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Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations

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Abstract

We present a technique for the rapid and reliable optimization of systems characterized by linear-functional outputs of coercive elliptic partial differential equations with affine (input) parameter dependence. The critical ingredients are: reduced-basis approximation to effect significant reduction in state-space dimensionality; a posteriori error bounds to provide rigorous error estimation and control; “offline/online” computational decompositions to permit rapid evaluation of output bounds, output bound gradients, and output bound Hessians in the limit of many queries; and reformulation of the approximate optimization statement to ensure (true) feasibility and control of suboptimality. To illustrate the method we consider the design of a three-dimensional thermal fin: Given volume and power objective-function weights, and root temperature “not-to-exceed” limits, the optimal geometry and heat transfer coefficient can be determined—with guaranteed feasibility—in only 2.3 seconds on a 500 MHz Pentium machine; note the latter includes only the online component of the calculations. Our method permits not only interactive optimal design at conception and manufacturing, but also real-time reliable adaptive optimal design in operation.

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Authors and Affiliations

  1. Massachusetts Institute of Technology, Dept. of Mechanical Engineering, Rm. 3-266, Cambridge, MA, 02139, USA

    A. T. Patera

  2. Operations Research R and D, 100 SAS Campus Dr, Room R5329, Cary, NC, 27513-8617, USA

    I. B. Oliveira

Authors
  1. I. B. Oliveira
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  2. A. T. Patera
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Correspondence to A. T. Patera.

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Oliveira, I.B., Patera, A.T. Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations. Optim Eng 8, 43–65 (2007). https://doi.org/10.1007/s11081-007-9002-6

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  • DOI: https://doi.org/10.1007/s11081-007-9002-6

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