案,得到一种改进的混合迭代优化方法。通过比较相同计算代价下两种方法的目标函数下降比率的大小,新方案使混合方法总是执行下降最快的迭代方法。Marmousi和Overthrust模型的数值试验表明,在保证反演质量的同时,改进方法的收敛速度明显快于L-BFGS法,比Enriched法有小幅提升。它也改进了HFN法效率低的不足。
The Hessian information of the objective function in the full waveform inversion plays an important role in accelerating convergence, but it is usually not feasible to directly calculate the Hessian matrix and its inverse. The limited memory BFS (Finite Memory Broyden-Fletcher-Goldfarb-Shannon, L-BFGS) method or the Hessian-free inexact Newton (HFN) method can use approximate Hessian information, but its use is limited. Both of these methods provide Hessian information to each other so they can be mixed and iterated. The performance of the hybrid approach depends on the efficient conversion between the two. In this paper, we design a new scheme of dynamic transformation based on the iterative method of the target function declining ratio (decreasing percentage), and get an improved hybrid iterative optimization method. By comparing the reduction rates of the objective functions of the two methods under the same computational cost, the new scheme makes the hybrid method always perform the fastest decreasing iteration method. The numerical experiments of Marmousi and Overthrust model show that the convergence rate of the improved method is obviously faster than that of the L-BFGS method, while slightly improving the Enriched method. It also improves the low efficiency of the HFN method.