On Kronecker preconditioning based on aggregation

Numerical solution of high order Partial Differential Equations on structured grids is required in several scientific fields such as Control, Computational Finance or Quantum Mechanics. However, the numerical solution of the resulting sparse linear systems for such problems is subject to the well-known curse of dimensionality that affects, especially, memory and computational requirements. These issues can be partly mitigated by utilizing preconditioned iterative methods coupled with efficient parameterized Kronecker based preconditioners that avoid explicit formation. For large numbers of dimensions, computation of the preconditioner parameters is a computationally expensive task requiring the solution of a large nonlinear optimization problem. This can be improved by introducing aggregation, per dimension, to limit the search space for the optimal parameters, coupled with a block Jacobi preconditioner. Moreover, different forms of the Kronecker based preconditioner are proposed and discussed in detail. The effectiveness and applicability of the proposed scheme is assessed by solving several model problems. Furthermore, comparative results with other schemes are also provided.