Unified Framework for Training Point Selection and Error Estimation for Surrogate Models

A unified framework for surrogate model training point selection and error estimation is proposed. Building auxiliary local surrogate models over subdomains of the global surrogate model forms the basis of the proposed framework. A discrepancy function, defined as the absolute difference between response predictions from local and global surrogate models for randomly chosen test candidates, drives the framework, thereby not requiring any additional exact function evaluations. The benefits of this new approach are demonstrated with analytical test functions and the construction of a two-dimensional aerodynamic database. The results show that the proposed training point selection approach improves the convergence monotonicity and produces more accurate surrogate models compared to random and quasi-random training point selection strategies. The introduced root-mean-square discrepancy and maximum absolute discrepancy exhibit close agreement with the actual root-mean-square error and maximum absolute error, respectively, and are therefore proposed as a measure for the approximation accuracy of surrogate models in applications of practical interest. Multivariate interpolation and regression is employed to build local surrogates, whereas kriging and polynomial chaos expansions serve as global surrogate models in demonstrating the applicability of the proposed framework

Multi-Fidelity Sparse Polynomial Chaos and Kriging Surrogate Models Applied to Analytical Benchmark Problems

In this article, multi-fidelity kriging and sparse polynomial chaos expansion (SPCE) surrogate models are constructed. In addition, a novel combination of the two surrogate approaches into a multi-fidelity SPCE-Kriging model will be presented. Accurate surrogate models, once obtained, can be employed for evaluating a large number of designs for uncertainty quantification, optimization, or design space exploration. Analytical benchmark problems are used to show that accurate multi-fidelity surrogate models can be obtained at lower computational cost than high-fidelity models. The benchmarks include non-polynomial and polynomial functions of various input dimensions, lower dimensional heterogeneous non-polynomial functions, as well as a coupled spring-mass-system. Overall, multi-fidelity models are more accurate than high-fidelity ones for the same cost, especially when only a few high-fidelity training points are employed. Full-order PCEs tend to be a factor of two or so worse than SPCES in terms of overall accuracy. The combination of the two approaches into the SPCE-Kriging model leads to a more accurate and flexible method overall

A general framework for the optimal control of unsteady flows with applications,” AIAA Paper

This paper presents a general framework to derive a discrete adjoint method for the optimal control of unsteady flows. First, we present the complete formulation of the timedependent optimal design problem and outline how to derive the discrete set of adjoint equations in a general approach. After that we present results that demonstrate the application of the theory to one-and two-dimensional inverse pulseshape designs, the data assimilation problem in a shock-tube, the drag minimization of viscous flow around a rotating cylinder, and the remote inverse design of a turbulent flow around a NACA0012 airfoil at a high angle of attack

Multi-Fidelity Sparse Polynomial Chaos and Kriging Surrogate Models Applied to Analytical Benchmark Problems

In this article, multi-fidelity kriging and sparse polynomial chaos expansion (SPCE) surrogate models are constructed. In addition, a novel combination of the two surrogate approaches into a multi-fidelity SPCE-Kriging model will be presented. Accurate surrogate models, once obtained, can be employed for evaluating a large number of designs for uncertainty quantification, optimization, or design space exploration. Analytical benchmark problems are used to show that accurate multi-fidelity surrogate models can be obtained at lower computational cost than high-fidelity models. The benchmarks include non-polynomial and polynomial functions of various input dimensions, lower dimensional heterogeneous non-polynomial functions, as well as a coupled spring-mass-system. Overall, multi-fidelity models are more accurate than high-fidelity ones for the same cost, especially when only a few high-fidelity training points are employed. Full-order PCEs tend to be a factor of two or so worse than SPCES in terms of overall accuracy. The combination of the two approaches into the SPCE-Kriging model leads to a more accurate and flexible method overall

Nomenclature

This paper presents a general framework to derive a discrete adjoint method for the optimal control of unsteady flows. First, we present the complete formulation of the timedependent optimal design problem and outline how to derive the discrete set of adjoint equations in a general approach. After that we present results that demonstrate the application of the theory to one- and two-dimensional inverse pulseshape designs, the data assimilation problem in a shock-tube, the drag minimization of viscous flow around a rotating cylinder, and the remote inverse design of a turbulent flow around a NACA0012 airfoil at a high angle of attack