Preconditioning Large Scale Iterative Solution of Ax = b Using a Statistical Method with Application to Matrix-Free Spectral Solution of Helmholtz Equation

where the residual vector R contains high-order derivatives of the dependent variable u(t, x1, x2, . . . , xn) with respect to independent variables. Equation (1) is an abstract representation of a generic Partial Differential Equation (PDE) in the residual form where the linear operator L accounts for a linear combination of the k partial derivative in the i direction, i.e. ∂/∂xi . When discretized using a suitable discretization method like Finite-Differences, Finite Elements or Finite Volume [5], the resulting equations can be written in the following semi-discrete form