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Generalized Nonlinear Least Squares Method for the Calibration of Complex Computer Code Using a Gaussian Process

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Summary
This summary is machine-generated.

The new generalized approximated nonlinear least squares (GALS) and max-minG algorithms improve parameter estimation in complex computer models by addressing correlated residuals and surrogate model uncertainty. These methods offer reduced bias and variance compared to traditional approaches.

Keywords:
Krigingbest linear unbiased predictorbig datacode tuningcombined datacomputer experimentsiteratively re-weighted least squaresnumerical optimization

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Area of Science:

  • Computational modeling
  • Statistical inference
  • Scientific computing

Background:

  • Approximated nonlinear least squares (ALS) is used for parameter estimation in complex, time-consuming computer codes.
  • ALS calibrates codes by minimizing differences between observations and model outputs using surrogates like Gaussian processes.
  • Correlated or heteroscedastic residuals and surrogate uncertainty can distort ALS tuning and increase estimation variance.

Purpose of the Study:

  • To propose a generalized approximated nonlinear least squares (GALS) method to address limitations of the standard ALS.
  • To introduce an iterative version of GALS, the max-minG algorithm, for enhanced parameter estimation.
  • To compare the performance of GALS and max-minG against ALS and iteratively re-weighted ALS (IRWALS).

Main Methods:

  • GALS constructs a covariance matrix for residuals, incorporating its inverse into the minimization objective.
  • The max-minG algorithm iteratively re-estimates parameters using maximum likelihood estimation and GALS.
  • Four methods (ALS, IRWALS, GALS, max-minG) were compared using five test functions and a nuclear fusion simulator.

Main Results:

  • Both GALS and max-minG demonstrated smaller bias and variance in parameter estimates compared to ALS and IRWALS.
  • The max-minG algorithm showed superior performance over GALS and other methods for complex test functions.
  • Proposed methods resolved abnormal residual patterns observed with the standard ALS method in a nuclear fusion simulation.

Conclusions:

  • The proposed GALS and max-minG algorithms offer significant improvements in parameter estimation accuracy and stability for complex computer models.
  • Max-minG is particularly effective for computationally intensive and complex simulation scenarios.
  • These advanced methods provide robust solutions for challenges posed by correlated residuals and surrogate model uncertainties in scientific computing.