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<p>Extended Full-Waveform Inversion (FWI) methods are generating considerable interest because of their capability to mitigate the cycle-skipping pathology. The Wavefield Reconstruction Inversion (WRI), which is a well-documented subclass of the extended FWI, extends the inversion search space by considering the wavefield as an extra unknown parameter during inversion. This modification involves the computation of the so-called Data-Assimilated Wavefields (DAW). The DAW is the solution to an ill-conditioned Hermitian Positive Definite linear system. In the case of large-scale 2D/3D models, explicit solutions of the normal equations associated with DAW (in the frequency domain) can be challenging, especially for conventional direct solvers due to the time and memory complexity. In this paper, we explore various iterative methods to solve the DAW of large-scale two-dimensional problems. More specifically, we employ the preconditioned Conjugate Gradient method with Symmetric Successive Over-Relaxation, Incomplete Cholesky Factorization, and Additive Schwarz (AS) preconditioner. To solve the wavefield reconstruction simultaneously for multiple (sources) Right-Hand Sides (RHSs), commonly, direct solvers are the method of choice since they solve all the RHSs at once. Block iterative methods, such as Block Conjugate Gradient (BCG), can be a substitute for direct solvers for multiple RHSs systems. However, the success of the BCG algorithm requires the orthogonality of the RHSs columns to avoid rank deficiency happening during the iterations of the algorithm. To cope with this issue, we propose a cheap approach, the so-called source sketching technique, which increases the orthogonality of the RHSs and dimension reduction, thereby not only preventing the breakdown of the block iterative methods but also reducing the computational complexity. We implemented the proposed method for the DAW problem in the Iteratively-Refined WRI (IR-WRI). The results obtained by numerical examples show high performance of the suggested method for large-scale 2D frequency-domain waveform inversion with an AS preconditioner.</p>

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Frequency-Domain Wavefield Reconstruction Inversion Using Preconditioned Block Conjugate Gradient

  • 期刊论文

  • 发表日期：

    2026年6月19日

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  • Amir Jamasb,
  • Mohammad Ali Riahi,
  • Arash Rezaei Nevisi

摘要

Extended Full-Waveform Inversion (FWI) methods are generating considerable interest because of their capability to mitigate the cycle-skipping pathology. The Wavefield Reconstruction Inversion (WRI), which is a well-documented subclass of the extended FWI, extends the inversion search space by considering the wavefield as an extra unknown parameter during inversion. This modification involves the computation of the so-called Data-Assimilated Wavefields (DAW). The DAW is the solution to an ill-conditioned Hermitian Positive Definite linear system. In the case of large-scale 2D/3D models, explicit solutions of the normal equations associated with DAW (in the frequency domain) can be challenging, especially for conventional direct solvers due to the time and memory complexity. In this paper, we explore various iterative methods to solve the DAW of large-scale two-dimensional problems. More specifically, we employ the preconditioned Conjugate Gradient method with Symmetric Successive Over-Relaxation, Incomplete Cholesky Factorization, and Additive Schwarz (AS) preconditioner. To solve the wavefield reconstruction simultaneously for multiple (sources) Right-Hand Sides (RHSs), commonly, direct solvers are the method of choice since they solve all the RHSs at once. Block iterative methods, such as Block Conjugate Gradient (BCG), can be a substitute for direct solvers for multiple RHSs systems. However, the success of the BCG algorithm requires the orthogonality of the RHSs columns to avoid rank deficiency happening during the iterations of the algorithm. To cope with this issue, we propose a cheap approach, the so-called source sketching technique, which increases the orthogonality of the RHSs and dimension reduction, thereby not only preventing the breakdown of the block iterative methods but also reducing the computational complexity. We implemented the proposed method for the DAW problem in the Iteratively-Refined WRI (IR-WRI). The results obtained by numerical examples show high performance of the suggested method for large-scale 2D frequency-domain waveform inversion with an AS preconditioner.