## Surrogate-Assisted Evolutionary Optimization Frameworks for High-Fidelity Engineering Design Problems (2004)

Venue: | In Knowledge Incorporation in Evolutionary Computation |

Citations: | 18 - 5 self |

### BibTeX

@INPROCEEDINGS{Ong04surrogate-assistedevolutionary,

author = {Y. S. Ong and P. B. Nair and A. J. Keane and K. W. Wong},

title = {Surrogate-Assisted Evolutionary Optimization Frameworks for High-Fidelity Engineering Design Problems},

booktitle = {In Knowledge Incorporation in Evolutionary Computation},

year = {2004},

pages = {307--332},

publisher = {Springer Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Over the last decade, Evolutionary Algorithms (EAs) have emerged as a powerful paradigm for global optimization of multimodal functions. More recently, there has been significant interest in applying EAs to engineering design problems. However, in many complex engineering design problems where high-fidelity analysis models are used, each function evaluation may require a Computational Structural Mechanics (CSM), Computational Fluid Dynamics (CFD) or Computational Electro-Magnetics (CEM) simulation costing minutes to hours of supercomputer time. Since EAs typically require thousands of function evaluations to locate a near optimal solution, the use of EAs often becomes computationally prohibitive for this class of problems. In this paper, we present frameworks that employ surrogate models for solving computationally expensive optimization problems on a limited computational budget. In particular, the key factors responsible for the success of these frameworks are discussed. Experimental results obtained on benchmark test functions and real-world complex design problems are presented.

### Citations

9805 | The Nature of Statistical Learning Theory
- Vapnik
- 1995
(Show Context)
Citation Context ...he form ˆy = ˆ f(x), (3) such that y = ˆy + ɛ, where ɛ represents the approximation error. There exist a variety of techniques for constructing surrogate models; see, for example, the texts by Va=-=pnik [38]-=- and Bishop [39] for excellent expositionssSurrogate-Assisted Evolutionary Search 7 of this area. One popular approach in the design optimization literature is least-squares regression using low-order... |

5289 |
Neural Networks for Pattern Recognition
- Bishop
- 1995
(Show Context)
Citation Context ...(x), (3) such that y = ˆy + ɛ, where ɛ represents the approximation error. There exist a variety of techniques for constructing surrogate models; see, for example, the texts by Vapnik [38] and Bish=-=op [39]-=- for excellent expositionssSurrogate-Assisted Evolutionary Search 7 of this area. One popular approach in the design optimization literature is least-squares regression using low-order polynomials, al... |

2180 |
Genetic algorithms + data structures = evolution programs
- Michalewicz
- 1996
(Show Context)
Citation Context ...hastic optimizers which have attracted much attention in recent years include simulated annealing; tabu search; genetic algorithms; evolutionary programming and evolution strategies [15], [16], [17], =-=[18]-=-. These stochastic methods have been successfully applied to mechanical and aerodynamic problems, including turbine blade design [19], multi-disciplinary rotor blade design [20], multi-level aircraft ... |

1706 |
The Grid: Blueprint for a New Computing Infrastructure
- Foster, Kesselman
- 1998
(Show Context)
Citation Context ...ple compute nodes. It would be important for any surrogate-assisted evolutionary framework to retain or further extend the intrinsic parallelism of traditional evolutionary algorithms. Grid Computing =-=[58]-=- has recently been perceived as the enabling technology for collaborative design and the embarrassing parallelism in the evolutionary search [59]. The benefits of Grid computing in the context of evol... |

1292 | Handbook of genetic algorithms
- Davis
- 1991
(Show Context)
Citation Context ...n stochastic optimizers which have attracted much attention in recent years include simulated annealing; tabu search; genetic algorithms; evolutionary programming and evolution strategies [15], [16], =-=[17]-=-, [18]. These stochastic methods have been successfully applied to mechanical and aerodynamic problems, including turbine blade design [19], multi-disciplinary rotor blade design [20], multi-level air... |

501 |
Evolution and Optimum Seeking
- Schwefel
- 1995
(Show Context)
Citation Context ...entional numerical methods commonly used in engineering design include steepest-descent methods, conjugategradient, quadratic programming, pattern search methods and linear approximation methods [4], =-=[5]-=-, [6].s4 Y. S. Ong, P. B. Nair, A. J. Keane and K. W. Wong Gradient-based optimization algorithms make use of line searches to locate a new iterate and hence the issue of range of validity of the appr... |

393 |
Design and Analysis of Computer Experiments
- Sacks, Welch, et al.
(Show Context)
Citation Context ...surrogate models of deterministic computer models is Bayesian interpolation, which is sometimes referred to as design and analysis of computer experiments (DACE) modeling in the statistics literature =-=[40]-=-, Gaussian process regression in the neural networks literature [41] and Kriging in the geostatistics literature. Artificial neural networks, including Multi-layer Perceptions, Radial Basis Functions ... |

255 |
Multivariate Adaptive Regression Splines” (with discussion
- Friedman
- 1991
(Show Context)
Citation Context ...s literature [41] and Kriging in the geostatistics literature. Artificial neural networks, including Multi-layer Perceptions, Radial Basis Functions (RBF) Networks and multivariate regression splines =-=[42]-=- have also been employed for constructing surrogate models in engineering design optimization. A comprehensive review of different approximation concepts is provided in [43] and a comparison of variou... |

230 | Gaussian processes for regression
- Williams, Rasmussen
- 1996
(Show Context)
Citation Context ...olation, which is sometimes referred to as design and analysis of computer experiments (DACE) modeling in the statistics literature [40], Gaussian process regression in the neural networks literature =-=[41]-=- and Kriging in the geostatistics literature. Artificial neural networks, including Multi-layer Perceptions, Radial Basis Functions (RBF) Networks and multivariate regression splines [42] have also be... |

187 | On evolution, search, optimization, genetic algorithms and martial arts: Toward memetic algorithms
- MOSCATO
- 1989
(Show Context)
Citation Context ... solve a wide variety of engineering design problems and experimental studies show that they not only often find better solutions than simple GAs, but also that they may search more efficiently [31], =-=[32]-=-, [33], [34], [35]. In diverse contexts, hybrid EA-LSs are also known as Memetic Algorithms. There are two basic strategies for using Memetic Algorithms: Lamarckian learning forces the genotype to ref... |

174 | A cooperative coevolutionary approach to function optimization
- Potter, Jong
- 1994
(Show Context)
Citation Context ...gate-Assisted Coevolutionary Search In recent years, coevolutionary computation has been applied with a great degree of success to function optimization, neural network training, and concept learning =-=[46]-=-, [47]. Its success lies in the ability to apply divide-and-conquer strategies. For example, in the context of optimization, the variables in the original problem are decomposed into a number of subse... |

159 | A Rigorous Framework for Optimization of Expensive Functions by Surrogates
- Booker, Dennis, et al.
- 1999
(Show Context)
Citation Context ...se of approximation models in non-gradient based numerical optimization methods algorithms such as pattern search algorithms have also been proposed in the literature; see, for example, Booker et al. =-=[8]-=- and Serafini [9]. One important reason these frameworks have been widely accepted and used is attributed to the theoretical guarantee of convergence to a local optima of the exact problem. Surrogate-... |

117 | The approximation power of moving least-squares
- Levin
- 1998
(Show Context)
Citation Context ..., [2], [37]. This idea of constructing local models is similar in spirit to the multipoint approximation technique proposed by Toropov et al. [51] and the moving least-squares approximation technique =-=[52]-=-. RBF Local Surrogate-Assisted Hybrid Genetic Algorithm The essential backbone of the framework proposed in [1] is a parallel evolutionary algorithm coupled with a feasible sequential quadratic progra... |

96 | The Design and Analysis of a Computational Model of Cooperative Coevolution
- Potter
- 1997
(Show Context)
Citation Context ...andles only a subset of the original design variables. However, while this divide-and-conquer approach enables us to tackle the curse of dimensionality, a well-known property of coevolutionary search =-=[49]-=- is that high epistatic interactions between the variables can lead to a significant degradation of the convergence rate. In the GA literature, epistasis refers to the variable interdependencies or li... |

70 | Comparative Studies of Metamodeling Techniques under Multiple Modeling Criteria
- Jin, Chen, et al.
- 2001
(Show Context)
Citation Context ... surrogate models in engineering design optimization. A comprehensive review of different approximation concepts is provided in [43] and a comparison of various techniques can be found in [14], [44], =-=[45]-=-. Here, we will be primarily concerned with approximating deterministic computer models that we assume do not suffer from numerically induced convergence or discretization noise, and hence perfectly i... |

52 |
Evolutionary Optimization of Computationally Expensive Problems Via Surrogate Modelling
- 9Ong, Nair, et al.
(Show Context)
Citation Context ...reasingly optimal with respect to prespecified design criteria. In recent years, Evolutionary Algorithms (EAs) have been applied with a great degree of success to complex design optimization problems =-=[1]-=-, [2], [3]. Their popularity lies in their ease of implementation and the ability to locate close the globally optimum designs. However, for many real-life design problems, thousands of calls to the a... |

46 |
Optimal Engineering Design: Principles and Applications
- Siddall
- 1982
(Show Context)
Citation Context ... Conventional numerical methods commonly used in engineering design include steepest-descent methods, conjugategradient, quadratic programming, pattern search methods and linear approximation methods =-=[4]-=-, [5], [6].s4 Y. S. Ong, P. B. Nair, A. J. Keane and K. W. Wong Gradient-based optimization algorithms make use of line searches to locate a new iterate and hence the issue of range of validity of the... |

39 | Evolving neural networks with collaborative species
- Potter, Jong
- 1995
(Show Context)
Citation Context ...ssisted Coevolutionary Search In recent years, coevolutionary computation has been applied with a great degree of success to function optimization, neural network training, and concept learning [46], =-=[47]-=-. Its success lies in the ability to apply divide-and-conquer strategies. For example, in the context of optimization, the variables in the original problem are decomposed into a number of subsets. Su... |

32 |
Design of optimal aerodynamic shapes using stochastic optimization methods and computational intelligence
- Giannakoglou
- 2002
(Show Context)
Citation Context ...ailable and used in the engineering optimization community. Optimization algorithms in the literature can be broadly classified into three categories: (1) conventional numerical optimization methods, =-=(2)-=- stochastic optimization methods and (3) hybrid methods. In this section, we present a brief overview of surrogateassisted optimization strategies. In particular, we consider a general nonlinear progr... |

31 |
Global convergence of a class of trust region methods for nonconvex minimization in Hilbert space
- Toint
- 1998
(Show Context)
Citation Context ...tionally prohibitive for many complex design problems. Convergence analysis of trust-region algorithms when only inexact gradient information is available has been considered by Carter [54] and Toint =-=[55]-=-. Leveraging these results, Arian et al. [56] presented a theoretical analysis for unconstrained optimization using surrogates to show that under mild assumptions, convergence can still be guaranteed.... |

29 |
On the global convergence of trust region algorithms using inexact gradient information
- Carter
- 1991
(Show Context)
Citation Context ...ould be computationally prohibitive for many complex design problems. Convergence analysis of trust-region algorithms when only inexact gradient information is available has been considered by Carter =-=[54]-=- and Toint [55]. Leveraging these results, Arian et al. [56] presented a theoretical analysis for unconstrained optimization using surrogates to show that under mild assumptions, convergence can still... |

29 | Trust-region proper orthogonal decomposition for flow control
- Arian, Fahl, et al.
- 2000
(Show Context)
Citation Context ...problems. Convergence analysis of trust-region algorithms when only inexact gradient information is available has been considered by Carter [54] and Toint [55]. Leveraging these results, Arian et al. =-=[56]-=- presented a theoretical analysis for unconstrained optimization using surrogates to show that under mild assumptions, convergence can still be guaranteed. In particular, the condition the surrogate m... |

24 | Transductive inference for estimating values of functions
- Chapelle, Vapnik, et al.
- 1999
(Show Context)
Citation Context ...ional Space Structure. local learning technique may be regarded as an instance of the transductive inference paradigm, which has been the focus of recent research in statistical learning theory [38], =-=[50]-=-. Traditionally, surrogate models are constructed using inductive inference, which involves using a training dataset to estimate a functional dependency and then using the computed model to predict th... |

15 | A Framework for Managing Models in Nonlinear Optimization of Computationally Expensive Functions
- Serafini
- 1999
(Show Context)
Citation Context ...on models in non-gradient based numerical optimization methods algorithms such as pattern search algorithms have also been proposed in the literature; see, for example, Booker et al. [8] and Serafini =-=[9]-=-. One important reason these frameworks have been widely accepted and used is attributed to the theoretical guarantee of convergence to a local optima of the exact problem. Surrogate-assisted conventi... |

14 |
Meta-lamarckian in memetic algorithm
- Ong, Keane
- 2004
(Show Context)
Citation Context ...ety of engineering design problems and experimental studies show that they not only often find better solutions than simple GAs, but also that they may search more efficiently [31], [32], [33], [34], =-=[35]-=-. In diverse contexts, hybrid EA-LSs are also known as Memetic Algorithms. There are two basic strategies for using Memetic Algorithms: Lamarckian learning forces the genotype to reflect the result of... |

13 |
Gelatt C, Vecchi M
- Kirkpatrick
- 1983
(Show Context)
Citation Context ...paces. Modern stochastic optimizers which have attracted much attention in recent years include simulated annealing; tabu search; genetic algorithms; evolutionary programming and evolution strategies =-=[15]-=-, [16], [17], [18]. These stochastic methods have been successfully applied to mechanical and aerodynamic problems, including turbine blade design [19], multi-disciplinary rotor blade design [20], mul... |

10 |
Genetic Algorithms in Multidisciplinary Rotor Blade Design," Paper AIAA-95-1144-CP
- Hajela, Lee
- 1995
(Show Context)
Citation Context ...gies [15], [16], [17], [18]. These stochastic methods have been successfully applied to mechanical and aerodynamic problems, including turbine blade design [19], multi-disciplinary rotor blade design =-=[20]-=-, multi-level aircraft wing design [3], military airframe preliminary design [21] and large flexible space structures design [22]. However, a well-known drawback of EAs in complex engineering design o... |

8 |
Aircraft Wing Design Using GA-Based Multi-Level Strategies
- 32Keane, Petruzzelli
- 2000
(Show Context)
Citation Context ...optimal with respect to prespecified design criteria. In recent years, Evolutionary Algorithms (EAs) have been applied with a great degree of success to complex design optimization problems [1], [2], =-=[3]-=-. Their popularity lies in their ease of implementation and the ability to locate close the globally optimum designs. However, for many real-life design problems, thousands of calls to the analysis co... |

8 | Passive Vibration Suppression of Flexible Space Structures via Optimal Geometric Redesign
- 6Nair, Keane
(Show Context)
Citation Context ...uding turbine blade design [19], multi-disciplinary rotor blade design [20], multi-level aircraft wing design [3], military airframe preliminary design [21] and large flexible space structures design =-=[22]-=-. However, a well-known drawback of EAs in complex engineering design optimization is the need for a large number of calls to the computationally expensive analysis solver in order to locate a near op... |

7 |
Combining Genetic and Deterministic Algorithms for Locating Actuators on Space Structures
- Furuya, Haftka
- 1996
(Show Context)
Citation Context ...l numerical optimization algorithms. Hence earlier research efforts related to evolutionary optimization have focused on the use of problem specific knowledge to increase the computational efficiency =-=[23]-=-,[24]. Even though such problem specific heuristics can be effectively used to achieve performance improvements, there are finite limits to the improvements achievable by such techniques. Robinson and... |

5 |
Evolutionary search of approximated N -dimensional landscapes
- Liang, Yao, et al.
- 2000
(Show Context)
Citation Context ...landscape, but the solution that is found is not encoded back into the genetic string. A strategy for coupling ES with local search and quadratic response surface methods was proposed in Liang et al. =-=[36]-=-. However the use of the exact analysis codes to perform local searches results in significantly high computational costs. Further, when working with multimodal high dimensional problems the accuracy ... |

4 |
Shape Optimisation Of Turbine Blade Firtrees
- Song
- 2002
(Show Context)
Citation Context ...d surrogate models. This approach uses both the expensive and approximate models throughout the search, with an empirical criterion to decide the frequency at which each model should be used. In Song =-=[30]-=-, a real-coded GA was coupled with Kriging in firtree structural optimization. 2.3 Hybrid Evolutionary Optimization Evolutionary algorithms are capable of exploring and exploiting promising regions of... |

3 | Design of an Aircraft Brake Component Using an Interactive Multidisciplinary Design Optimization Framework
- Stelmack, Batill, et al.
- 1999
(Show Context)
Citation Context ...f the exact problem. Surrogate-assisted conventional numerical optimization methods have been applied with much success to complex engineering design optimization problems, see for example, [8], [9], =-=[10]-=-, [11], [12], [13]. A more detailed survey of the state-of-the-art can be found in Simpson et al. [14] 2.2 Evolutionary Optimization Conventional numerical optimization methods have the known advantag... |

3 |
A case for multi-level optimisation in aeronautical design
- Robinson, Keane
- 1999
(Show Context)
Citation Context ...ce improvements, there are finite limits to the improvements achievable by such techniques. Robinson and Keane demonstrated the use of variable-fidelity analysis models in EAs for aeronautical design =-=[25]-=-. A computational framework for integrating a class of single-point approximation models with GAs was also proposed in [26]. However, these frameworks are restricted to a special class of approximatio... |

3 |
Zilinskas
- unknown authors
- 1992
(Show Context)
Citation Context ...ace. They can, however, take a relatively long time to locate the exact local optimum in a region of convergence (and may sometimes not find the optimum with sufficient precision). Torn and Zilinskas =-=[31]-=- observe that two competing goals govern the design of global search methods: exploration is important to ensure global reliability; i.e., every part of the domain is searched enough to provide a reli... |

3 | Utilizing Lamarckian Evolution and the Baldwin Effect in Hybrid Genetic Algorithms
- Houck, Joines, et al.
- 1996
(Show Context)
Citation Context ... a wide variety of engineering design problems and experimental studies show that they not only often find better solutions than simple GAs, but also that they may search more efficiently [31], [32], =-=[33]-=-, [34], [35]. In diverse contexts, hybrid EA-LSs are also known as Memetic Algorithms. There are two basic strategies for using Memetic Algorithms: Lamarckian learning forces the genotype to reflect t... |

3 | Gridifying aerodynamic design problem using GridRPC
- Ho, Ong, et al.
- 2003
(Show Context)
Citation Context ...of traditional evolutionary algorithms. Grid Computing [58] has recently been perceived as the enabling technology for collaborative design and the embarrassing parallelism in the evolutionary search =-=[59]-=-. The benefits of Grid computing in the context of evolutionary design optimization are expected to be numerous. Besides the ability to tap into vast compute power, it provide access to almost limitle... |

2 |
A (2002) Multi-Point Cubic Surrogate Functions for Sequential Approximate Optimization
- Canfield
(Show Context)
Citation Context ...problem. Surrogate-assisted conventional numerical optimization methods have been applied with much success to complex engineering design optimization problems, see for example, [8], [9], [10], [11], =-=[12]-=-, [13]. A more detailed survey of the state-of-the-art can be found in Simpson et al. [14] 2.2 Evolutionary Optimization Conventional numerical optimization methods have the known advantage of their e... |

2 |
Sendho B (2002) A Framework for Evolutionary Optimization with Approximate Fitness Functions
- Jin, Olhofer
(Show Context)
Citation Context ...The same problem was revisited by El-Beltagy et al. [28], where it is argued that the issue of balancing the concerns of optimization with those of design of experiments must be addressed. Jin et al. =-=[29]-=- presented a framework for coupling ES and neural network-based surrogate models. This approach uses both the expensive and approximate models throughout the search, with an empirical criterion to dec... |

1 |
Lewis R M, Torczon V
- Alexandrov, Dennis
- 1998
(Show Context)
Citation Context ...ue of range of validity of the approximation models or the control of approximation errors is directly addressed by using ad hoc move limits or a trust region framework. As shown by Alexandrov et al. =-=[7]-=-, the trust-region strategy for adaptively controlling the move limits guarantees convergence under some mild assumptions on the accuracy of the surrogate model. Other general surrogate-assisted frame... |

1 |
Grandhi R V (2000) Ultipoint Approximation Development: Thermal Structural Optimization Case Study
- Xu
(Show Context)
Citation Context ...exact problem. Surrogate-assisted conventional numerical optimization methods have been applied with much success to complex engineering design optimization problems, see for example, [8], [9], [10], =-=[11]-=-, [12], [13]. A more detailed survey of the state-of-the-art can be found in Simpson et al. [14] 2.2 Evolutionary Optimization Conventional numerical optimization methods have the known advantage of t... |

1 |
Redhe M (2003) Response Surface Methods and Pareto Optimization in Crashworthiness Design
- Andersson
(Show Context)
Citation Context ...m. Surrogate-assisted conventional numerical optimization methods have been applied with much success to complex engineering design optimization problems, see for example, [8], [9], [10], [11], [12], =-=[13]-=-. A more detailed survey of the state-of-the-art can be found in Simpson et al. [14] 2.2 Evolutionary Optimization Conventional numerical optimization methods have the known advantage of their efficie... |

1 |
A J, Ghosh D, Giunta A A, Koch P N, Yang R J (2002) Approximation Methods in Multidisciplinary Analysis and Optimization: A Panel Discussion
- Simpson, Booker
(Show Context)
Citation Context ...with much success to complex engineering design optimization problems, see for example, [8], [9], [10], [11], [12], [13]. A more detailed survey of the state-of-the-art can be found in Simpson et al. =-=[14]-=- 2.2 Evolutionary Optimization Conventional numerical optimization methods have the known advantage of their efficiency, however, they are very sensitive to the starting point selection and are very l... |

1 |
Sendho B (2000) Optimisation of a stator blade used in a transonic compressor cascade with evolution strategies
- Olhofer, Arima, et al.
(Show Context)
Citation Context ...volutionary programming and evolution strategies [15], [16], [17], [18]. These stochastic methods have been successfully applied to mechanical and aerodynamic problems, including turbine blade design =-=[19]-=-, multi-disciplinary rotor blade design [20], multi-level aircraft wing design [3], military airframe preliminary design [21] and large flexible space structures design [22]. However, a well-known dra... |

1 |
A H, Bonham Ch R(2000) Multi objective satisfaction within an interactive evolutionary design environment
- Parmee, Cvetkovi6, et al.
(Show Context)
Citation Context ...ied to mechanical and aerodynamic problems, including turbine blade design [19], multi-disciplinary rotor blade design [20], multi-level aircraft wing design [3], military airframe preliminary design =-=[21]-=- and large flexible space structures design [22]. However, a well-known drawback of EAs in complex engineering design optimization is the need for a large number of calls to the computationally expens... |

1 |
Haftka R T, Gurdal Z, Watson L T
- Nagendra
- 1996
(Show Context)
Citation Context ...erical optimization algorithms. Hence earlier research efforts related to evolutionary optimization have focused on the use of problem specific knowledge to increase the computational efficiency [23],=-=[24]-=-. Even though such problem specific heuristics can be effectively used to achieve performance improvements, there are finite limits to the improvements achievable by such techniques. Robinson and Kean... |

1 |
A J, Shimpi R P
- Nair, Keane
- 1998
(Show Context)
Citation Context ... the use of variable-fidelity analysis models in EAs for aeronautical design [25]. A computational framework for integrating a class of single-point approximation models with GAs was also proposed in =-=[26]-=-. However, these frameworks are restricted to a special class of approximation models that are domain specific. For more general surrogate-assisted evolutionary frameworks, several efforts have been m... |

1 |
Quagliarella D
- Vicini
- 1999
(Show Context)
Citation Context ...e variety of engineering design problems and experimental studies show that they not only often find better solutions than simple GAs, but also that they may search more efficiently [31], [32], [33], =-=[34]-=-, [35]. In diverse contexts, hybrid EA-LSs are also known as Memetic Algorithms. There are two basic strategies for using Memetic Algorithms: Lamarckian learning forces the genotype to reflect the res... |

1 |
D M, Zhang Z K (2003) Global Convergence Unconstrained And Bound Constrained Surrogate-Assisted Evolutionary Search
- Ong, Lum, et al.
(Show Context)
Citation Context ... parallel hybrid EA framework that leverages surrogate models for solving computationally expensive design problems with general constraints was proposed by the authors in [1] and further extended in =-=[37]-=- to incorporate gradient information. 3 Surrogate Modeling Surrogate models or metamodels are (often statistical) models that are built to approximate computationally expensive simulation codes. Surro... |

1 |
Haftka R T
- Barthelemy
- 1993
(Show Context)
Citation Context ...variate regression splines [42] have also been employed for constructing surrogate models in engineering design optimization. A comprehensive review of different approximation concepts is provided in =-=[43]-=- and a comparison of various techniques can be found in [14], [44], [45]. Here, we will be primarily concerned with approximating deterministic computer models that we assume do not suffer from numeri... |