A certified RB method for PDE-constrained parametric optimization problems

A. Manzoni, S. Pagani
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引用次数: 2

Abstract

Abstract We present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization problems (such as optimal control and optimal design) governed by elliptic PDEs and involving possibly non-convex cost functionals, assuming that the control functions are described in terms of a parameter vector. At each optimization step, the high-fidelity approximation of state and adjoint problems is replaced by a certified RB approximation, thus yielding a very efficient solution through an “optimize-then-reduce” approach. We develop a posteriori error estimates for the solutions of state and adjoint problems, the cost functional, its gradient and the optimal solution. We confirm our theoretical results in the case of optimal control/design problems dealing with potential and thermal flows.
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PDE约束参数优化问题的一种证明RB方法
摘要提出了一种有效求解pde约束参数优化问题的证明约简基框架。我们考虑由椭圆偏微分方程控制的优化问题(如最优控制和最优设计),并可能涉及非凸代价函数,假设控制函数用参数向量来描述。在每个优化步骤中,状态和伴随问题的高保真近似值被经过认证的RB近似值所取代,从而通过“先优化后减少”的方法产生非常有效的解决方案。我们对状态和伴随问题的解、代价泛函、代价泛函的梯度和最优解进行了后验误差估计。我们在处理势流和热流的最优控制/设计问题的情况下证实了我们的理论结果。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
3
审稿时长
16 weeks
期刊介绍: Communications in Applied and Industrial Mathematics (CAIM) is one of the official journals of the Italian Society for Applied and Industrial Mathematics (SIMAI). Providing immediate open access to original, unpublished high quality contributions, CAIM is devoted to timely report on ongoing original research work, new interdisciplinary subjects, and new developments. The journal focuses on the applications of mathematics to the solution of problems in industry, technology, environment, cultural heritage, and natural sciences, with a special emphasis on new and interesting mathematical ideas relevant to these fields of application . Encouraging novel cross-disciplinary approaches to mathematical research, CAIM aims to provide an ideal platform for scientists who cooperate in different fields including pure and applied mathematics, computer science, engineering, physics, chemistry, biology, medicine and to link scientist with professionals active in industry, research centres, academia or in the public sector. Coverage includes research articles describing new analytical or numerical methods, descriptions of modelling approaches, simulations for more accurate predictions or experimental observations of complex phenomena, verification/validation of numerical and experimental methods; invited or submitted reviews and perspectives concerning mathematical techniques in relation to applications, and and fields in which new problems have arisen for which mathematical models and techniques are not yet available.
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