Efficient Helmholtz equation solver for frequency domain waveform inversion based on the decomposition into one-way wave equations

Rehman Ali; Feiyu Wang; Trevor Mitcham; Nebojsa Duric

doi:10.1117/12.3006288

1 April 2024 Efficient Helmholtz equation solver for frequency domain waveform inversion based on the decomposition into one-way wave equations

Rehman Ali, Feiyu Wang, Trevor Mitcham, Nebojsa Duric

Author Affiliations +

Proceedings Volume 12932, Medical Imaging 2024: Ultrasonic Imaging and Tomography; 1293216 (2024) https://doi.org/10.1117/12.3006288
Event: SPIE Medical Imaging, 2024, San Diego, California, United States

Conference Poster

Abstract

When using a ring array to perform ultrasound tomography, the most computationally intensive component of frequency-domain full waveform inversion (FWI) is the Helmholtz equation solver. The Helmholtz equation is an elliptic partial differential equation (PDE) whose discretization leads to a large system of equations; in many cases, the solution of this large system is itself the inverse problem and requires an iterative method. Our current solution relies on discretizing the 2D Helmholtz equation based on a 9-point stencil and using the resulting block tridiagonal structure to efficiently compute a block LU factorization. Conceptually, the L and U systems are equivalent to a forward and backward wave propagation along one of the spatial dimensions of the grid, resulting in a direct non-iterative solution to the Helmholtz equation based on a single forward and backward sweep. Based on this observation, the PDE representation of the Helmholtz equation is split into two one-way wave equations prior to discretization. The numerical implementations of these one-way wave equations are highly parallelizable and lend themselves favorably to accelerated GPU implementations. We consider the Fourier split-step and phase-shift-plus-interpolation (PSPI) methods from seismic imaging as numerical solutions to the one-way wave equations. We examine how each scheme affects the numerical accuracy of the final Helmholtz equation solution and present its impact on FWI with breast imaging examples.

(2024) Published by SPIE. Downloading of the abstract is permitted for personal use only.

Citation Download Citation

Rehman Ali, Feiyu Wang, Trevor Mitcham, and Nebojsa Duric "Efficient Helmholtz equation solver for frequency domain waveform inversion based on the decomposition into one-way wave equations", Proc. SPIE 12932, Medical Imaging 2024: Ultrasonic Imaging and Tomography, 1293216 (1 April 2024); https://doi.org/10.1117/12.3006288

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