Multilevel Phase Reconstruction for a Rapidly Decreasing Interpolating Function

Abstract

Let \(\mathcal {S}(\mathbb {R})\) be the Schwartz space of all complex–valued, rapidly decreasing functions. Let \(f \in \mathcal {S}(\mathbb {R})\) be an interpolating function with the Fourier transform \(\hat {f}\). In this paper, we consider the following phase retrieval problem: Recover f, if only finitely many values of |f| and \(|\hat {f}|\) are given. This problem leads to a well structured nonlinear system. The solution of the nonlinear system is an ill–posed inverse problem which we solve by iteratively regularized Gauss–Newton method. Applying a new multilevel strategy, we construct sufficiently good initial guesses. We close with some numerical tests for undisturbed and noisy data, which illustrate the performance of our method.