H²-matrix preconditioners for integral and elliptic partial differential equations

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The recently developed H-matrix method can be used to construct efficient solvers for elliptic partial differential equations and integral equations. This approach is particularly robust when considering differential equations with discontinuous or anisotropic coefficients, e.g., appearing in simulations of composite materials. An H-matrix usually requires O(n k log n) units of storage, where n is the matrix dimension and k depends on the condition number of the linear system and the desired accuracy of the approximation.H²-matrices combine the H-matrix technique with multilevel structures in order to reduce the storage requirements to O(n k) and handle significantly larger linear systems. Numerical experiments show that H²-matrices indeed require less storage than their H-matrix counterparts, but the setup time of currently existing algorithms compares unfavorably to those for H-matrices.This research project focuses on developing a new, more efficient method for the construction of H²-matrix solvers. The approach is based on a new algorithm that performs local low-rank updates for H²-matrices in linear, i.e., optimal, complexity. This flexible new algorithm can be used to construct preconditioners for differential and integral equations more efficiently and it is also expected to pave the way for the development of efficient techniques for other arithmetic operations in the set of H²-matrices.