(679c) Application of Singular Value Decomposition to Enhance Proper Generalized Decomposition In Mildly Non-Linear 2-D Partial Differential Equation Problems

We consider the recently-developed Proper Generalized Decomposition (PGD) method [1] for the solution of certain classes of partial differential equations (PDEs).  The PGD method approximates the PDE solution by a summation in which each term is a product of single-variable functions in each dimension.  By applying a weighted residuals approach similar to the Galerkin method used in traditional Finite Element Methods, successive terms may be computed iteratively by repeatedly solving 1-D problems in each dimension.  Use of the PGD method has therefore the potential to drastically reduce the computational requirements in high dimensional problems.  Example applications include the Fokker-Planck equations in polymer solution models [1] and the chemical master equation [2].  A potential limitation, however, is that its application requires solving non-linear systems which admit multiple solutions, even when the original system of PDEs is linear.  A recent theoretical analysis [3] considers this issue in detail and highlights the inherent difficulties that arise when trying to assess the convergence properties of PGD.  It is readily shown that the term computed within a particular iteration of the PGD is not necessarily optimal in the sense that it leads to the best possible convergence of the sum to the true solution.  Thus, we are interested in assessing the performance of PGD in real-world examples and exploring enhancements to improve its convergence characteristics, especially in association to high accuracy spectral approximations.  By decomposing the original multidimensional problem to a series of n-by-m dimensional problems, all the way to 2-D problems at the end, we can explore a powerful tool, the singular value decomposition (SVD).  The SVD when applied to a matrix representing the solution in an n-by-m dimensional space yields an optimum expression of the solution in terms of a sum of products of solutions in n and m dimensions, resepectively, in the sense that this approximation has optimum convergence with number of terms to the actual solution.  As such the SVD can provide a benchmark against which the solutions obtained with various variants of PGD can be compared.  Additionally, the SVD suggests a strategy for increasing the efficiency of PGD by periodically reducing the obtained sums through its application.  We studied the PGD method applied to the 2-D pseudo-conformal map of a sinusoidally varying undulating channel on a straight channel.  The problem is mildly nonlinear, with the non-linearity appearing only in one of the boundary conditions.  We used pseudospectral discretization methods to solve the 1-D problems and perform the required integrations.  We obtained solutions approaching machine accuracy on a 64x64 point pseudospectral grid having sums with an order of 100 terms.  This is compared against a much more compact sum obtained using SVD on the full solution of only 15 terms.  Various convergence acceleration options are discussed.

References

[1] A. Ammar, B. Mkdad, F. Chinesta and R. Keunings, J. Non-Newtonian Fluid Mech., 139:153-176 (2006).

[2] F. Chinesta, A. Ammar, E. Cueto, European J. Computational Mechanics, 19:53-64 (2010).

[3] C. Le Bris, T. Lelievre, Y. Maday, Constr. Approx. 30:621-651 (2009).