Low Variance Sketched Finite Elements for Elliptic Equations

We present a direct solver for parameter-dependent linear systems of large dimension arising from the application of the finite element method on elliptic boundary value problems. The solver is particularly suited to the many-parameter-query context, typically encountered in uncertainty quantification and inverse problems, when computations must be done on-the-fly and/or without substantial computational resources. Our approach involves low-dimensional subspace projection and randomized sketching of the induced equations, that exploits the positive definite structure of the coefficients matrix and the prior knowledge of this matrix for a uniformly distributed parameter. To suppress the random component of the error in the projected solution we introduce an estimator based on control variates that reduces the variance of the sketched system whilst preserving its well-posedness.