On the convergence of the MG/OPT method

Borzì, Alfio

doi:10.1002/pamm.200510342

Cited by 20 publications

(13 citation statements)

References 6 publications

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“…In the context of quantum control, these are, for example, entanglement and the nature of quantum measurements. Eventually, a new research domain, mathematical quantum control theory [14][15][16][17][18][19][20][21][22][23] has emerged (see also [11,24], Sect. 19.3).…”

Section: General Aspects and Mathematics Of Optimal Controlmentioning

confidence: 99%

Training Schrödinger’s cat: quantum optimal control

et al. 2015

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Abstract. It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a dynamical system from a given initial state into a desired target state with minimized expenditure of energy and resources. As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantumenhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation. In this communication, state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium of experts in optimal control theory and applications to spectroscopy, imaging, as well as quantum dynamics of closed and open systems. We address key challenges and sketch a roadmap for future developments. ForewordThe authors of this paper represent the QUAINT consortium, a European Coordination Action on Optimal Control of Quantum Systems, funded by the European Commission Framework Programme 7, Future Emerging Technologies FET-OPEN programme and the Virtual Facility for Quantum Control (VF-QC). This consortium has considerable expertise in optimal control theory and its applications to quantum systems, both in existing areas, such as spectroscopy and imaging, and in emerging quantum technologies, such as quantum information processing, quantum communication, quantum simulation a e-mail: fwm@lusi.uni-sb.de and quantum sensing. The list of challenges for quantum control has been gathered by a broad poll of leading researchers across the communities of general and mathematical control theory, atomic, molecular-, and chemical physics, electron and nuclear magnetic resonance spectroscopy, as well as medical imaging, quantum information, communication and simulation. 144 experts in these fields have provided feedback and specific input on the state of the art, mid-term and long-term goals. Those have been summarized in this document, which can be viewed as a perspectives paper, providing a roadmap for the future development of quantum control. Because such an endeavour can hardly ever be complete (there are many additional areas of quantum control applications, such as spintronics, nano-optomechanical technologies etc.), this roadmap

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Section: General Aspects and Mathematics Of Optimal Controlmentioning

confidence: 99%

Training Schrödinger’s cat: quantum optimal control

et al. 2015

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“…Second, the pre-and postsmoothing steps at each level, (4.2) and (4.4), can be computed by applying a fixed number of iterations of an optimization routine as suggested in [31]. Alternatively, Borzì [8] and Borzì and Schulz [12] suggest applying the pre-and postsmoothing optimization routines until the following sufficient decrease conditions are satisfied:…”

Section: Collocation Error Boundsmentioning

confidence: 99%

“…) is a descent direction, then Borzì [8] and Borzì and Schulz [12] suggest the following a priori choice for the line-search parameter λ k :…”

Section: Exists and Is Lipschitz Continuous With Lipschitz Constantmentioning

confidence: 99%

A Multilevel Stochastic Collocation Algorithm for Optimization of PDEs with Uncertain Coefficients

Kouri¹

2014

SIAM/ASA J. Uncertainty Quantification

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Abstract. In this work, we apply the MG/OPT framework to a multilevel-in-sample-space discretization of optimization problems governed by PDEs with uncertain coefficients. The MG/OPT algorithm is a template for the application of multigrid to deterministic PDE optimization problems. We employ MG/OPT to exploit the hierarchical structure of sparse grids in order to formulate a multilevel stochastic collocation algorithm. The algorithm is provably first-order convergent under standard assumptions on the hierarchy of discretized objective functions as well as on the optimization routines used as pre-and postsmoothers. We present explicit bounds on the total number of PDE solves and an upper bound on the error for one V-cycle of the MG/OPT algorithm applied to a linear quadratic control problem. We provide numerical results that confirm the theoretical bound on the number of PDE solves and show a dramatic reduction in the total number of PDE solves required to solve these optimization problems when compared with standard optimization routines applied to a fixed sparse-grid discretization of the same problem.

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“…Therefore, within the hierarchy of levels defined by the multigrid strategy, the approach of MSQP allows to construct a coarse-grid correction step that can be understood from the geometrical [40] and optimization [5,6,29,33,41] point of views, whereas the idea in [7,12] of coarsening by taking a subset of shape parameters appears based on heuristic consideration.…”

Section: Msqp: a Multigrid Shape Optimization Algorithmmentioning

confidence: 99%

Recent Results in Shape Optimization and Optimal Control for PDEs

Falcone

Verani

2014

Lecture Notes in Computational Science and Engineering

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In this paper we will present some recent advances in the numerical approximation of two classical problems: shape optimization and optimal control for evolutive partial differential equations. For shape optimization we present two novel techniques which have shown to be rather efficient on some applications. The first technique is based on multigrid methods whereas the second relies on an adaptive sequential quadratic programming. With respect to the optimal control of evolutive problems, the approximation is based on the coupling between a POD representation of the dynamical system and the classical Dynamic Programming approach. We look for an approximation of the value function characterized as the weak solution (in the viscosity sense) of the corresponding Hamilton-Jacobi equation. Several tests illustrate the main features of the above methods.

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On the convergence of the MG/OPT method

Abstract: Global convergence of the MG/OPT method for optimization is discussed.

Cited by 20 publications

References 6 publications

Training Schrödinger’s cat: quantum optimal control

Training Schrödinger’s cat: quantum optimal control

A Multilevel Stochastic Collocation Algorithm for Optimization of PDEs with Uncertain Coefficients

Recent Results in Shape Optimization and Optimal Control for PDEs

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