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We present an adaptive grid matrix-free operator approximation scheme basedon a "product-convolution" interpolation of convolution operators. This schemeis appropriate for operators that are locally translation-invariant, even ifthese operators are high-rank or full-rank. Such operators arise in Schurcomplement methods for solving partial differential equations (PDEs), asHessians in PDE-constrained optimization and inverse problems, as integraloperators, as covariance operators, and as Dirichlet-to-Neumann maps.Constructing the approximation requires computing the impulse responses of theoperator to point sources centered on nodes in an adaptively refined grid ofsample points. A randomized a-posteriori error estimator drives the adaptivity.Once constructed, the approximation can be efficiently applied to vectors usingthe fast Fourier transform. The approximation can be efficiently converted tohierarchical matrix ($H$-matrix) format, then inverted or factorized usingscalable $H$-matrix arithmetic. The quality of the approximation degradesgracefully as fewer sample points are used, allowing cheap lower qualityapproximations to be used as preconditioners. This yields an automated methodto construct preconditioners for locally translation-invariant Schurcomplements. We directly address issues related to boundaries and prove thatour scheme eliminates boundary artifacts. We test the scheme on a spatiallyvarying blurring kernel, on the non-local component of an interface Schurcomplement for the Poisson operator, and on the data misfit Hessian for anadvection dominated advection-diffusion inverse problem. Numerical results showthat the scheme outperforms existing methods.

Scalable matrix-free adaptive product-convolution approximation for locally translation-invariant operators