Matrix Probing and its Conditioning

Abstract

When a matrix A with n columns is known to be well-approximated by a linear combination of basis matrices B1, . . . , Bp, we can apply A to a random vector and solve a linear system to recover this linear combination. The same technique can be used to obtain an approximation to A[superscript −1]. A basic question is whether this linear system is well-conditioned. This is important for two reasons: a well-conditioned system means (1) we can invert it and (2) the error in the reconstruction can be controlled. In this paper, we show that if the Gram matrix of the Bj ’s is sufficiently well-conditioned and each Bj has a high numerical rank, then n [infinity symbol] p log[superscript 2] n will ensure that the linear system is well-conditioned with high probability. Our main application is probing linear operators with smooth pseudodifferential symbols such as the wave equation Hessian in seismic imaging [L. Demanet et al., Appl. Comput. Harmonic Anal., 32 (2012), pp. 155–168]. We also demonstrate numerically that matrix probing can produce good preconditioners for inverting elliptic operators in variable media.