Loïc Giraldi

To be or not to be intrusive? The solution of parametric and stochastic equations — the "plain vanilla" Galerkin case ∗

In parametric equations---stochastic equations are a special case---one may want to approximate the solution such that it is easy to evaluate its dependence on the parameters. Interpolation in the parameters is an obvious possibility---in this context often labeled as a collocation method. In the frequent situation where one has a “solver” for a given fixed parameter value, this may be used “nonintrusively” as a black-box component to compute the solution at all the interpolation points independently of each other. By extension, all other methods, and especially simple Galerkin methods, which produce some kind of coupled system, are often classed as “intrusive.” We show how, for such “plain vanilla” Galerkin formulations, one may solve the coupled system in a nonintrusive way, and even the simplest form of block-solver has comparable efficiency. This opens at least two avenues for possible speed-up: first, to benefit from the coupling in the iteration by using more sophisticated block-solvers and, second,...

Low-Rank Approximate Inverse for Preconditioning Tensor-Structured Linear Systems

In this paper, we propose an algorithm for the construction of low-rank approximations of the inverse of an operator given in low-rank tensor format. The construction relies on an updated greedy algorithm for the minimization of a suitable distance to the inverse operator. It provides a sequence of approximations that are defined as the projections of the inverse operator in an increasing sequence of linear subspaces of operators. These subspaces are obtained by the tensorization of bases of operators that are constructed from successive rank-one corrections. In order to handle high-order tensors, approximate projections are computed in low-rank hierarchical Tucker subsets of the successive subspaces of operators. Some desired properties such as symmetry or sparsity can be imposed on the approximate inverse operator during the correction step, where an optimal rank-one correction is searched as the tensor product of operators with the desired properties. Numerical examples illustrate the ability of this a...

To Be or Not to be Intrusive? The Solution of Parametric and Stochastic Equations---Proper Generalized Decomposition

A numerical method is proposed to compute a low-rank Galerkin approximation to the solution of a parametric or stochastic equation in a nonintrusive fashion. The considered nonlinear problems are associated with the minimization of a parameterized differentiable convex functional. We first introduce a bilinear parameterization of fixed-rank tensors and employ an alternating minimization scheme for computing the low-rank approximation. In keeping with the idea of nonintrusiveness, at each step of the algorithm the minimizations are carried out with a quasi-Newton method to avoid the computation of the Hessian. The algorithm is made nonintrusive through the use of numerical integration. It only requires the evaluation of residuals at specific parameter values. The algorithm is then applied to two numerical examples.