Random sketching of operators with application to learning preconditioners

We propose a new random sketching approach for embedding high-dimensional Hilbert-Schmidt operators, using random input-output pairs. Such operator can then be approximated in a low-dimensional subspace of operators by solving a small least-squares problem. To achieve computational efficiency, we introduce a structured random map, composed of three random matrices. We provide rigorous conditions under which subspaces of operators are accurately embedded with high probability. The framework is flexible, as the random matrices may be adapted to the operator structure and the computational environment. As an application, we consider the construction of preconditioners for high-dimensional linear equations. We derive a rigorous characterization of preconditioner quality through the discrepancy between the preconditioned operator and an optimal baseline, which can be tailored to a linear approximation space for the solution. We show that this quantity can be efficiently minimized within the proposed framework, especially for parameter separable linear equations. We then establish rigorous high-probability bounds on the quasi-optimality error of the preconditioned Galerkin projection and on the accuracy of a preconditioned residual-based error estimator when the sketch dimensions are sufficiently large. Numerical experiments on an acoustic wave scattering benchmark demonstrate the effectiveness of the method.