Analysis of a Greedy Reconstruction Algorithm

Abstract

This paper is devoted to the development and convergence analysis of greedy reconstruction algorithms based on the strategy presented in [Y. Maday and J. Salomon, Joint Proceedings of the 48th IEEE Conference on Decision and Control and the 28th Chinese Control Conference, 2009, pp. 375-379]. These procedures allow the design of a sequence of control functions that ease the identification of unknown operators in nonlinear dynamical systems. The original strategy of greedy reconstruction algorithms is based on an offline/online decomposition of the reconstruction process and an ansatz for the unknown operator obtained by an a priori chosen set of linearly independent matrices. In the previous work [S. Buchwald, G. Ciaramella and J. Salomon, SIAM J. Control Optim., 59(6), pp. 4511-4537], convergence results were obtained in the case of linear identification problems. We tackle here the more general case of nonlinear systems. More precisely, we introduce a new greedy algorithm based on the linearized system. Then, we show that the controls obtained with this new algorithm lead to the local convergence of the classical Gauss-Newton method applied to the online nonlinear identification problem. We then extend this result to the controls obtained on nonlinear systems where a local convergence result is also proved. The main convergence results are obtained for the reconstruction of drift operators in dynamical systems with linear and bilinear control structures.

A greedy reconstruction algorithm for the identification of spin distribution

“…For pedagogical purposes, we have limited the mathematical derivation of the algorithm to its strict minimum. The interested reader can find mathematical details about the algorithms in [56] and [57] for the standard and optimized GRA, respectively.…”

“…if for consecutive iterations the rank of W does not increases. These reasons are at the origin of an optimized algorithm [57]. OGRA takes as input a set Φ, possibly larger than (ϕ k ) K k=1 with linearly dependent elements, and returns as output not only a set of K control functions, but also a set of linearly independent functions ( ϕ k ) K k=1 .…”

“…The integers K and K are not necessarily equal and may be smaller than K (in contrast to GRA). The extension of the OGR method of [57] to the distribution reconstruction problem is detailed below, where we use the map h S defined as where S = (ϕ 1 , • • • , ϕ k ). Note that for the fitting-step problem, we do not have any constraint for the choice of coefficients β.…”

“…Greedy algorithms find generally a sub-optimal solution, but in a computational time which may be very small compared to the one of a global optimization procedure. Such algorithms have been recently applied to the identification of quantum systems [56,57] and we propose to adapt them to the reconstruction of probability distribution. For the sake of clarity, we focus in this study on a specific example, although our algorithm applies to a large variety of systems.…”

Greedy reconstruction algorithm for the identification of spin distribution

“…In a previous work [30], we introduced a Greedy Reconstruction Algorithm (GRA) to identify in a systematic way the probability distribution of one given Hamiltonian parameter. This was based on the framework presented in [31,18]. In particular, we focused on an ensemble of spin 1/2 particles in Nuclear Magnetic Resonance (NMR) subjected to an inhomogeneous radio-frequency magnetic field [32,6,33,8,34,35], where the algorithm was successfully applied to identify the distribution of the scaling factor corresponding to the sample inhomogeneity.…”