Solving diffusive equations by proper generalized decomposition with preconditioner

Proper Generalized Decomposition (PGD) approximates a function by a series of modes, each of them taking a variable-separated form. This allows drastic reduction in numerical complexity, particularly suits high dimensional problems and has the potential to tackle with the curse of dimensionality. In this paper, we formulate residual functionals for stepwise PGD to approximate function, to solve diffusive equation with/without preconditioning, respectively. The discrete counterparts are also presented. We prove that in the discrete counterpart for function approximation, namely, matrix or tensor approximation, the stepwise PGD and multi-modal approximation by Tensor Decomposition (TD) are equivalent. In case of matrix approximation, both give the same result as that by Singular Value Decomposition (SVD). Furthermore, in view of difficulties in convergence and accuracy arising in diffusive equation solving by PGD, we propose to minimize a preconditioned residual functional instead, leading to a Preconditioned PGD (PPGD) approach. Numerical tests for the heat equation in one and two space dimension(s) demonstrate accurate resolution with a small number of modes. It remains effective even at the presence of inhomogeneity or nonlinearity, and is extended to treat linear wave equation. The proposed PPGD may shed insights to design effective PGD algorithms for solving higher dimensional transient partial differential equations.