Full-waveform inversion (FWI) has emerged as a powerful strategy
for estimating the subsurface model parameters by iteratively
minimizing the difference between the synthetic data and observed
data. The gradient-based methods promise to converge globally but
suffer from slow convergence rate. The Newton-type methods provide
a quadratic convergence, but the computation, storage and
inversion of the Hessian are beyond the current computation
ability for large-scale inverse problem. The Hessian-free (HF)
optimization method represents an attractive alternative to these
above-mentioned optimization methods. At each iteration, it
obtains the search direction by approximately solving the Newton
linear system using a conjugate-gradient (CG) algorithm with a
matrix-free fashion. One problem of the HF optimization method is
that the CG algorithm requires many iterations. The main goal of
this paper is to accelerate the HF FWI by preconditioning the CG
algorithm. In this research, different preconditioning schemes for
the HF Gauss-Newton optimization method are developed. The
preconditioners are designed as Hessian approximations (e.g.,
diagonal pseudo-Hessian and diagonal Gauss-Newton Hessian) or its
inverse approximations. We also developed a new pseudo diagonal
Gauss-Newton Hessian approximation for preconditioning based on
the reciprocal property of the Greenâ€™s function. Furthermore, a
quasi-Newton *
l*
-BFGS inverse Hessian approximation preconditioner
with the diagonal Hessian approximation as initial guess is
proposed and developed. Several numerical examples are solved to
demonstrate the effectiveness of the preconditioning schemes. It
is concluded that the quasi-Newton *
l*
-BFGS preconditioning scheme
with the pseudo diagonal Gauss-Newton Hessian as initial guess
shows the best performances in speeding up the HF Gauss-Newton
FWI, improving the convergence rate and reducing the computation
burden.

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