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\begin{document}

\title{Multi-solver spectral-element and adjoint methods}



\author[1]{Yujiang Xie}%
\author[1]{Catherine A. Rychert}%
\author[1]{Nicholas Harmon}%
\author[2]{Qinya Liu}%
\author[3]{Dirk Gajewski}%
\affil[1]{University of Southampton}%
\affil[2]{University of Toronto}%
\affil[3]{University of Hamburg}%


\vspace{-1em}



  
  \date{June 14, 2024}


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\begin{abstract}
The spectral-element method (SEM) for simulating wave propagation is
widely used with adjoint methods for full-waveform inversion. Typically,
SEM is used to compute forward and adjoint wavefields, which is then
applied to evaluate the Fr\selectlanguage{ngerman}échet derivatives for updating the seismic
structural model. The Hessian is rarely computed as the high
computational and storage costs, although it can improve the accuracy of
the model update and model convergence. Instead the approximate Hessian
is determined, which is obtained with less computational effort. We
present a method for simultaneously constructing Fréchet and Hessian
kernels on the fly, which we call Multi-solver spectral-element and
adjoint methods (Multi-SEM). Rather than storing all the wavefields,
Multi-SEM is computed on the fly and requires only about a 2-fold
computational cost when compared to the computation of Fréchet kernels.
Numerical examples demonstrate the functionality of the method and the
computer codes are provided with this contribution.%
\end{abstract}\selectlanguage{ngerman}%



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