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

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This paper builds on work by Haylock and O'Hagan which developed a Bayesian approach to uncertainty analysis. The generic problem is to make posterior inference about the output of a complex computer code, and the specific problem of uncertainty analysis is to make inference when the "true" values of the input parameters are unknown. Given the distribution of the input parameters (which is often a subjective distribution derived from expert opinion), we wish to make inference about the implied distribution of the output. The computer code is sufficiently complex that the time to compute the output for any input configuration is substantial. The Bayesian approach was shown to improve dramatically on the classical approach, which is based on drawing a sample of values of the input parameters and thereby obtaining a sample from the output distribution. We review the basic Bayesian approach to the generic problem of inference for complex computer codes, and present some recent advances---inference about the distribution of quantile functions of the uncertainty distribution, calibration of models, and the use of runs of the computer code at different levels of complexity to make efficient use of the quicker, cruder, versions of the code. The emphasis is on practical applications. Keywords: COMPUTATIONAL EXPERIMENT; SIMULATION; GAUSSIAN PROCESS; SENSITIVITY ANALYSIS; UNCERTAINTY DISTRIBUTION; CALIBRATION; MULTI-LEVEL CODES; MODEL INADEQUACY. 1. INTRODUCTION 1.1. Complex computer codes

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Abstract — The development of optimized motions of humanoid robots that guarantee a fast and also stable walking is an important task especially in the context of autonomous soccer playing robots in RoboCup. We present a walking motion optimization approach for the humanoid robot prototype HR18 which is equipped with a low dimensional parameterized walking trajectory generator, joint motor controller and an internal stabilization. The robot is included as hardware-in-the-loop to define a low dimensional black-box optimization problem. In contrast to previously performed walking optimization approaches we apply a sequential surrogate optimization approach using stochastic approximation of the underlying objective function and sequential quadratic programming to search for a fast and stable walking motion. This is done under the conditions that only a small number of physical walking experiments should have to be carried out during the online optimization process. For the identified walking motion for the considered 55 cm tall humanoid robot we measured a forward walking speed of more than 30 cm/sec. With a modified version of the robot even more than 40 cm/sec could be achieved in permanent operation.

this paper deals with the unconstrained global optimization problem, minimize f(x) where x = (x 1 ; : : : ; x k ). This includes the class of problems with simple constraints like a i x i b i , since these problems can be transformed to unconstrained global optimization problems. Throughout we assume without loss of generality that the extremum of interest is a minimum.

For design problems involving computation-intensive analysis or simulation processes, approximation models are usually introduced to reduce computation time. Most approximation-based optimization methods make step-by-step improvements to the approximation model by adjusting the limits of the design variables. In this work, a new approximation-based optimization method for computation-intensive design problems — the adaptive response surface method (ARSM), is presented. The ARSM creates quadratic approximation models for the computation-intensive design objective function in a gradually reduced design space. The ARSM was designed to avoid being trapped by local optimum and to identify the global design optimum with a modest number of objective function evaluations. Extensive tests on the ARSM as a global optimization scheme using benchmark problems, as well as an industrial design application of the method, are presented. Advantages and limitations of the approach are also discussed.

Consider a function f : B ! R, where B is a compact subset of R m , and consider a "simulation" used to estimate f(x); x 2 B with the following properties: The simulation can switch from one x 2 B to another in zero time, and a simulation at x lasting t units of time yields a random variable with mean f(x) and variance v(x)=t. With such a simulation we can divide T units of time into as many separate simulations as we like. Therefore, in principle we can design an "experiment" that spends ø (A) units of time simulating points in each A 2 B, where B is the Borel oe-field on B and ø is an arbitrary finite measure on (B; B). We call a design specified by a measure ø a "generalized design". We propose an approximation for f based on the data from a generalized design. When ø is discrete, the approximation, f , reduces to a "Kriging"- like estimator. We study discrete designs in detail, including asymptotics (as the length of the simulation increases) and a numerical procedure for findin...

Gaussian random fields as a very flexible class of models for computer experiments. This article considers a subclass of these models that are exactly once mean square differentiable. In particular, the likelihood function is determined in closed form, and under mild conditions the sieve maximum likelihood estimators for the parameters of the covariance function are shown to be weakly consistent with respect to fixed-domain asymptotics.

We consider prediction and uncertainty analysis for systems which are approximated using complex mathematical codes. Such models, implemented as computer codes, are often generic in the sense that by suitable choice of some of the model's input parameters the code can be used to predict behaviour of the system in a variety of specific applications. However, in any specific application the values of necessary parameters may be unknown. In this case, physical observations of the system in the specific context are used to learn about the unknown parameters. The process of fitting the model to the observed data by adjusting the parameters is known as calibration. Calibration is typically effected by ad hoc fitting, and after calibration the model is used, with the fitted input values, to predict future behaviour of the system. We present a Bayesian calibration technique which improves on this traditional approach in two respects. First, the predictions allow for remaining uncertainty over ...