Analysis and Computation of Hyperbolic PDEs with Random Data

Hyperbolic partial differential equations (PDEs) are mathematical models of wave phenomena, with applications in a wide range of scientific and engineering fields such as electromagnetic radiation, geosciences, fluid and solid mechanics, aeroacoustics, and general relativity. The theory of hyperbolic problems, including Friedrichs and Kreiss theories, has been well developed based on energy estimates and the method of Fourier and Laplace transforms [8, 16]. Moreover, stable numerical methods, such as the finite difference method [14], the finite volume method [17], the finite element method [6], the spectral method [4], and the boundary element method [11], have been proposed to compute approximate solutions of hyperbolic problems. However, the development of the theory and numerics for hyperbolic PDEs has been based on the assumption that all input data, such as coefficients, initial data, boundary and force terms, and computational domain, are exactly known.