Bayesian Target‐Vector Optimization for Efficient Parameter Reconstruction

Abstract

Parameter reconstructions are indispensable in metrology. Here, the objective is to explain K experimental measurements by fitting to them a parameterized model of the measurement process. The model parameters are regularly determined by least-square methods, that is, by minimizing the sum of the squared residuals between the K model predictions and the K experimental observations, 𝝌 2 . The model functions often involve computationally demanding numerical simulations. Bayesian optimization methods are specifically suited for minimizing expensive model functions. However, in contrast to least-square methods such as the Levenberg-Marquardt algorithm, they only take the value of 𝝌 2 into account, and neglect the K individual model outputs. A Bayesian target-vector optimization scheme with improved performance over previous developments, that considers all K contributions of the model function and that is specifically suited for parameter reconstruction problems which are often based on hundreds of observations is presented. Its performance is compared to established methods for an optical metrology reconstruction problem and two synthetic least-squares problems. The proposed method outperforms established optimization methods. It also enables to determine accurate uncertainty estimates with very few observations of the actual model function by using Markov chain Monte Carlo sampling on a trained surrogate model.

Impact study of numerical discretization accuracy on parameter reconstructions and model parameter distributions

“…This approach can be adopted to solve least squares parameter reconstruction problems. In [19,34] K independent GPs are trained on the K outputs of the vector valued forward model function f( p). At each iteration in the optimization an appropriate utility function then generates new sample candidates that can lead to an improvement over the currently best χ 2 value.…”

“…After they are trained they can serve as a cheap-toevaluate predictor for the actual model, and can therefore also be used as an approximation of them. We use the Bayesian target vector optimization (BTVO) scheme [19] for performing the parameter reconstructions. Afterwards we apply an MCMC sampler to infer the full model parameter distribution from the GP surrogates trained during the parameter reconstruction.…”

“…In section 3 we investigate the impact of a proper or improper choice of numerical parameters. We revisit a parameter reconstruction problem, in which a line grating is investigated in a Grazing Incidence x-ray Fluorescence (GIXRF) experiment [19,20]. We first perform a conventional convergence A schematic of the complete three-step reconstruction process.…”

“…We briefly review the theoretical fundamentals for parameter reconstructions using the BTVO method [19]. Moreover, we briefly discuss ways of controlling the level of accuracy in finite element method (FEM) simulations.…”

Elementary, my dear Zernike: model order reduction for accelerating optical dimensional microscopy

“…Due to the sophistication of the model, we require a method of solving the potentially multi-dimensional inverse problem with relatively few model evaluations. Recently, the Bayesian target vector optimization (BTVO) method has been shown to both efficiently solve the inverse problem and provide informa- * e-mail: phillip.manley@jcmwave.com tion on the uncertainties of the predicted dimensional parameters such as line widths and particle diameters [4].…”