Fast OMP for Exact Recovery and Sparse Approximation

Orthogonal Matching Pursuit (OMP) has been a powerful method in sparse signal recovery and approximation. However OMP suffers computational issue when the signal has large number of non-zeros. This paper advances OMP in two fronts: it offers a fast algorithm for the orthogonal projection of the input signal at each iteration, and a new selection criterion for making the greedy choice, which reduces the number of iterations it takes to recover the signal. The proposed modifications to OMP directly reduce the computational complexity. Experiment results show significant improvement over the classical OMP in computation time. The paper also provided a sufficient condition for exact recovery under the new greedy choice criterion. For general signals that may not have sparse representations, the paper provides a bound for the approximation error. The approximation error is at the same order as OMP but is obtained within fewer iterations and less time.

Learning Domain-Independent Green's Function For Elliptic Partial Differential Equations

Green's function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green's function is a non-trivial exercise, especially for a PDE defined on a complex domain or a PDE with variable coefficients. In this paper, we propose a novel boundary integral network to learn the domain-independent Green's function, referred to as BIN-G. We evaluate the Green's function in the BIN-G using a radial basis function (RBF) kernel-based neural network. We train the BIN-G by minimizing the residual of the PDE and the mean squared errors of the solutions to the boundary integral equations for prescribed test functions. By leveraging the symmetry of the Green's function and controlling refinements of the RBF kernel near the singularity of the Green function, we demonstrate that our numerical scheme enables fast training and accurate evaluation of the Green's function for PDEs with variable coefficients. The learned Green's function is independent of the domain geometries, forcing terms, and boundary conditions in the boundary integral formulation. Numerical experiments verify the desired properties of the method and the expected accuracy for the two-dimensional Poisson and Helmholtz equations with variable coefficients.