Model reduction of a parametrized scalar hyperbolic conservation law using displacement interpolation

We propose a model reduction technique for parametrized partial differential equations arising from scalar hyperbolic conservation laws. The key idea of the technique is to construct basis functions that are local in parameter and time space via displacement interpolation. The construction is motivated by the observation that the derivative of solutions to hyperbolic conservation laws satisfy a contractive property with respect to the Wasserstein metric [Bolley et al. J. Hyperbolic Differ. Equ. 02 (2005), pp. 91-107]. We will discuss the approximation properties of the displacement interpolation, and show that it can naturally complement linear interpolation. Numerical experiments illustrate that we can successfully achieve the model reduction of a parametrized Burgers' equation, and that the reduced order model is suitable for performing typical tasks in uncertainty quantification.