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59
A Rigorous Framework for Optimization of Expensive Functions by Surrogates
, 1998
"... 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 approxima ..."
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Cited by 132 (17 self)
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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. # ...
A generalized discrepancy and quadrature error bound
 Math. Comp
, 1998
"... Abstract. An error bound for multidimensional quadrature is derived that includes the KoksmaHlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which dep ..."
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Cited by 91 (11 self)
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Abstract. An error bound for multidimensional quadrature is derived that includes the KoksmaHlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which depends only on the quadrature rule, is defined as a generalized discrepancy. The generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the L pstar discrepancy and Pα that arises in the study of lattice rules.
Latin Supercube Sampling for Very High Dimensional Simulations
, 1997
"... This paper introduces Latin supercube sampling (LSS) for very high dimensional simulations, such as arise in particle transport, finance and queuing. LSS is developed as a combination of two widely used methods: Latin hypercube sampling (LHS), and QuasiMonte Carlo (QMC). In LSS, the input variables ..."
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Cited by 69 (7 self)
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This paper introduces Latin supercube sampling (LSS) for very high dimensional simulations, such as arise in particle transport, finance and queuing. LSS is developed as a combination of two widely used methods: Latin hypercube sampling (LHS), and QuasiMonte Carlo (QMC). In LSS, the input variables are grouped into subsets, and a lower dimensional QMC method is used within each subset. The QMC points are presented in random order within subsets. QMC methods have been observed to lose effectiveness in high dimensional problems. This paper shows that LSS can extend the benefits of QMC to much higher dimensions, when one can make a good grouping of input variables. Some suggestions for grouping variables are given for the motivating examples. Even a poor grouping can still be expected to do as well as LHS. The paper also extends LHS and LSS to infinite dimensional problems. The paper includes a survey of QMC methods, randomized versions of them (RQMC) and previous methods for extending Q...
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 68 (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...
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 67 (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
Hypercube Sampling and the Propagation of Uncertainty in Analyses of Complex Systems
, 2002
"... ..."
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 37 (8 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
"... . Hybrids of equidistribution and Monte Carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter. This paper studies the variance of one such hybrid, scrambled nets, by applying a multidimensional multiresolution (w ..."
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Cited by 28 (1 self)
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. Hybrids of equidistribution and Monte Carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter. This paper studies the variance of one such hybrid, scrambled nets, by applying a multidimensional multiresolution (wavelet) analysis to the integrand. The integrand is assumed to be measurable and square integrable but not necessarily of bounded variation. In simple Monte Carlo, every nonconstant term of the multiresolution contributes to the variance of the estimated integral. For scrambled nets, certain lowdimensional and coarse terms do not contribute to the variance. For any integrand in L 2 , the sampling variance tends to zero faster under scrambled net quadrature than under Monte Carlo sampling, as the number of function evaluations n tends to infinity. Some finite n results bound the variance under scrambled net quadrature by a small constant multiple of the Monte Carlo variance, uniformly ove...
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 25 (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
Generating QuasiRandom Paths for Stochastic Processes
 SIAM Review
, 1998
"... The need to numerically simulate stochastic processes arises in many fields. Frequently this is done by discretizing the process into small time steps and applying pseudorandom sequences to simulate the randomness. This paper address the question of how to use quasiMonte Carlo methods to improve t ..."
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Cited by 23 (0 self)
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The need to numerically simulate stochastic processes arises in many fields. Frequently this is done by discretizing the process into small time steps and applying pseudorandom sequences to simulate the randomness. This paper address the question of how to use quasiMonte Carlo methods to improve this simulation. Special techniques must be applied to avoid the problem of high dimensionality which arises when a large number of time steps are required. Two such techniques, the generalized Brownian bridge and particle reordering, are described here. These methods are applied to a problem from finance, the valuation of a 30 year bond with monthly coupon payments assuming a mean reverting stochastic interest rate. When expressed as an integral, this problem is nominally 360 dimensional. The analysis of the integrand presented here explains the effectiveness of the quasirandom sequences on this high dimensional problem and suggests methods of variance reduction which can be used in conjunc...