ABSTRACT

In this dissertation, the deflation preconditioner is adapted along with variation in CSLP for the Helmholtz equation. The equation finds applications in areas where a wave phenomenon is modelled. This work focuses on seismology and seismic imaging. The high frequency wave propagation, which is modelled by the Helmholtz equation, is considered. The conventional methods face difficulty as the problem is ill-conditioned and the severity of conditioning is proportional to the number of grid points. Sparsity patterns and properties of the linear system make the Krylov subspace methods inevitable. Their vulnerability to preconditioning is not a hidden secret. The standard matrix preconditioner had enjoyed success to some extent. Lately, CSLP has been rewarding in terms of time and memory. However, high frequency and a greater number of grid points, a prerequisite of Discretization accuracy, exhaust the CSLP for any choice of shifts within. The part of the spectrum around origin has hampered the convergence, which is noticed in small frequency problems as remarkable. Deflation preconditioner projects this near null part of eigenvalues. This research is an attempt to adapt the deflation preconditioner, to make multilevel solver more viable and adaptive. Also, adaption in CSLP in a shift at various levels has been proposed and propositions are validated by numerical and graphical experiments. The parallel implementation, at two levels, is performed. Results show the worth of the effort/
Keywords: Helmholtz Equation, Seismic Waves, Complex Shifted Laplace
Preconditioner, Krylov Subspace, Deflation, Adaptive Deflation, Multilevel,
Coarse grid, Fine Grid.