Parametric PDEs: Sparse or Low-Rank Approximations?
For researchers in computational PDEs, this work provides a rigorous comparison of approximation strategies, clarifying when low-rank methods outperform sparse polynomials or vice versa.
The paper compares sparse polynomial expansions, low-rank approximations, and hierarchical tensor decompositions for solving elliptic PDEs with many parameters, showing that the optimal method depends on the problem type, with computational costs varying significantly.
We consider adaptive approximations of the parameter-to-solution map for elliptic operator equations depending on a large or infinite number of parameters, comparing approximation strategies of different degrees of nonlinearity: sparse polynomial expansions, general low-rank approximations separating spatial and parametric variables, and hierarchical tensor decompositions separating all variables. We describe corresponding adaptive algorithms based on a common generic template and show their near-optimality with respect to natural approximability assumptions for each type of approximation. A central ingredient in the resulting bounds for the total computational complexity are new operator compression results for the case of infinitely many parameters. We conclude with a comparison of the complexity estimates based on the actual approximability properties of classes of parametric model problems, which shows that the computational costs of optimized low-rank expansions can be significantly lower or higher than those of sparse polynomial expansions, depending on the particular type of parametric problem.