Chebyshev HOPGD with sparse grid sampling for parameterized linear systems

TL;DR

This work tackles efficient solution of parameterized linear systems $A(\mu_1,\mu_2)x(\mu_1,\mu_2)=b$ with nonlinear dependence on two parameters. It builds a reduced order model by combining companion linearization and a preconditioned Krylov BiCG for snapshot generation with a sparse-grid, higher-order PGD (HOPGD) tensor decomposition to obtain a separable representation, which is then interpolated to cover the parameter domain. The resulting model enables fast evaluation and even parameter estimation across $(\mu_1,\mu_2)$, demonstrated through parameterized Helmholtz and advection-diffusion PDEs with both sparse-grid and full-grid snapshot strategies. The approach is robust to snapshot selection, can regenerate snapshots if decomposition fails, and achieves competitive accuracy with significantly reduced online costs, offering a practical tool for multi-parameter PDE solvers and inverse problems.

Abstract

We consider approximating solutions to parameterized linear systems of the form $A(μ_1,μ_2) x(μ_1,μ_2) = b$. Here the matrix $A(μ_1,μ_2) \in \mathbb{R}^{n \times n}$ is nonsingular, large, and sparse and depends nonlinearly on the parameters. Specifically, the system arises from a discretization of a partial differential equation and $x(μ_1,μ_2) \in \mathbb{R}^n$, $b \in \mathbb{R}^n$. 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 $x(μ_1,μ_2)$, 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.

Figures (16)

• Figure 3.1: Example of tensor matrix $X(\mu_1,\mu_2)$ consisting of 9 snapshot solutions (dots connected by vertical lines) and reduced order model $X^m(\mu_1,\mu_2)$ of rank $m$ as in \ref{['tensor']}, where $\Phi_n^k \in \mathbb{R}^n$, $\bm{f_1^k}\coloneqqF_1^k(\mu_1^1),\ldots,F_1^k(\mu_1^5) \in \mathbb{R}^5$, $\bm{f_2^k}\coloneqqF_2^k(\mu_2^1),\ldots,F_2^k(\mu_2^5) \in \mathbb{R}^5$. Approximation is sum of rank-one tensors.
• Figure 3.2: Sampling on a sparse grid in the parameter space. Nodes \ref{['nodes']} correspond to the 9 snapshot solutions in Figure \ref{['tikz1']} with $\mu_1 \in [a_1,b_1]$, $\mu_2 \in [a_2,b_2]$, $\mu_1^* = (a_1+b_1)/2$, $\mu_2^* = (a_2 + b_2)/2$. All snapshots generated via two executions of Preconditioned Chebyshev BiCG.
• Figure 5.1: Locations of $13$ snapshots $(\mu_1^i,\mu_2^*)$ and $(\mu_1^*,\mu_2^j)$ used to construct $X^m$ in \ref{['tensor']} to approximate \ref{['model-prob']}, sampling on a sparse grid in the parameter space.
• Figure 5.2: Convergence of Preconditioned Chebyshev BiCG for generating the snapshot solutions corresponding to the nodes in Figure \ref{['fig1']} with $\sigma=1.48$. Simulation in Figure \ref{['chebyplot1']} gives all solutions for $(\mu_1,\mu_2^*)$, $\mu_1 \in [1,2]$, simulation in Figure \ref{['chebyplot2']} gives all solutions for $(\mu_1^*,\mu_2)$, $\mu_2 \in [1,2]$.
• Figure 5.3: Function evaluations $F_1^1(\mu_1^i)$ and $F_2^1(\mu_2^j)$ produced by Algorithm \ref{['alg:ChebyshevHOPGD']} to approximate \ref{['model-prob']}, plotted with corresponding interpolations $\bm{F}_1^1(\mu_1)$ and $\bm{F}_2^1(\mu_2)$ as in \ref{['int']} with $\mu_1^i$ and $\mu_2^j$ as in Figure \ref{['fig1']}.

Theorems & Definitions (6)

• Remark 2.1: Choice of target parameter
• Remark 3.1: Evaluation of the model at a point in space
• Remark 3.2: Separated expression in more than two parameters
• Remark 3.3: Comparison of HOPGD to similar methods
• Remark 3.4: Offline and online stages
• Remark 3.5: Sources of error