arXiv subjects

Siobhán Correnty

Publications and source records attributed to Siobhán Correnty.

4 recordsLinked to original sources

arXiv2025

Chebyshev HOPGD with sparse grid sampling for parameterized linear systems

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.

math.NA↗

arXiv2023

Preconditioned Chebyshev BiCG for parameterized linear systems

We consider the problem of approximating the solution to $A(μ) x(μ) = b$ for many different values of the parameter $μ$. Here we assume $A(μ)$ is large, sparse, and nonsingular with a nonlinear dependence on $μ$. Our method is based on a companion linearization derived from an accurate Chebyshev interpolation of $A(μ)$ on the interval $[-a,a]$, $a \in \mathbb{R}$. The solution to the linearization is approximated in a preconditioned BiCG setting for shifted systems, where the Krylov basis matrix is formed once. This process leads to a short-term recurrence method, where one execution of the algorithm produces the approximation to $x(μ)$ for many different values of the parameter $μ\in [-a,a]$ simultaneously. In particular, this work proposes one algorithm which applies a shift-and-invert preconditioner exactly as well as an algorithm which applies the preconditioner inexactly. The competitiveness of the algorithms are illustrated with large-scale problems arising from a finite element discretization of a Helmholtz equation with parameterized material coefficient. The software used in the simulations is publicly available online, and thus all our experiments are reproducible.

math.NA↗

arXiv2022

Preconditioned infinite GMRES for parameterized linear systems

We are interested in obtaining approximate solutions to parameterized linear systems of the form $A(μ) x(μ) = b$ for many values of the parameter $μ$. Here $A(μ)$ is large, sparse, and nonsingular, with a nonlinear analytic dependence on $μ$. Our approach is based on a companion linearization for parameterized linear systems. The companion matrix is similar to the operator in the infinite Arnoldi method, and we use this to adapt the flexible GMRES setting. In this way, our method returns a function $\tilde{x}(μ)$ which is cheap to evaluate for different $μ$, and the preconditioner is applied only approximately. This novel approach leads to increased freedom to carry out the action of the operation inexactly, which provides performance improvement over the method infinite GMRES, without a loss of accuracy in general. We show that the error of our method is estimated based on the magnitude of the parameter $μ$, the inexactness of the preconditioning, and the spectrum of the linear companion matrix. Numerical examples from a finite element discretization of a Helmholtz equation with a parameterized material coefficient illustrate the competitiveness of our approach. The simulations are reproducible and publicly available online.

math.NA↗

arXiv2021

Infinite GMRES for parameterized linear systems

We consider linear parameter-dependent systems $A(μ) x(μ) = b$ for many different $μ$, where $A$ is large and sparse, and depends nonlinearly on $μ$. Solving such systems individually for each $μ$ would require great computational effort. In this work we propose to compute a partial parameterization $\tilde{x} \approx x(μ)$ where $\tilde{x}(μ)$ is cheap to compute for many different $μ$. Our methods are based on the observation that a companion linearization can be formed where the dependence on $μ$ is only linear. In particular, we develop methods which combine the well-established Krylov subspace method for linear systems, GMRES, with algorithms for nonlinear eigenvalue problems (NEPs) to generate a basis for the Krylov subspace. Within this new approach, the basis matrix is constructed in three different ways, using a tensor structure and exploiting that certain problems have low-rank properties. We show convergence factor bounds obtained similarly to those for the method GMRES for linear systems. More specifically, a bound is obtained based on the magnitude of the parameter $μ$ and the spectrum of the linear companion matrix, which corresponds to the reciprocal solutions to the corresponding NEP. Numerical experiments illustrate the competitiveness of our methods for large-scale problems. The simulations are reproducible and publicly available online.

math.NA↗