The Block Conjugate Gradient for Multiple Right Hand Sides in a Direct Current Resistivity Inversion

In geophysical applications direct current (DC) resistivity surveys are collected by injecting current and recording voltage potentials over a field site of interest. These data sets can provide valuable non-invasive images of the sub-surface. The optimization problem that arrises in producing these images requires multiple solves of a linear system with numerous right hand sides (RHSs). Due to the three dimensional nature of the DC resistivity problem, the system to solve is typically too large for direct methods, and iterative methods must be used; in this case conjugate gradient (CG) is ideal. However, standard iterative solvers have no way to share information between the different solves. Information sharing techniques between the RHSs is necessary for speedy convergence and are why direct methods are so attractive. Standard iterative solvers can be modified to block methods that share information by solving multiple RHSs at once. The extension from CG to the corresponding block method, block conjugate gradient (BLCG), is clearly laid out and practical implementation issues are discussed. BLCG can have have numerical instabilities if linear dependencies exist between the different RHSs. This is dealt with through initial deflation and orthogonalization of the RHSs and an unequal convergence scheme. Numerical experiments were ran and suggest that using BLCG can have significant gains over using standard CG. These gains are most dramatic if initial deflation of the RHSs is possible and enforced.