Exploring rational approximations of fractional power operators for preconditioning

Abstract We define and analyze preconditioners for the Riesz operator $$-(- \Delta )^{\frac{\alpha }{2}}$$ , $$\alpha \in (1,2]$$ commonly used in fractional models, such as anomalous diffusion. For $$\alpha$$ close to 2 there are various effective preconditioners at disposal with linear computational cost. Seminal results on treatment of the case $$\alpha$$ near 1 that still maintains linear computational complexity has been obtained approximating the Riesz operator as a fractional power of a d