Low-Rank Approximations for Parametric Non-Symmetric Elliptic Problems - Chacón Rebollo, Tomás; Gómez Mármol, Macarena; Sánchez Muñoz, Isabel (Frontiers in Physics, 2022)
In this work we obtain low-rank approximations for the solution of parametric non symmetric elliptic partial differential equations. We prove the existence of optimal approximation subspaces that minimize the error between the solution and an approximation on this subspace, with respect to the mean parametric quadratic norm associated to any pre-set norm in the space of solutions. Using a low-rank tensorized decomposition, we build an expansion of approximating solutions with summands on finite-dimensional optimal subspaces, and prove the strong convergence of the truncated expansion. For rank-one approximations, similar to the PGD expansion, we prove the linear convergence of the Power Iteration method to compute the modes of the series, for data small enough. We present some numerical results in good agreement with this theoretical analysis.