Development and validation of a dynamic metamodel based on stochastic radial basis functions and uncertainty quantification

A dynamic radial basis function (DRBF) metamodel is derived and validated, based on stochastic RBF and uncertainty quantification (UQ). A metric for assessing metamodel efficiency is developed and used. The validation includes comparisons with a dynamic implementation of Kriging (DKG) and static metamodels for both deterministic test functions (with dimensionality ranging from two to six) and industrial UQ problems with analytical and numerical benchmarks, respectively. DRBF extends standard RBF using stochastic kernel functions defined by an uncertain tuning parameter whose distribution is arbitrary and whose effects on the prediction are determined using UQ methods. Auto-tuning based on curvature, adaptive sampling based on prediction uncertainty, parallel infill, and multiple response criteria are used. Industrial problems are two UQ applications in ship hydrodynamics using high-fidelity computational fluid dynamics for the high-speed Delft catamaran with stochastic operating and environmental conditions: (1) calm water resistance, sinkage and trim with variable Froude number; and (2) mean value and root mean square of resistance and heave and pitch motions with variable regular head wave. The number of high-fidelity evaluations required to achieve prescribed error levels is considered as the efficiency metric, focusing on fitting accuracy and UQ variables. DKG is found more efficient for fitting low-dimensional test functions and one-dimensional UQ, whereas DRBF has a greater efficiency for fitting higher-dimensional test functions and two-dimensional UQ.

[1]  G. Matheron.  Principles of geostatistics , 1963 .

[2]  D. Shepard.  A two-dimensional interpolation function for irregularly-spaced data , 1968, ACM National Conference.

[3]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .

[4]  M. Piccioni,et al.  Random tunneling by means of acceptance-rejection sampling for global optimization , 1989 .

[5]  G. Wahba.  Spline models for observational data , 1990 .

[6]  Stephen A. Billings,et al.  Radial basis function network configuration using genetic algorithms , 1995, Neural Networks.

[7]  T Haftka Raphael,et al.  Multidisciplinary aerospace design optimization: survey of recent developments , 1996 .

[8]  Jaroslaw Sobieszczanski-Sobieski,et al.  Multidisciplinary aerospace design optimization - Survey of recent developments , 1996 .

[9]  Donald R. Jones,et al.  Efficient Global Optimization of Expensive Black-Box Functions , 1998, J. Glob. Optim..

[10]  A. J. Booker,et al.  A rigorous framework for optimization of expensive functions by surrogates , 1998 .

[11]  Martin D. Buhmann,et al.  Radial Basis Functions , 2021, Encyclopedia of Mathematical Geosciences.

[12]  Xiaoping Du,et al.  The use of metamodeling techniques for optimization under uncertainty , 2001 .

[13]  Hans-Martin Gutmann,et al.  A Radial Basis Function Method for Global Optimization , 2001, J. Glob. Optim..

[14]  T. Simpson,et al.  Comparative studies of metamodelling techniques under multiple modelling criteria , 2001 .

[15]  Johan A. K. Suykens,et al.  Least Squares Support Vector Machines , 2002 .

[16]  Thomas J. Santner,et al.  The Design and Analysis of Computer Experiments , 2003, Springer Series in Statistics.

[17]  Tomas Jansson,et al.  Using surrogate models and response surfaces in structural optimization – with application to crashworthiness design and sheet metal forming , 2003 .

[18]  Haralambos Sarimveis,et al.  A new algorithm for developing dynamic radial basis function neural network models based on genetic algorithms , 2004, Comput. Chem. Eng..

[19]  Ren-Jye Yang,et al.  Metamodeling development for vehicle frontal impact simulation , 2001, DAC 2001.

[20]  T. Simpson,et al.  Use of Kriging Models to Approximate Deterministic Computer Models , 2005 .

[21]  M. Eldred,et al.  The promise and peril of uncertainty quantification using response surface approximations , 2005 .

[22]  Achille Messac,et al.  Extended Radial Basis Functions: More Flexible and Effective Metamodeling , 2004 .

[23]  Runze Li,et al.  Analysis of Computer Experiments Using Penalized Likelihood in Gaussian Kriging Models , 2005, Technometrics.

[24]  Zelda B. Zabinsky,et al.  A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems , 2005, J. Glob. Optim..

[25]  Marc C. Kennedy,et al.  Case studies in Gaussian process modelling of computer codes , 2006, Reliab. Eng. Syst. Saf..

[26]  Robert B. Gramacy,et al.  Ja n 20 08 Bayesian Treed Gaussian Process Models with an Application to Computer Modeling , 2009 .

[27]  Christine A. Shoemaker,et al.  A Stochastic Radial Basis Function Method for the Global Optimization of Expensive Functions , 2007, INFORMS J. Comput..

[28]  H. Bijl,et al.  A Probabilistic Radial Basis Function Approach for Uncertainty Quantification , 2007 .

[29]  J. Burkardt,et al.  LATINIZED, IMPROVED LHS, AND CVT POINT SETS IN HYPERCUBES , 2007 .

[30]  Jürgen Pilz,et al.  Why do we need and how should we implement Bayesian kriging methods , 2008 .

[31]  Shapour Azarm,et al.  An accumulative error based adaptive design of experiments for offline metamodeling , 2009 .

[32]  M. Rais-Rohani,et al.  Ensemble of metamodels with optimized weight factors , 2008 .

[33]  Martin D. Buhmann,et al.  Radial Basis Functions: Theory and Implementations: Preface , 2003 .

[34]  Daniele Peri,et al.  Self-Learning Metamodels for Optimization1 , 2009 .

[35]  S. Lucidi,et al.  New global optimization methods for ship design problems , 2009 .

[36]  Herbert K. H. Lee,et al.  Bayesian Guided Pattern Search for Robust Local Optimization , 2009, Technometrics.

[37]  Ke Meng,et al.  Self-adaptive radial basis function neural network for short-term electricity price forecasting , 2009 .

[38]  Andy J. Keane,et al.  Recent advances in surrogate-based optimization , 2009 .

[39]  Rommel G. Regis,et al.  Stochastic radial basis function algorithms for large-scale optimization involving expensive black-box objective and constraint functions , 2011, Comput. Oper. Res..

[40]  Kyung K. Choi,et al.  Metamodeling Method Using Dynamic Kriging for Design Optimization , 2011 .

[41]  Xiao Jian Zhou,et al.  Ensemble of surrogates with recursive arithmetic average , 2011 .

[42]  Daniele Peri,et al.  CFD-based multiobjective stochastic optimization of a waterjet propelled high speed ship , 2012 .

[43]  Jean Baccou,et al.  Construction and Efficient Implementation of Adaptive Objective-Based Designs of Experiments , 2013, Mathematical Geosciences.

[44]  Matteo Diez,et al.  URANS study of Delft catamaran total/added resistance, motions and slamming loads in head sea including irregular wave and uncertainty quantification for variable regular wave and geometry , 2013 .

[45]  Daniele Peri,et al.  Simulation based design optimization of waterjet propelled Delft catamaran , 2013 .

[46]  Wei He,et al.  FRAMEWORK FOR CONVERGENCE AND VALIDATION OF STOCHASTIC UNCERTAINTY QUANTIFICATION AND RELATIONSHIP TO DETERMINISTIC VERIFICATION AND VALIDATION , 2013 .

[47]  M. Diez,et al.  Uncertainty quantification of Delft catamaran resistance, sinkage and trim for variable Froude number and geometry using metamodels, quadrature and Karhunen–Loève expansion , 2014 .