Chebyshev HOPGD with sparse grid sampling for parameterized linear systems

We consider approximating solutions to parameterized linear systems of the form $A(\mu_1,\mu_2) x(\mu_1,\mu_2) = b$. Here the matrix $A(\mu_1,\mu_2) \in \mathbb{R}^{n \times n}$ is nonsingular, large, and sparse and depends nonlinearly on the parameters. Specifically, the system arises from a discretization of a partial differential equation and $x(\mu_1,\mu_2) \in \mathbb{R}^n$, $b \in \mathbb{R}^n$. The treatment of linear systems with nonlinear dependence on a single parameter has been well-studied, and robust methods combining companion linearization, Krylov subspace methods, and Chebyshev interpolation have enabled fast solution for multiple parameter values at the cost of a single iteration. Solution of systems depending nonlinearly on multiple parameters is more challenging. This work overcomes those additional challenges by combining companion linearization, the Krylov subspace method preconditioned bi-conjugate gradient (BiCG), and a decomposition of a tensor matrix of precomputed solutions, called snapshots. This produces a reduced order model of $x(\mu_1,\mu_2)$, and this model can be evaluated inexpensively for many values of the parameters. An interpolation of the model is used to produce approximations on the entire parameter space. In addition this method can be used to solve a parameter estimation problem. This approach allows us to achieve similar computational savings as for the one-parameter case; we can solve for many parameter pairs at the cost of many fewer applications of an efficient iterative method. The technique is presented for dependence on two parameters, but the strategy can be extended to more parameters using the same approach. Numerical examples of a parameterized Helmholtz equation show the competitiveness of our approach.