For my Master's thesis I worked on discontinuity detection in the outputs of computational simulations. Discontinuities often arise in computational simulations as a result of phase transitions or bifurcations. When a simulation output does not depend smoothly on its inputs, the error and convergence rate of many approximation methods deteriorate substantially. I developed a method for efficiently localizing discontinuities in the input space in order to split the input space into two separate regions. The approach involves the following ingredients
- Locating and defining the discontinuity with polynomial annihilation
- Approximating the discontinuity with kernel-based support vector machines (SVMs)
- Refining the discontinuity with active learning
The overall approach results in an anytime algorithm that avoids structured grids and exploits the geometry structure of the separating surface.
A.A. Gorodetsky and Y.M. Marzouk. (2014) Efficient Localization of Discontinuities in Complex Computational Simulations. SIAM Journal on Scientific Computing 36(6), A2584–A2610. (download)