A short note on rank-2 relaxation for waveform inversion

This note is a first attempt to perform waveform inversion by utilizing recent developments in semidefinite relaxations for polynomial equations to mitigate non-convexity. The approach consists in reformulating the inverse problem as a set of constraints on a low-rank moment matrix in a higher-dimensional space. While this idea has mostly been a theoretical curiosity so far, the novelty of this note is the suggestion that a modified adjoint-state method enables algorithmic scalability of the relaxed formulation to standard 2D community models in geophysical imaging. Numerical experiments show that the new formulation leads to a modest increase in the basin of attraction of least-squares waveform inversion.TOTAL (Firm)Belgian National Foundation for Scientific ResearchMIT International Science and Technology InitiativesUnited States. Air Force. Office of Scientific ResearchUnited States. Office of Naval ResearchNational Science Foundation (U.S.

A convex approach to blind deconvolution with diverse inputs

This note considers the problem of blind identification of a linear, time-invariant (LTI) system when the input signals are unknown, but belong to sufficiently diverse, known subspaces. This problem can be recast as the recovery of a rank-1 matrix, and is effectively relaxed using a semidefinite program (SDP). We show that exact recovery of both the unknown impulse response, and the unknown inputs, occurs when the following conditions are met: (1) the impulse response function is spread in the Fourier domain, and (2) the N input vectors belong to generic, known subspaces of dimension K in ℝL. Recent results in the well-understood area of low-rank recovery from underdetermined linear measurements can be adapted to show that exact recovery occurs with high probablility (on the genericity of the subspaces) provided that K,L, and N obey the information-theoretic scalings, namely L ≳ K and N ≳ 1 up to log factors.Fonds national de la recherche scientifique (Belgium)MIT International Science and Technology InitiativesUnited States. Air Force. Office of Scientific ResearchUnited States. Office of Naval ResearchNational Science Foundation (U.S.)Total S