  We introduce a new ADI-based low rank solver for $AX-XB=F$, where $F$ has
rapidly decaying singular values. Our approach results in both theoretical and
practical gains, including (1) the derivation of new bounds on singular values
for classes of matrices with high displacement rank, (2) a practical algorithm
for solving certain Lyapunov and Sylvester matrix equations with high rank
right-hand sides, and (3) a collection of low rank Poisson solvers that achieve
spectral accuracy and optimal computational complexity.
