  The complex-shifted Laplacian systems arising in a wide range of
applications. In this work, we propose an absolute-value based preconditioner
for solving the complex-shifted Laplacian system. In our approach, the
complex-shifted Laplacian system is equivalently rewritten as a $2\times 2$
block real linear system. With the Toeplitz structure of uniform-grid
discretization of the constant-coefficient Laplacian operator, the absolute
value of the block real matrix is fast invertible by means of fast sine
transforms. For more general coefficient function, we then average the
coefficient function and take the absolute value of the averaged matrix as our
preconditioner. With assumptions on the complex shift, we theoretically prove
that the eigenvalues of the preconditioned matrix in absolute value are upper
and lower bounded by constants independent of matrix size, indicating a
matrix-size independent linear convergence rate of MINRES solver.
Interestingly, numerical results show that the proposed preconditioner is still
efficient even if the assumptions on the complex shift are not met. The fast
invertibility of the proposed preconditioner and the robust convergence rate of
the preconditioned MINRES solver lead to a linearithmic (nearly optimal)
complexity of the proposed solver. The proposed preconditioner is compared with
several state-of-the-art preconditioners via several numerical examples to
demonstrate the efficiency of the proposed preconditioner.
