Modern Solvers for Helmholtz Problems

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This edited volume provides a comprehensive overview of advanced solvers for the Helmholtz equation, structured into three parts: developments and analysis, practical methods and implementations, and industrial applications. The Helmholtz equation is crucial in various scientific and engineering fields involving wave propagation, such as seismic inversion, ultrasound imaging, sonar detection, and harbor wave analysis. Although the equation appears straightforward, it poses significant challenges in solving. Numerical methods are essential for approximating solutions, beginning with discretization through techniques like the Finite Difference Method, Finite Element Method, Discontinuous Galerkin Method, and Boundary Element Method. The resulting linear systems can be large, especially as problem size increases with frequency, necessitating detailed seismic images that lead to larger numerical challenges. Fast and robust iterative solvers are essential for tackling these three-dimensional problems. However, standard iterative methods often require excessive iterations to converge. Consequently, new methods have been developed to address these issues. The book targets researchers from academia and industry, as well as graduate students, with a prerequisite understanding of partial differential equations and numerical linear algebra.