Least-squares proper generalized decompositions for weakly coercive elliptic problems

Proper generalised decompositions (PGDs) are a family of methods for efficiently solving high-dimensional PDEs which seek to find a low-rank approximation to the solution of the PDE a priori. Convergence of PGD algorithms can only be proven for problems which are continuous, symmetric and strongly coercive. In the particular case of problems which are only weakly coercive we have the additional issue that weak coercivity estimates are not guaranteed to be inherited by the low-rank PGD approximation. This can cause stability issues when employing a Galerkin PGD approximation of weakly coercive problems. In this paper we propose the use of PGD algorithms based on least-squares formulations which always lead to symmetric and strongly coercive problems and hence provide stable and provably convergent algorithms. Taking the Stokes problem as a prototypical example of a weakly coercive problem, we develop and compare rigorous least-squares PGD algorithms based on continuous least-squares estimates for two different reformulations of the problem. We show that these least-squares PGD provide a much stabler algorithm than an equivalent Galerkin PGD and provide proofs of convergence of the algorithms

Reduced order modeling for physically-based augmented reality

In this work we explore the possibilities of reduced order modeling for augmented reality applications. We consider parametric reduced order models based upon separate (affine) parametric dependence so as to speedup the associated data assimilation problems, which involve in a natural manner the minimization of a distance functional. The employ of reduced order methods allows for an important reduction in computational cost, thus allowing to comply with the stringent real time constraints of video streams, i.e., around 30 Hz. Examples are included that show the potential of the proposed technique in different situations