Generalized pointwise bias error bounds for response surface approximations

Goel, Tushar; Haftka, Raphael T.; Papila, Melih; Shyy, Wei

doi:10.1002/nme.1532

Cited by 5 publications

(7 citation statements)

References 37 publications

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“…There has recently been considerable attention given to RSM based approximation optimization in areas of mechanical and aerospace design optimization [2][3][4][5][6][7]. RSM is also an efficient approach that contributes to the probabilistic design such as robust and/or reliability-based design optimization [8,9].…”

Section: Introductionmentioning

confidence: 99%

A response surface based sequential approximate optimization using constraint-shifting analogy

Han

Lee

2009

J Mech Sci Technol

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When a traditional response surface method (RSM) is used as a meta-model for inequality constraint functions, an approximate optimal solution is sometimes actually infeasible in a case where it is active at the constraint boundary. The paper proposes a new RSM that ensures the constraint feasibility with respect to an approximate optimal solution. Constraint-shifting is suggested in order to secure the constraint feasibility during the sequential approximate optimization process. A central composite design is used as a tool for design of experiments. The proposed approach is verified through a mathematical function problem and engineering optimization problems to support the proposed strategies.

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Section: Introductionmentioning

confidence: 99%

A response surface based sequential approximate optimization using constraint-shifting analogy

Han

Lee

2009

J Mech Sci Technol

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“…Traditional variance-based designs minimize the effect of noise and attempt to obtain uniformity (stability) over design space but they do not address bias errors. Recently Goel et al [3] have demonstrated that prediction variance is not an appropriate tool to estimate errors when bias errors are dominant.…”

Section: Introduction and Literature Reviewmentioning

confidence: 99%

“…They assumed that the true model is a higher degree polynomial than the approximating polynomial and it satisfies the given data exactly. Goel et al [3] generalized this bias error bounds estimation method to account for inconsistencies between the assumed true model and actual data, and rank deficiencies in the matrix of equations used in linear regression. They demonstrated that the bounds can be used to develop adaptive design of experiment to reduce the effect of bias errors in region of interest.…”

Section: Introduction and Literature Reviewmentioning

confidence: 99%

Pointwise RMS bias error estimates for design of experiments

Goel

Haftka

Shyy

et al. 2006

44th AIAA Aerospace Sciences Meeting and Exhibit

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Response surface approximations offer an effective way to solve complex problems. However limitations on computational expense pose restrictions to generate ample data and a simple model is typically used for approximation leaving the possibility of errors due to insufficient model known as bias errors. This paper presents a method to estimate pointwise RMS bias errors in response surface models. Prior to generation of data, RMS bias error estimates can be used to construct design of experiments (DOEs) to minimize the maximal RMS bias errors or compare different DOEs. It is demonstrated that for high dimensional design spaces, central composite design gives a reasonable trade-off between variance and bias errors under conventional assumption of quadratic model and cubic true function. The results indicate that compared to bounds on bias error, RMS estimates are less affected by increase in the dimensionality of the problem. Latin Hypercube Sampling (LHS), though very popular, may yield large unsampled regions which can be successfully detected by implementing an alternate criterion like maximum standard error or maximum RMS bias error in design space. We further demonstrate that poor LHS designs can be eliminated by using more than one DOE. With the help of a non-polynomial example problem, it is demonstrated that the proposed method is correctly able to identify the regions of high errors even when assumed true model is not accurate.

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“…The encouraging results from the examples indicate that it may be worthwhile studying these strategies more rigorously and in more detail. In the past, the majority of the work related to the construction of EDs was done by considering only one design objective [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. When noise is the dominant source of error, there are a number of EDs that minimize the effect of variance (noise) on the resulting approximation, for example, the D-optimal design that minimizes the variance associated with the estimates of coefficients of the response surface model.…”

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confidence: 99%

Pitfalls of using a single criterion for selecting experimental designs

Goel

Haftka

Shyy

et al. 2007

Numerical Meth Engineering

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SUMMARYFor surrogate construction, a good experimental design (ED) is essential to simultaneously reduce the effect of noise and bias errors. However, most EDs cater to a single criterion and may lead to small gains in that criterion at the expense of large deteriorations in other criteria. We use multiple criteria to assess the performance of different popular EDs. We demonstrate that these EDs offer different trade-offs, and that use of a single criterion is indeed risky. In addition, we show that popular EDs, such as Latin hypercube sampling (LHS) and D-optimal designs, often leave large regions of the design space unsampled even for moderate dimensions. We discuss a few possible strategies to combine multiple criteria and illustrate them with examples. We show that complementary criteria (e.g. bias handling criterion for variancebased designs and vice versa) can be combined to improve the performance of EDs. We demonstrate improvements in the trade-offs between noise and bias error by combining a model-based criterion, like the D-optimality criterion, and a geometry-based criterion, like LHS. Next, we demonstrate that selecting an ED from three candidate EDs using a suitable error-based criterion helped eliminate potentially poor designs. Finally, we show benefits from combining the multiple criteria-based strategies, that is, generation of multiple EDs using the D-optimality and LHS criteria, and selecting one design using a pointwise bias error criterion. The encouraging results from the examples indicate that it may be worthwhile studying these strategies more rigorously and in more detail.

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Generalized pointwise bias error bounds for response surface approximations

Cited by 5 publications

References 37 publications

A response surface based sequential approximate optimization using constraint-shifting analogy

A response surface based sequential approximate optimization using constraint-shifting analogy

Pointwise RMS bias error estimates for design of experiments

Pitfalls of using a single criterion for selecting experimental designs

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