Noise Filtering and Uncertainty Quantification in Surrogate based Optimization
Résumé
Dense layers of solid particles surrounding a high energy explosive generate jet-like structures at later times after detonation. Conjectures as to the cause of these jet structures include inhomogeneities in the initial distribution of particles. We characterize this variation as particle volume fraction (PVF), defined as volume of particles divided by the volume of gas and particles in a computational cell. We explore what trimodal sinusoidal initial PVF variation would lead to the observed jet formation. This is done by looking for mode shape parameters that amplify most rapidly via optimization. Because the initial perturbations are small they take time to develop, which places a large computational burden on the simulation. We therefore use large initial imperfections that develop into finger-like structures more rapidly. To reduce further the computational cost of the optimization we build a surrogate model. An initial hurdle was to select an objective function that would measure the growth of the initial perturbations. After substantial analysis and numerical experimentation, we settled on the departure from cylindrical symmetry in the particle distribution. The variables considered are the parameters of a trimodal sinusoidal perturbation (amplitudes, wavelengths, and phases). We observed substantial noise in the objective function due to a combination of randomness in the initial position of the particles and the use of Cartesian coordinates for a cylindrically symmetric problem. Since a noisy function is more difficult to optimize, the noise was reduced by a Fourier filter we have developed. We present a novel technique to measure uncertainties using the problem dihedral symmetries. Although it can be applied to the general case in nine variables (3 amplitudes, 3 wave-numbers and 3 phases) we present a simplified problem in three variables. If the amplitude for each of the three modes is kept the same and there is no phase shift, the order of the wave-numbers does not matter, i.e. the case with wave-numbers (k1, k2, k3) should have the same output than its permutations (k1, k3, k2), (k2, k1, k3), (k2, k3, k1), (k3, k1, k2), (k3, k2, k1). Therefore, for each point simulated, we have five extra validation points that we call permutation points, ready to be used to compute uncertainty. We found range-normalized errors up to 33%.