Convergence of sum-up rounding schemes for cloaking problems governed by the Helmholtz equation

Leyffer, Sven; Manns, Paul; Winckler, Malte

doi:10.1007/s10589-020-00262-3

Cited by 11 publications

(7 citation statements)

References 33 publications

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“…The SUR technique is proposed by the work of Sager et al [50] and is widely used in integer control optimization problems. To the best of our knowledge, most work using SUR rounds either a continuous-time control function [63,64] or controls of the continuous relaxation with the same time discretization [19,20,65,66]. In our problem, the time of solving the continuous relaxation is the major part of the overall computational time and significantly increases when the number of time steps T and scenarios S is high.…”

Section: Sum-up Rounding Techniquementioning

confidence: 99%

Binary Control Pulse Optimization for Quantum Systems

Fei

Brady

Larson

et al. 2023

Quantum

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Quantum control aims to manipulate quantum systems toward specific quantum states or desired operations. Designing highly accurate and effective control steps is vitally important to various quantum applications, including energy minimization and circuit compilation. In this paper we focus on discrete binary quantum control problems and apply different optimization algorithms and techniques to improve computational efficiency and solution quality. Specifically, we develop a generic model and extend it in several ways. We introduce a squared L2-penalty function to handle additional side constraints, to model requirements such as allowing at most one control to be active. We introduce a total variation (TV) regularizer to reduce the number of switches in the control. We modify the popular gradient ascent pulse engineering (GRAPE) algorithm, develop a new alternating direction method of multipliers (ADMM) algorithm to solve the continuous relaxation of the penalized model, and then apply rounding techniques to obtain binary control solutions. We propose a modified trust-region method to further improve the solutions. Our algorithms can obtain high-quality control results, as demonstrated by numerical studies on diverse quantum control examples.

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Section: Sum-up Rounding Techniquementioning

confidence: 99%

Binary Control Pulse Optimization for Quantum Systems

Fei

Brady

Larson

et al. 2023

Quantum

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“…The derivation of the IPMs follows [32] and, more specifically, [9]. We first observe that problems ( Ppen red ) and (Ppen) can be rewritten as (23) min…”

Section: The Interior Point Frameworkmentioning

confidence: 99%

“…Finally, other methods for MIPDECO problems such as Sum-up-Rounding strategies [22,23], derivative-free approaches [24], and sophisticated rounding techniques [25], might become too costly when adapted to tackle the large-scale problems considered in this article.…”

Section: Introductionmentioning

confidence: 99%

An improved penalty algorithm using model order reduction for MIPDECO problems with partial observations

et al. 2022

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This work addresses optimal control problems governed by a linear time-dependent partial differential equation (PDE) as well as integer constraints on the control. Moreover, partial observations are assumed in the objective function. The resulting problem poses several numerical challenges due to the mixture of combinatorial aspects, induced by integer variables, and large scale linear algebra issues, arising from the PDE discretization. Since classical solution approaches such as the branch-and-bound framework are typically overwhelmed by such large-scale problems, this work extends an improved penalty algorithm proposed by the authors, to the time-dependent setting. The main contribution is a novel combination of an interior point method, preconditioning, and model order reduction yielding a tailored local optimization solver at the heart of the overall solution procedure. A thorough numerical investigation is carried out both for the heat equation as well as a convection-diffusion problem demonstrating the versatility of the approach.

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“…A rich class of instances of (P) are mixed-integer PDE-constrained optimization problems, where J = j•S, where j is the objective of the optimization and S the control-to-state operator of an underlying partial differential equation (PDE). Such problems arise in many different areas such as topology optimization [14,10], optimum experimental design [28], and gas network optimization [7,9].…”

Section: Introductionmentioning

confidence: 99%

On Convergence of Binary Trust-Region Steepest Descent

Manns¹,

Kirches²,

Leyffer³

et al. 2022

Preprint

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Combinatorial integral approximation and binary trust-region steepest descent are two approaches for the solution of optimal control problems with discrete (binary in the latter case) control inputs. While the former method approximates solutions of a continuous relaxation with discrete controls, the latter produces a sequence of discrete and objective-decreasing controls by manipulating the level sets of the control inputs. The main difference in their analyses is an additional compactness assumption on the reduced objective functional for the combinatorial integral decomposition that implies an approximation of stationary points or (local) minimizers of the continuous relaxation and their objective function values. We prove that by imposing a suitable compactness assumption on the derivative of the reduced objective, the iterates produced by binary trust-region steepest descent have the same convergence properties. The methods achieve comparable results under comparable structural assumptions on the problem. We provide computational results to illustrate qualitative and quantitative similarities and differences of the iterates of both methods.

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Convergence of sum-up rounding schemes for cloaking problems governed by the Helmholtz equation

Cited by 11 publications

References 33 publications

Binary Control Pulse Optimization for Quantum Systems

Binary Control Pulse Optimization for Quantum Systems

An improved penalty algorithm using model order reduction for MIPDECO problems with partial observations

On Convergence of Binary Trust-Region Steepest Descent

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