Mr. Feiyu Wang Profile

Feiyu Wang

at Univ of Rochester

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Proceedings Article | 1 April 2024 Poster + Paper

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

Proceedings Volume 12932, 1293216 (2024) https://doi.org/10.1117/12.3006288

KEYWORDS: Wave equations, Imaging arrays, Signal attenuation, Wave propagation, Ultrasound tomography, Tissues, Matrices, Visualization, Transducers, Partial differential equations

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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.

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