We present a statistical model of empirical optimization that admits the creation of algorithms with explicit and intuitively defined desiderata. Because No Free Lunch theoremsdictate that no optimizationalgorithmcan beconsideredmoreefficientthan any other when considering all possible functions, the desired function class plays a prominent role in the model. In particular, this provides a direct way to answer the traditionallydifficultquestionofwhatalgorithmisbest matchedtoaparticularclassof functions. Among the benefits ofthe modelarethe abilityto specifythe function class inastraightforwardmanner, anatural waytospecifynoisyordynamicfunctions,and anew source of insight into No Free Lunch theorems for optimization.

The computer graphics and animation fields are filled with applications that require the setting of tricky parameters. In many cases, the models are complex and the parameters unintuitive for non-experts. In this paper, we present an optimization method for setting parameters of a procedural fluid animation system by showing the user examples of different parametrized animations and asking for feedback. Our method employs the Bayesian technique of bringing in “prior ” belief based on previous runs of the system and/or expert knowledge, to assist users in finding good parameter settings in as few steps as possible. To do this, we introduce novel extensions to Bayesian optimization, which permit effective learning for parameter-based procedural animation applications. We show that even when users are trying to find a variety of different target animations, the system can learn and improve. We demonstrate the effectiveness of our method compared to related active learning methods. We also present a working application for assisting animators in the challenging task of designing curl-based velocity fields, even with minimal domain knowledge other than identifying when a simulation “looks right”.

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In this paper, we compare the global accuracy of different strategies to build response surfaces by varying sampling methods and modeling techniques. The final application of the response surfaces being aerodynamic shape optimization, the test cases are issued from CFD simulations of aerodynamic coefficients. For comparisons, a robust strategy for model fit using a new efficient initialization technique followed by a gradient optimization was applied. Firstly, a study of different sampling methods proves that including ’a posteriori ’ information on the function to sample distribution can improve accuracy over classical space filling methods like Latin Hypercube Sampling. Secondly, comparing Kriging and gradient enhanced Kriging on two to six dimensional test cases shows that interpolating gradient vectors drastically improves response surface accuracy. Although direct and indirect Cokriging have equivalent formulations, the indirect Cokriging outperforms the direct approach. The slow linear phase of error convergence when increasing sample size is not avoided by Cokriging. Thus, the number of samples needed to have a globally accurate surface stays generally out of reach for problems considering more than four design variables.

In this paper, we compare the global accuracy of different strategies to build response surfaces by varying sampling methods and modeling techniques. The final application of the response surfaces being aerodynamic shape optimization, the test cases are issued from CFD simulations of aerodynamic coefficients. For comparisons, a robust strategy for model fit using a new efficient initialization technique followed by a gradient optimization was applied. Firstly, a study of different sampling methods proves that including ’a posteriori ’ information on the function to sample distribution can improve accuracy over classical space filling methods like Latin Hypercube Sampling. Secondly, comparing Kriging and gradient enhanced Kriging on two to six dimensional test cases shows that interpolating gradient vectors drastically improves response surface accuracy. Although direct and indirect Cokriging have equivalent formulations, the indirect Cokriging outperforms the direct approach. The slow linear phase of error convergence when increasing sample size is not avoided by Cokriging. Thus, the number of samples needed to have a globally accurate surface stays generally out of reach for problems considering more than four design variables.

This paper deals with aerodynamic shape optimization using a high-fidelity solver. Because of the computational cost needed to solve the Reynolds-averaged Navier–Stokes equations, the performance of the shape must be improved using very few objective function evaluations, despite the high number of design variables. In our framework, the reference algorithm is a quasi-Newton gradient optimizer. An adjoint method inexpensively computes the sensitivities of the functions, with respect to design variables, to build the gradient of the objective function. As usual, aerodynamic functions show numerous local optima when the shape varies, and a more global optimizer is expected to be beneficial. Consequently, a kriging-based optimizer is set up and described. It uses an original sampling refinement process that adds up to three points per iteration by using a balancing between function minimization and error minimization. To efficiently apply this algorithm to high-dimensional problems, the same sampling process is reused to form a cokriging (gradient-enhanced model) based optimizer. A comparative study is then described on two drag-minimization problems depending on 6 and 45 design variables. This study was conducted using an original set of performance criteria, characterizing the strength and weakness of each optimizer in terms of improvement, cost, exploration, and exploitation. Nomenclature A = amplitude of the Hicks–Henne bump function

This paper deals with aerodynamic shape optimization using an high fidelity solver. Due to the computational cost and restitution time needed to solve the RANS equations, this type of optimization framework must improve the solution using very few objective function evaluations despite the high number of design variables. The choice of the optimizer is thus largely based on its speed of convergence. The quickest optimization algorithms use gradient information to converge along a descent path departing from the baseline shape to a local optimum. Within the past few decades, numerous design problems were successfully solved using this method. In our framework, the reference algorithm uses a quasi-Newton gradient method and an adjoint method to inexpensively compute the sensitivities of the functions with respect to shape variables. As usual aerodynamic functions show numerous local optima when varying shape, a more global optimizer can be beneficial at the cost of more function evaluations. More recently, the use of expensive global optimizers became possible by implementing response surfaces between optimizer and CFD code. In this way, a Kriging based optimizer is described. This optimizer proceeds in iteratively refining at up to three points per iteration by using a balancing between function minimization and error minimization. It is compared to the reference algorithm on two drag minimization problems. The test cases are 2D and 3D lifting bodies parameterized with six to more than forty design variables driving deformation of meshes with Hicks-Henne bumps. The new optimizer effectively proves to converge to lower function values without prohibitively increasing the cost. However, response surfaces are known to become inefficient when dimension increases. In order to efficiently apply this response surface based optimizer on such problems, a Cokriging method is used to interpolate gradient information at sample locations.