Iterative Born Solver for the Acoustic Helmholtz Equation with Heterogeneous Sound Speed and Density

Abstract

Efficient numerical solution of the acoustic Helmholtz equation in heterogeneous media remains challenging, particularly for large-scale problems with spatially-varying density - a limitation that restricts applications in biomedical acoustics and seismic imaging. We present a fast iterative solver that extends the Convergent Born Series method to handle arbitrary variations in sound speed, density, and absorption simultaneously. Our approach reformulates the Helmholtz equation as a first-order system and applies Vettenburg and Vellekoop's universal split-preconditioner, yielding a matrix-free algorithm that leverages Fast Fourier Transforms for computational efficiency. Unlike existing Born series methods, our solver accommodates heterogeneous density without requiring expensive matrix decompositions or pre-processing steps, making it suitable for large-scale 3D problems with minimal memory overhead. The method provides both forward and adjoint solutions, enabling its application for inverse problems. We validate accuracy through comparison against an analytical solution and demonstrate the solver's practical utility through transcranial ultrasound simulations. The solver achieves convergence for strong scattering scenarios, offering a computationally efficient alternative to time-domain methods and matrix-based Helmholtz solvers for applications ranging from medical ultrasound treatment planning to seismic exploration.