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A rigorous framework for optimization of expensive functions by surrogates
 Structural Optimization
, 1999
"... Abstract. The goal of the research reported here is to develop rigorous optimization algorithms to apply to some engineering design problems for which direct application of traditional optimization approaches is not practical. This paper presents and analyzes a framework for generating a sequence of ..."
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Cited by 164 (16 self)
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Abstract. The goal of the research reported here is to develop rigorous optimization algorithms to apply to some engineering design problems for which direct application of traditional optimization approaches is not practical. This paper presents and analyzes a framework for generating a sequence of approximations to the objective function and managing the use of these approximations as surrogates for optimization. The result is to obtain convergence to a minimizer of an expensive objective function subject to simple constraints. The approach is widely applicable because it does not require, or even explicitly approximate, derivatives of the objective. Numerical results are presented for a 31variable helicopter rotor blade design example and for a standard optimization test example. Key words. approximation concepts, surrogate optimization, response surfaces, pattern search methods, derivativefree optimization, design and analysis of computer experiments (DACE), computational engineering. Subject classication. Applied & Numerical Mathematics 1. Introduction. The
Computer Experiments
, 1996
"... Introduction Deterministic computer simulations of physical phenomena are becoming widely used in science and engineering. Computers are used to describe the flow of air over an airplane wing, combustion of gasses in a flame, behavior of a metal structure under stress, safety of a nuclear reactor, a ..."
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Cited by 84 (5 self)
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Introduction Deterministic computer simulations of physical phenomena are becoming widely used in science and engineering. Computers are used to describe the flow of air over an airplane wing, combustion of gasses in a flame, behavior of a metal structure under stress, safety of a nuclear reactor, and so on. Some of the most widely used computer models, and the ones that lead us to work in this area, arise in the design of the semiconductors used in the computers themselves. A process simulator starts with a data structure representing an unprocessed piece of silicon and simulates the steps such as oxidation, etching and ion injection that produce a semiconductor device such as a transistor. A device simulator takes a description of such a device and simulates the flow of current through it under varying conditions to determine properties of the device such as its switching speed and the critical voltage at which it switches. A circuit simulator takes a list of devices and the
Valuation of Mortgage Backed Securities Using Brownian Bridges to Reduce Effective Dimension
, 1997
"... The quasiMonte Carlo method for financial valuation and other integration problems has error bounds of size O((log N) k N \Gamma1 ), or even O((log N) k N \Gamma3=2 ), which suggests significantly better performance than the error size O(N \Gamma1=2 ) for standard Monte Carlo. But in hig ..."
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Cited by 74 (13 self)
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The quasiMonte Carlo method for financial valuation and other integration problems has error bounds of size O((log N) k N \Gamma1 ), or even O((log N) k N \Gamma3=2 ), which suggests significantly better performance than the error size O(N \Gamma1=2 ) for standard Monte Carlo. But in high dimensional problems this benefit might not appear at feasible sample sizes. Substantial improvements from quasiMonte Carlo integration have, however, been reported for problems such as the valuation of mortgagebacked securities, in dimensions as high as 360. We believe that this is due to a lower effective dimension of the integrand in those cases. This paper defines the effective dimension and shows in examples how the effective dimension may be reduced by using a Brownian bridge representation. 1 Introduction Simulation is often the only effective numerical method for the accurate valuation of securities whose value depends on the whole trajectory of interest Mathematics Departmen...
hypercube sampling and the propagation of uncertainty in analyses of complex systems, Reliability Engineering and System Safety 81
, 2003
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Using Approximations to Accelerate Engineering Design Optimization
 Proceedings of the 7th AIAA/USAF/NASA/ISSMO Multidisciplinary Analysis & Optimization Symposium (held at Saint Louis, Missouri), Paper 984800
, 1998
"... Optimization problems that arise in engineering design are often characterized by several features that hinder the use of standard nonlinear optimization techniques. Foremost among these features is that the functions used to dene the engineering optimization problem usually require the solution of ..."
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Cited by 40 (0 self)
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Optimization problems that arise in engineering design are often characterized by several features that hinder the use of standard nonlinear optimization techniques. Foremost among these features is that the functions used to dene the engineering optimization problem usually require the solution of dierential equations, a process which is itself computationally intensive. Within a standard nonlinear optimization algorithm, the solution of these dierential equations is required for each iteration of the algorithm. To mitigate such expense, an attractive alternative is to replace the computationally intensive objective with a less expensive surrogate. In conformance with engineering practice, we draw a crucial distinction between surrogate models and surrogate approximations. Surrogate models are auxiliary simulations that are less physically faithful, but also less computationally expensive, than the expensive simulation that is regarded as \truth. " An instructive example is the use of an equivalentplate analysis method in lieu of a nite element analysis, e.g. to analyze a wingbox of a highspeed civil transport. Surrogate models exist independently of the expensive simulation and can provide new information about the physical phenomenon of interest without requiring additional runs
Methods for Approximating Integrals in Statistics with Special Emphasis on Bayesian Integration Problems
 Statistical Science
"... This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain method ..."
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Cited by 36 (5 self)
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This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain methods. Each method is discussed giving an outline of the basic supporting theory and particular features of the technique. Conclusions are drawn concerning the relative merits of the methods based on the discussion and their application to three examples. The following broad recommendations are made. Asymptotic methods should only be considered in contexts where the integrand has a dominant peak with approximate ellipsoidal symmetry. Importance sampling, and preferably adaptive importance sampling, based on a multivariate Student should be used instead of asymptotics methods in such a context. Multiple quadrature, and in particular subregion adaptive integration, are the algorithms of choice for...
Filter Pattern Search Algorithms for Mixed Variable Constrained Optimization Problems
 SIAM Journal on Optimization
, 2004
"... A new class of algorithms for solving nonlinearly constrained mixed variable optimization problems is presented. This class combines and extends the AudetDennis Generalized Pattern Search (GPS) algorithms for bound constrained mixed variable optimization, and their GPSfilter algorithms for gene ..."
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Cited by 35 (7 self)
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A new class of algorithms for solving nonlinearly constrained mixed variable optimization problems is presented. This class combines and extends the AudetDennis Generalized Pattern Search (GPS) algorithms for bound constrained mixed variable optimization, and their GPSfilter algorithms for general nonlinear constraints. In generalizing existing algorithms, new theoretical convergence results are presented that reduce seamlessly to existing results for more specific classes of problems. While no local continuity or smoothness assumptions are required to apply the algorithm, a hierarchy of theoretical convergence results based on the Clarke calculus is given, in which local smoothness dictate what can be proved about certain limit points generated by the algorithm. To demonstrate the usefulness of the algorithm, the algorithm is applied to the design of a loadbearing thermal insulation system. We believe this is the first algorithm with provable convergence results to directly target this class of problems.
Monte Carlo variance of scrambled net quadrature
 SIAM J. Numer. Anal
, 1997
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Snobfit  Stable Noisy Optimization by Branch and Fit
"... this paper produces a userspeci ed number of suggested evaluation points in each step; proceeds by successive partitioning of the box (branch) and building local quadratic models ( t); combines local and global search and allows the user to determine which of both should be emphasized; h ..."
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Cited by 28 (5 self)
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this paper produces a userspeci ed number of suggested evaluation points in each step; proceeds by successive partitioning of the box (branch) and building local quadratic models ( t); combines local and global search and allows the user to determine which of both should be emphasized; handles local search from the best point with the aid of trust regions; allows for hidden constraints and assigns to such points a function value based on the function values of nearby feasible points
Sampling Strategies for Computer Experiments: Design and Analysis
, 2001
"... Computerbased simulation and analysis is used extensively in engineering for a variety of tasks. Despite the steady and continuing growth of computing power and speed, the computational cost of complex highfidelity engineering analyses and simulations limit their use in important areas like design ..."
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Cited by 23 (2 self)
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Computerbased simulation and analysis is used extensively in engineering for a variety of tasks. Despite the steady and continuing growth of computing power and speed, the computational cost of complex highfidelity engineering analyses and simulations limit their use in important areas like design optimization and reliability analysis. Statistical approximation techniques such as design of experiments and response surface methodology are becoming widely used in engineering to minimize the computational expense of running such computer analyses and circumvent many of these limitations. In this paper, we compare and contrast five experimental design types and four approximation model types in terms of their capability to generate accurate approximations for two engineering applications with typical engineering behaviors and a wide range of nonlinearity. The first example involves the analysis of a twomember frame that has three input variables and three responses of interest. The second example simulates the rollover potential of a semitractortrailer for different combinations of input variables and braking and steering levels. Detailed error analysis reveals that uniform designs provide good sampling for generating accurate approximations using different sample sizes while kriging models provide accurate approximations that are robust for use with a variety of experimental designs and sample sizes.