Mårten Gulliksson

Mårten Gulliksson, Sverker Edvardsson, Andreas Lind

We present a new algorithm which is named the Dynamical Functional Particle Method, DFPM. It is based on the idea of formulating a finite dimensional damped dynamical system whose stationary points are the solution to the original equations. The resulting Hamiltonian dynamical system makes it possible to apply efficient symplectic integrators. Other attractive properties of DFPM are that it has an exponential convergence rate, automatically includes a sparse formulation and in many cases can solve nonlinear problems without any special treatment. We study the convergence and convergence rate of DFPM. It is shown that for the discretized symmetric eigenvalue problems the computational complexity is given by $\mathcal{O}(N^{(d+1)/{d}})$, where \emph{d} is the dimension of the problem and \emph{N} is the vector size. An illustrative example of this is made for the 2-dimensional Schrödinger equation. Comparisons are made with the standard numerical libraries ARPACK and LAPACK. The conjugated gradient method and shifted power method are tested as well. It is concluded that DFPM is both versatile and efficient.

Mårten Gulliksson, Anna Oleynik

We consider the problem of finding sparse solutions to a system of underdetermined nonlinear system of equations. The methods are based on a Gauss-Newton approach with line search where the search direction is found by solving a linearized problem using only a subset of the columns in the Jacobian. The choice of columns in the Jacobian is made through a greedy approach looking at either maximum descent or an approach corresponding to orthogonal matching for linear problems. The methods are shown to be convergent and efficient and outperform the $\ell_1$ approach on the test problems presented.