Reduced Order Modeling for the Wavelet-Galerkin Approximation of Differential Equations
Galerkin methods are a common class of methods used to approximate ordinary and partial differential equations (ODE/PDE). Galerkin methods rely on the selection of a set of basis functions that are used to represent the solution of the differential equation. Typical basis functions include: piecewise linear and quadratic polynomials and sine/cosine functions for spectral methods.
From areas of image compression to speech recognition wavelets have had a profound impact on representing large and small scale datasets in the computational science realm. Wavelets also have a number of features that make them attractive functions to work with including: multi-resolution, compact support, differentiability and orthogonality.
Reduced Order Modeling (ROM) is a widely used method to reduce the computational cost solving differential equations when using standard techniques like the Finite Element Method (FEM). This research will demonstrate the viability of using ROM with the Wavelet Galerkin approach to solving ODE’s